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  • Simplex based Steiner tree instances yield large integrality gaps for the bidirected cut relaxation
    arXiv.cs.DM Pub Date : 2020-02-18
    Robert Vicari

    The bidirected cut relaxation is the characteristic representative of the bidirected relaxations ($\mathrm{\mathcal{BCR}}$) which are a well-known class of equivalent LP-relaxations for the NP-hard Steiner Tree Problem in Graphs (STP). Although no general approximation algorithm based on $\mathrm{\mathcal{BCR}}$ with an approximation ratio better than $2$ for STP is known, it is mostly preferred in integer programming as an implementation of STP, since there exists a formulation of compact size, which turns out to be very effective in practice. It is known that the integrality gap of $\mathrm{\mathcal{BCR}}$ is at most $2$, and a long standing open question is whether the integrality gap is less than $2$ or not. The best lower bound so far is $\frac{36}{31} \approx 1.161$ proven by Byrka et al. [BGRS13]. Based on the work of Chakrabarty et al. [CDV11] about embedding STP instances into simplices by considering appropriate dual formulations, we improve on this result by constructing a new class of instances and showing that their integrality gaps tend at least to $\frac{6}{5} = 1.2$. More precisely, we consider the class of equivalent LP-relaxations $\mathrm{\mathcal{BCR}}^{+}$, that can be obtained by strengthening $\mathrm{\mathcal{BCR}}$ by already known straightforward Steiner vertex degree constraints, and show that the worst case ratio regarding the optimum value between $\mathrm{\mathcal{BCR}}$ and $\mathrm{\mathcal{BCR}}^{+}$ is at least $\frac{6}{5}$. Since $\mathrm{\mathcal{BCR}}^{+}$ is a lower bound for the hypergraphic relaxations ($\mathrm{\mathcal{HYP}}$), another well-known class of equivalent LP-relaxations on which the current best $(\ln(4) + \varepsilon)$-approximation algorithm for STP by Byrka et al. [BGRS13] is based, this worst case ratio also holds for $\mathrm{\mathcal{BCR}}$ and $\mathrm{\mathcal{HYP}}$.

    更新日期:2020-02-20
  • On Tuza's conjecture for triangulations and graphs with small treewidth
    arXiv.cs.DM Pub Date : 2020-02-18
    Fábio Botler; Cristina G. Fernandes; Juan Gutiérrez

    Tuza (1981) conjectured that the size $\tau(G)$ of a minimum set of edges that intersects every triangle of a graph $G$ is at most twice the size $\nu(G)$ of a maximum set of edge-disjoint triangles of $G$. In this paper we present three results regarding Tuza's Conjecture. We verify it for graphs with treewidth at most $6$; we show that $\tau(G)\leq \frac{3}{2}\,\nu(G)$ for every planar triangulation $G$ different from $K_4$; and that $\tau(G)\leq\frac{9}{5}\,\nu(G) + \frac{1}{5}$ if $G$ is a maximal graph with treewidth 3.

    更新日期:2020-02-20
  • BB_Evac: Fast Location-Sensitive Behavior-Based Building Evacuation
    arXiv.cs.DM Pub Date : 2020-02-19
    Subhra Mazumdar; Arindam Pal; Francesco Parisi; V. S. Subrahmanian

    Past work on evacuation planning assumes that evacuees will follow instructions -- however, there is ample evidence that this is not the case. While some people will follow instructions, others will follow their own desires. In this paper, we present a formal definition of a behavior-based evacuation problem (BBEP) in which a human behavior model is taken into account when planning an evacuation. We show that a specific form of constraints can be used to express such behaviors. We show that BBEPs can be solved exactly via an integer program called BB_IP, and inexactly by a much faster algorithm that we call BB_Evac. We conducted a detailed experimental evaluation of both algorithms applied to buildings (though in principle the algorithms can be applied to any graphs) and show that the latter is an order of magnitude faster than BB_IP while producing results that are almost as good on one real-world building graph and as well as on several synthetically generated graphs.

    更新日期:2020-02-20
  • Interface Modeling for Quality and Resource Management
    arXiv.cs.DM Pub Date : 2020-02-19
    Martijn Hendriks; Marc Geilen; Kees Goossens; Rob de Jong; Twan Basten

    We develop an interface-modeling framework for quality and resource management that captures configurable working points of hardware and software components in terms of functionality, resource usage and provision, and quality indicators such as performance and energy consumption. We base these aspects on partially-ordered sets to capture quality levels, budget sizes, and functional compatibility. This makes the framework widely applicable and domain independent (although we aim for embedded and cyber-physical systems). The framework paves the way for dynamic (re-)configuration and multi-objective optimization of component-based systems for quality- and resource-management purposes.

    更新日期:2020-02-20
  • The Tree Stabbing Number is not Monotone
    arXiv.cs.DM Pub Date : 2020-02-19
    Wolfgang Mulzer; Johannes Obenaus

    Let $P \subseteq \mathbb{R}^2$ be a set of points and $T$ be a spanning tree of $P$. The \emph{stabbing number} of $T$ is the maximum number of intersections any line in the plane determines with the edges of $T$. The \emph{tree stabbing number} of $P$ is the minimum stabbing number of any spanning tree of $P$. We prove that the tree stabbing number is not a monotone parameter, i.e., there exist point sets $P \subsetneq P'$ such that \treestab{$P$} $>$ \treestab{$P'$}, answering a question by Eppstein \cite[Open Problem~17.5]{eppstein_2018}.

    更新日期:2020-02-20
  • Subexponential parameterized algorithms and kernelization on almost chordal graphs
    arXiv.cs.DM Pub Date : 2020-02-19
    Fedor V. Fomin; Petr A. Golovach

    We study the algorithmic properties of the graph class Chordal-ke, that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. We discover that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from Chordal-ke. We identify a large class of optimization problems on Chordal-ke that admit algorithms with the typical running time 2^{O(\sqrt{k}\log k)}\cdot n^{O(1)}. Examples of the problems from this class are finding an independent set of maximum weight, finding a feedback vertex set or an odd cycle transversal of minimum weight, or the problem of finding a maximum induced planar subgraph. On the other hand, we show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on Chordal-ke when parameterized by k but do not admit subexponential in k algorithms unless ETH fails. Besides subexponential time algorithms, the class of Chordal-ke graphs appears to be appealing from the perspective of kernelization (with parameter k). While it is possible to show that most of the weighted variants of optimization problems do not admit polynomial in k kernels on Chordal-ke graphs, this does not exclude the existence of Turing kernelization and kernelization for unweighted graphs. In particular, we construct a polynomial Turing kernel for Weighted Clique on Chordal-ke graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of Chordal-ke, namely, Interval-ke and Split-ke graphs.

    更新日期:2020-02-20
  • U-Bubble Model for Mixed Unit Interval Graphs and its Applications: The MaxCut Problem Revisited
    arXiv.cs.DM Pub Date : 2020-02-19
    Jan Kratochvíl; Tomáš Masařík; Jana Novotná

    Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs; a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a much richer class of graphs. In particular, mixed unit interval graphs are not claw-free, compared to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs. The original bubble model was used by Boyaci, Ekim, and Shalom for proving the polynomiality of the MaxCut problem on unit interval graphs. However, we found a significant mistake in the proof which seems to be hardly repairable. Moreover, we demonstrate the advantages of such a model by providing a subexponential-time algorithm solving the MaxCut problem on mixed unit interval graphs using our extended version of the bubble model. In addition, it gives us a polynomial-time algorithm for specific mixed unit interval graphs; that improves a state-of-the-art result even for unit interval graphs. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs. Clique-width is one of the most general structural graph parameters, where a large group of natural problems is still solvable in the tracktable time when an efficient representation is given. Unfortunately, the exact computation of the clique-width representation is NP-hard. Therefore, good upper-bounds on clique-width are highly appreciated.

    更新日期:2020-02-20
  • Automaticity and invariant measures of linear cellular automata
    arXiv.cs.DM Pub Date : 2018-11-03
    Eric Rowland; Reem Yassawi

    We show that spacetime diagrams of linear cellular automata $\Phi : {\mathbb F}_p^{\mathbb Z} \to {\mathbb F}_p^{\mathbb Z}$ with $(-p)$-automatic initial conditions are automatic. This extends existing results on initial conditions which are eventually constant. Each automatic spacetime diagram defines a $(\sigma, \Phi)$-invariant subset of ${\mathbb F}_p^{\mathbb Z}$, where $\sigma$ is the left shift map, and if the initial condition is not eventually periodic then this invariant set is nontrivial. For the Ledrappier cellular automaton we construct a family of nontrivial $(\sigma, \Phi)$-invariant measures on ${\mathbb F}_3^{\mathbb Z}$. Finally, given a linear cellular automaton $\Phi$, we construct a nontrivial $(\sigma, \Phi)$-invariant measure on ${\mathbb F}_p^{\mathbb Z}$ for all but finitely many $p$.

    更新日期:2020-02-20
  • Planar graphs have bounded queue-number
    arXiv.cs.DM Pub Date : 2019-04-09
    Vida Dujmović; Gwenaël Joret; Piotr Micek; Pat Morin; Torsten Ueckerdt; David R. Wood

    We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath, Leighton and Rosenberg from 1992. The key to the proof is a new structural tool called layered partitions, and the result that every planar graph has a vertex-partition and a layering, such that each part has a bounded number of vertices in each layer, and the quotient graph has bounded treewidth. This result generalises for graphs of bounded Euler genus. Moreover, we prove that every graph in a minor-closed class has such a layered partition if and only if the class excludes some apex graph. Building on this work and using the graph minor structure theorem, we prove that every proper minor-closed class of graphs has bounded queue-number. Layered partitions have strong connections to other topics, including the following two examples. First, they can be interpreted in terms of strong products. We show that every planar graph is a subgraph of the strong product of a path with some graph of bounded treewidth. Similar statements hold for all proper minor-closed classes. Second, we give a simple proof of the result by DeVos et al. (2004) that graphs in a proper minor-closed class have low treewidth colourings.

    更新日期:2020-02-20
  • Fine-grained complexity of the graph homomorphism problem for bounded-treewidth graphs
    arXiv.cs.DM Pub Date : 2019-06-19
    Karolina Okrasa; Paweł Rzążewski

    For graphs $G$ and $H$, a \emph{homomorphism} from $G$ to $H$ is an edge-preserving mapping from the vertex set of $G$ to the vertex set of $H$. For a fixed graph $H$, by \textsc{Hom($H$)} we denote the computational problem which asks whether a given graph $G$ admits a homomorphism to $H$. If $H$ is a complete graph with $k$ vertices, then \textsc{Hom($H$)} is equivalent to the $k$-\textsc{Coloring} problem, so graph homomorphisms can be seen as generalizations of colorings. It is known that \textsc{Hom($H$)} is polynomial-time solvable if $H$ is bipartite or has a vertex with a loop, and NP-complete otherwise [Hell and Ne\v{s}et\v{r}il, JCTB 1990]. In this paper we are interested in the complexity of the problem, parameterized by the treewidth of the input graph $G$. If $G$ has $n$ vertices and is given along with its tree decomposition of width $\mathrm{tw}(G)$, then the problem can be solved in time $|V(H)|^{\mathrm{tw}(G)} \cdot n^{\mathcal{O}(1)}$, using a straightforward dynamic programming. We explore whether this bound can be improved. We show that if $H$ is a \emph{projective core}, then the existence of such a faster algorithm is unlikely: assuming the Strong Exponential Time Hypothesis (SETH), the \textsc{Hom($H$)} problem cannot be solved in time $(|V(H)|-\epsilon)^{\mathrm{tw}(G)} \cdot n^{\mathcal{O}(1)}$, for any $\epsilon > 0$. This result provides a full complexity characterization for a large class of graphs $H$, as almost all graphs are projective cores. We also notice that the naive algorithm can be improved for some graphs $H$, and show a complexity classification for all graphs $H$, assuming two conjectures from algebraic graph theory. In particular, there are no known graphs $H$ which are not covered by our result. In order to prove our results, we bring together some tools and techniques from algebra and from fine-grained complexity.

    更新日期:2020-02-20
  • Clustered Variants of Hajós' Conjecture
    arXiv.cs.DM Pub Date : 2019-08-14
    Chun-Hung Liu; David R. Wood

    Haj\'os conjectured that every graph containing no subdivision of the complete graph $K_{s+1}$ is properly $s$-colorable. This result was disproved by Catlin. Indeed, the maximum chromatic number of such graphs is $\Omega(s^2/\log s)$. In this paper we prove that $O(s)$ colors are enough for a weakening of this conjecture that only requires every monochromatic component to have bounded size (so-called \emph{clustered} coloring). Our approach in this paper leads to more results. Say that a graph is an {\it almost $(\leq 1)$-subdivision} of a graph $H$ if it can be obtained from $H$ by subdividing edges, where at most one edge is subdivided more than once. We prove the following (where $s \geq 2$): \begin{enumerate} \item Graphs of bounded treewidth and with no almost $(\leq 1)$-subdivision of $K_{s+1}$ are $s$-choosable with bounded clustering. \item For every graph $H$, graphs with no $H$-minor and no almost $(\leq 1)$-subdivision of $K_{s+1}$ are $(s+1)$-colorable with bounded clustering. \item For every graph $H$ of maximum degree at most $d$, graphs with no $H$-subdivision and no almost $(\leq 1)$-subdivision of $K_{s+1}$ are $\max\{s+3d-5,2\}$-colorable with bounded clustering. \item For every graph $H$ of maximum degree $d$, graphs with no $K_{s,t}$ subgraph and no $H$-subdivision are $\max\{s+3d-4,2\}$-colorable with bounded clustering. \item Graphs with no $K_{s+1}$-subdivision are $\max\{4s-5,1\}$-colorable with bounded clustering. \end{enumerate} The first result shows that the weakening of Haj\'{o}s' conjecture is true for graphs of bounded treewidth in a stronger sense; the final result is the first $O(s)$ bound on the clustered chromatic number of graphs with no $K_{s+1}$-subdivision.

    更新日期:2020-02-20
  • The Computational Complexity of Finding Temporal Paths under Waiting Time Constraints
    arXiv.cs.DM Pub Date : 2019-09-13
    Arnaud Casteigts; Anne-Sophie Himmel; Hendrik Molter; Philipp Zschoche

    Computing a (shortest) path between two vertices in a graph is one of the most fundamental primitives in graph algorithms. In recent years, the study of paths in temporal graphs, that is, graphs where the vertex set is fixed but the edge set changes over time, gained more and more attention. Such a path is time-respecting, or temporal, if it uses edges over non-decreasing times. In this paper, we investigate a basic constraint for temporal paths, where the time spent at each vertex must not exceed a given duration $\Delta$, referred to as $\Delta$-restless temporal paths. This constraint arises naturally in the modeling of real-world processes like infectious diseases and packet routing in communication networks. While the reachability problem for temporal paths in general is known to be polynomial-time solvable, we show that the restless version of this problem becomes computationally hard even in very restrictive settings. For example, it is W[1]-hard when parameterized by feedback vertex number or pathwidth of the underlying graph. The main question thus becomes whether the problem is tractable in some natural settings. As of today, no reference set of parameters exist for temporal graphs. We explore several directions in this respect, presenting FPT algorithms for three kinds of parameters: (1) output-related parameters (here, the maximum length of the path), (2) classical parameters applied to the underlying graph (e.g., feedback edge number), and (3) a new parameter called timed feedback vertex number, which captures finer-grained temporal features of the input temporal graph, and which may be of interest beyond this work.

    更新日期:2020-02-20
  • List colouring of two matroids through reduction to partition matroids
    arXiv.cs.DM Pub Date : 2019-11-24
    Kristóf Bérczi; Tamás Schwarcz; Yutaro Yamaguchi

    In the list coloring problem for two matroids, we are given matroids $M_1=(S,{\cal I}_1)$ and $M_2=(S,{\cal I}_2)$ on the same ground set $S$, and the goal is to determine the smallest number $k$ such that given arbitrary lists $L_s$ of $k$ colors for $s\in S$, it is possible to choose a color from each list so that every monochromatic set is independent in both $M_1$ and $M_2$. When both $M_1$ and $M_2$ are partition matroids, Galvin's list coloring theorem for bipartite graphs gives the answer. One of the main open questions is to decide if there exists a constant $c$ such that if the coloring number is $k$ (i.e., the ground set can be partitioned into $k$ common independent sets), then the list coloring number is at most $c\cdot k$. We consider matroid classes that appear naturally in combinatorial optimization problems, namely graphic matroids, paving matroids and gammoids. We show that if both matroids are from these fundamental classes, then the list coloring number is at most twice the coloring number. The proof is based on a new approach that reduces a matroid to a partition matroid without increasing its coloring number too much, and might be of independent combinatorial interest. In particular, we show that if $M=(S,{\cal I})$ is a matroid in which $S$ can be partitioned into $k$ independent sets, then there exists a partition matroid $N=(S,{\cal J})$ with ${\cal J}\subseteq{\cal I}$ in which $S$ can be partitioned into (A) $k$ independent sets if $M$ is a transversal matroid, (B) $2k-1$ independent sets if $M$ is a graphic matroid, (C) $\lceil kr/(r-1)\rceil$ independent sets if $M$ is a paving matroid of rank $r$, and (D) $2k-2$ independent sets if $M$ is a gammoid. We extend our results by showing that the existence of a matroid $N$ with $\chi(N)\leq 2\chi(M)$ implies the existence of a matroid $N'$ with $\chi(N')\leq 2\chi(M')$ for every truncation $M'$ of $M$.

    更新日期:2020-02-20
  • Weighted Additive Spanners
    arXiv.cs.DM Pub Date : 2020-02-15
    Reyan Ahmed; Greg Bodwin; Faryad Darabi Sahneh; Stephen Kobourov; Richard Spence

    An $\alpha$-additive spanner of an undirected graph $G=(V, E)$ is a subgraph $H$ such that the distance between any two vertices in $G$ is stretched by no more than an additive factor of $\alpha$. It is previously known that unweighted graphs have 2-, 4-, and 6-additive spanners containing $\widetilde{O}(n^{3/2})$, $\widetilde{O}(n^{7/5})$, and $O(n^{4/3})$ edges, respectively. In this paper, we generalize these results to weighted graphs. We consider $\alpha=2W$, $4W$, $6W$, where $W$ is the maximum edge weight in $G$. We first show that every $n$-node graph has a subsetwise $2W$-spanner on $O(n |S|^{1/2})$ edges where $S \subseteq V$ and $W$ is a constant. We then show that for every set $P$ with $|P| = p$ vertex demand pairs, there are pairwise $2W$- and $4W$-spanners on $O(np^{1/3})$ and $O(np^{2/7})$ edges respectively. We also show that for every set $P$, there is a $6W$-spanner on $O(np^{1/4})$ edges where $W$ is a constant. We then show that every graph has additive $2W$- and $4W$-spanners on $O(n^{3/2})$ and $O(n^{7/5})$ edges respectively. Finally, we show that every graph has an additive $6W$-spanner on $O(n^{4/3})$ edges where $W$ is a constant.

    更新日期:2020-02-19
  • Cover Combinatorial Filters and their Minimization Problem
    arXiv.cs.DM Pub Date : 2020-02-15
    Yulin Zhang; Dylan A. Shell

    A recent research theme has been the development of automatic methods to minimize robots' resource footprints. In particular, the class of combinatorial filters (discrete variants of widely-used probabilistic estimators) has been studied and methods developed for automatically reducing their space requirements. This paper extends existing combinatorial filters by introducing a natural generalization that we dub cover combinatorial filters. In addressing the new---but still NP-complete---problem of minimization of cover filters, this paper shows that three of the concepts previously believed to be true about combinatorial filters (and actually conjectured, claimed, or assumed to be) are in fact false. For instance, minimization does not induce an equivalence relation. We give an exact algorithm for the cover filter minimization problem. Unlike prior work (based on graph coloring) we consider a type of clique-cover problem, involving a new conditional constraint, from which we can find more general relations. In addition to solving the more general problem, the algorithm we present also corrects flaws present in all prior filter reduction methods. The algorithm also forms a promising basis for practical future development as it involves a reduction to SAT.

    更新日期:2020-02-19
  • Density of Binary Compact Disc Packings
    arXiv.cs.DM Pub Date : 2020-02-17
    Thomas Fernique; Nicolas Bédaride

    A disc packing in the plane is compact if its contact graph is a triangulation. There are $9$ values of $r$ such that a compact packing by discs of radii $1$ and $r$ exists. We prove, for each of these $9$ values, that the maximal density over all the packings by discs of radii $1$ and $r$ is reached for a compact packing (we give it as well as its density).

    更新日期:2020-02-19
  • Semi-dynamic Algorithms for Strongly Chordal Graphs
    arXiv.cs.DM Pub Date : 2020-01-31
    Md. Zamilur Rahman; Asish Mukhopadhyay

    There is an extensive literature on dynamic algorithms for a large number of graph theoretic problems, particularly for all varieties of shortest path problems. Germane to this paper are a number fully dynamic algorithms that are known for chordal graphs. However, to the best of our knowledge no study has been done for the problem of dynamic algorithms for strongly chordal graphs. To address this gap, in this paper, we propose a semi-dynamic algorithm for edge-deletions and a semi-dynamic algorithm for edge-insertions in a strongly chordal graph, $G = (V, E)$, on $n$ vertices and $m$ edges. The query complexity of an edge-deletion is $O(d_u^2d_v^2 (n + m))$, where $d_u$ and $d_v$ are the degrees of the vertices $u$ and $v$ of the candidate edge $\{u, v\}$, while the query-complexity of an edge-insertion is $O(n^2)$.

    更新日期:2020-02-19
  • Spectrum preserving short cycle removal on regular graphs
    arXiv.cs.DM Pub Date : 2020-02-17
    Pedro Paredes

    We describe a new method to remove short cycles on regular graphs while maintaining spectral bounds (the nontrivial eigenvalues of the adjacency matrix), as long as the graphs have certain combinatorial properties. These combinatorial properties are related to the number and distance between short cycles and are known to happen with high probability in uniformly random regular graphs. Using this method we can show two results involving high girth spectral expander graphs. First, we show that given $d \geq 3$ and $n$, there exists an explicit distribution of $d$-regular $\Theta(n)$-vertex graphs where with high probability its samples have girth $\Omega(\log_{d - 1} n)$ and are $\epsilon$-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by $2\sqrt{d - 1} + \epsilon$ (excluding the single trivial eigenvalue of $d$). Then, for every constant $d \geq 3$ and $\epsilon > 0$, we give a deterministic poly$(n)$-time algorithm that outputs a $d$-regular graph on $\Theta(n)$-vertices that is $\epsilon$-near-Ramanujan and has girth $\Omega(\sqrt{\log n})$, based on the work of arXiv:1909.06988 .

    更新日期:2020-02-19
  • Quasi-majority Functional Voting on Expander Graphs
    arXiv.cs.DM Pub Date : 2020-02-18
    Nobutaka Shimizu; Takeharu Shiraga

    Consider a distributed graph where each vertex holds one of two distinct opinions. In this paper, we are interested in synchronous voting processes where each vertex updates its opinion according to a predefined common local updating rule. For example, each vertex adopts the majority opinion among 1) itself and two randomly picked neighbors in best-of-two or 2) three randomly picked neighbors in best-of-three. Previous works intensively studied specific rules including best-of-two and best-of-three individually. In this paper, we generalize and extend previous works of best-of-two and best-of-three on expander graphs by proposing a new model, quasi-majority functional voting. This new model contains best-of-two and best-of-three as special cases. We show that, on expander graphs with sufficiently large initial bias, any quasi-majority functional voting reaches consensus within $O(\log n)$ steps with high probability. Moreover, we show that, for any initial opinion configuration, any quasi-majority functional voting on expander graphs with higher expansion (e.g., Erd\H{o}s-R\'enyi graph $G(n,p)$ with $p=\Omega(1/\sqrt{n})$) reaches consensus within $O(\log n)$ with high probability. Furthermore, we show that the consensus time is $O(\log n/\log k)$ of best-of-$(2k+1)$ for $k=o(n/\log n)$.

    更新日期:2020-02-19
  • Generating random bigraphs with preferential attachment
    arXiv.cs.DM Pub Date : 2020-02-18
    Dominik GrzelakSoftware Technology Group at Technische Universität DresdenCentre for Tactile Internet with Human-in-the-Loop; Barbara PriwitzerFakultät Technik at Hochschule Reutlingen; Uwe AßmannSoftware Technology Group at Technische Universität DresdenCentre for Tactile Internet with Human-in-the-Loop

    The bigraph theory is a relatively young, yet formally rigorous, mathematical framework encompassing Robin Milner's previous work on process calculi, on the one hand, and provides a generic meta-model for complex systems such as multi-agent systems, on the other. A bigraph $F = \langle F^P, F^L\rangle$ is a superposition of two independent graph structures comprising a place graph $F^P$ (i.e., a forest) and a link graph $F^L$ (i.e., a hypergraph), sharing the same node set, to express locality and communication of processes independently from each other. In this paper, we take some preparatory steps towards an algorithm for generating random bigraphs with preferential attachment feature w.r.t. $F^P$ and assortative (disassortative) linkage pattern w.r.t. $F^L$. We employ parameters allowing one to fine-tune the characteristics of the generated bigraph structures. To study the pattern formation properties of our algorithmic model, we analyze several metrics from graph theory based on artificially created bigraphs under different configurations. Bigraphs provide a quite useful and expressive semantic for process calculi for mobile and global ubiquitous computing. So far, this subject has not received attention in the bigraph-related scientific literature. However, artificial models may be particularly useful for simulation and evaluation of real-world applications in ubiquitous systems necessitating random structures.

    更新日期:2020-02-19
  • The Post Correspondence Problem and equalisers for certain free group and monoid morphisms
    arXiv.cs.DM Pub Date : 2020-02-18
    Laura Ciobanu; Alan D. Logan

    A marked free monoid morphism is a morphism for which the image of each generator starts with a different letter, and immersions are the analogous maps in free groups. We show that the (simultaneous) PCP is decidable for immersions of free groups, and provide an algorithm to compute bases for the sets, called equalisers, on which the immersions take the same values. We also answer a question of Stallings about the rank of the equaliser. Analogous results are proven for marked morphisms of free monoids.

    更新日期:2020-02-19
  • Representations for the largest Extension of a closure system
    arXiv.cs.DM Pub Date : 2020-02-18
    Karima Ennaoui; Khaled Maafa; Lhouari Nourine

    We consider extension of a closure system on a finite set S as a closure system on the same set S containing the given one as a sublattice. A closure system can be represented in different ways, e.g. by an implicational base or by the set of its meet-irreducible elements. When a closure system is described by an implicational base, we provide a characterization of the implicational base for the largest extension. We also show that the largest extension can be handled by a small modification of the implicational base of the input closure system. This answers a question asked in [12]. Second, we are interested in computing the largest extension when the closure system is given by the set of all its meet-irreducible elements. We give an incremental polynomial time algorithm to compute the largest extension of a closure system, and left open if the number of meet-irreducible elements grows exponentially.

    更新日期:2020-02-19
  • Discrete Line Fields on Surfaces
    arXiv.cs.DM Pub Date : 2020-02-18
    Tiago Novello; João Paixão; Carlos Tomei; Thomas Lewiner

    Vector fields and line fields, their counterparts without orientations on tangent lines, are familiar objects in the theory of dynamical systems. Among the techniques used in their study, the Morse--Smale decomposition of a (generic) field plays a fundamental role, relating the geometric structure of phase space to a combinatorial object consisting of critical points and separatrices. Such concepts led Forman to a satisfactory theory of discrete vector fields, in close analogy to the continuous case. In this paper, we introduce discrete line fields. Again, our definition is rich enough to provide the counterparts of the basic results in the theory of continuous line fields: a Euler-Poincar\'e formula, a Morse--Smale decomposition and a topologically consistent cancellation of critical elements, which allows for topological simplification of the original discrete line field.

    更新日期:2020-02-19
  • Fuzzy Simultaneous Congruences
    arXiv.cs.DM Pub Date : 2020-02-18
    Max A. Deppert; Klaus Jansen; Kim-Manuel Klein

    We introduce a very natural generalization of the well-known problem of simultaneous congruences. Instead of searching for a positive integer $s$ that is specified by $n$ fixed remainders modulo integer divisors $a_1,\dots,a_n$ we consider remainder intervals $R_1,\dots,R_n$ such that $s$ is feasible if and only if $s$ is congruent to $r_i$ modulo $a_i$ for some remainder $r_i$ in interval $R_i$ for all $i$. This problem is a special case of a 2-stage integer program with only two variables per constraint which is is closely related to directed Diophantine approximation as well as the mixing set problem. We give a hardness result showing that the problem is NP-hard in general. Motivated by the study of the mixing set problem and a recent result in the field of real-time systems we investigate the case of harmonic divisors, i.e. $a_{i+1}/a_i$ is an integer for all $i

    更新日期:2020-02-19
  • Performance Guarantees of Distributed Algorithms for QoS in Wireless Ad Hoc Networks
    arXiv.cs.DM Pub Date : 2018-10-06
    Ashwin Ganesan

    Consider a wireless network where each communication link has a minimum bandwidth quality-of-service requirement. Certain pairs of wireless links interfere with each other due to being in the same vicinity, and this interference is modeled by a conflict graph. Given the conflict graph and link bandwidth requirements, the objective is to determine, using only localized information, whether the demands of all the links can be satisfied. At one extreme, each node knows the demands of only its neighbors; at the other extreme, there exists an optimal, centralized scheduler that has global information. The present work interpolates between these two extremes by quantifying the tradeoff between the degree of decentralization and the performance of the distributed algorithm. This open problem is resolved for the primary interference model, and the following general result is obtained: if each node knows the demands of all links in a ball of radius $d$ centered at the node, then there is a distributed algorithm whose performance is away from that of an optimal, centralized algorithm by a factor of at most $(2d+3)/(2d+2)$. The tradeoff between performance and complexity of the distributed algorithm is also analyzed. It is shown that for line networks under the protocol interference model, the row constraints are a factor of at most $3$ away from optimal. Both bounds are best possible.

    更新日期:2020-02-19
  • Complexity of packing common bases in matroids
    arXiv.cs.DM Pub Date : 2019-03-08
    Kristóf Bérczi; Tamás Schwarcz

    One of the most intriguing unsolved questions of matroid optimization is the characterization of the existence of $k$ disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures that can be formulated as special cases, such as Woodall's conjecture on packing disjoint dijoins in a directed graph, or Rota's beautiful conjecture on rearrangements of bases. In the present paper we prove that the problem is difficult under the rank oracle model, i.e., we show that there is no algorithm which decides if the common ground set of two matroids can be partitioned into $k$ common bases by using a polynomial number of independence queries. Our complexity result holds even for the very special case when $k=2$. Through a series of reductions, we also show that the abstract problem of packing common bases in two matroids includes the NAE-SAT problem and the Perfect Even Factor problem in directed graphs. These results in turn imply that the problem is not only difficult in the independence oracle model but also includes NP-complete special cases already when $k=2$, one of the matroids is a partition matroid, while the other matroid is linear and is given by an explicit representation.

    更新日期:2020-02-19
  • Zeros and approximations of Holant polynomials on the complex plane
    arXiv.cs.DM Pub Date : 2019-05-08
    Katrin Casel; Philipp Fischbeck; Tobias Friedrich; Andreas Göbel; J. A. Gregor Lagodzinski

    We present fully polynomial approximation schemes for a broad class of Holant problems with complex edge weights, which we call Holant polynomials. We transform these problems into partition functions of abstract combinatorial structures known as polymers in statistical physics. Our method involves establishing zero-free regions for the partition functions of polymer models and using the most significant terms of the cluster expansion to approximate them. Results of our technique include new approximation and sampling algorithms for a diverse class of Holant polynomials in the low-temperature regime and approximation algorithms for general Holant problems with small signature weights. Additionally, we give randomised approximation and sampling algorithms with faster running times for more restrictive classes. Finally, we improve the known zero-free regions for a perfect matching polynomial.

    更新日期:2020-02-19
  • Decision list compression by mild random restrictions
    arXiv.cs.DM Pub Date : 2019-09-24
    Shachar Lovett; Kewen Wu; Jiapeng Zhang

    A decision list is an ordered list of rules. Each rule is specified by a term, which is a conjunction of literals, and a value. Given an input, the output of a decision list is the value corresponding to the first rule whose term is satisfied by the input. Decision lists generalize both CNFs and DNFs, and have been studied both in complexity theory and in learning theory. The size of a decision list is the number of rules, and its width is the maximal number of variables in a term. We prove that decision lists of small width can always be approximated by decision lists of small size, where we obtain sharp bounds. This in particular resolves a conjecture of Gopalan, Meka and Reingold (Computational Complexity, 2013) on DNF sparsification. An ingredient in our proof is a new random restriction lemma, which allows to analyze how DNFs (and more generally, decision lists) simplify if a small fraction of the variables are fixed. This is in contrast to the more commonly used switching lemma, which requires most of the variables to be fixed.

    更新日期:2020-02-19
  • Polynomial-Time Data Reduction for Weighted Problems Beyond Additive Goal Functions
    arXiv.cs.DM Pub Date : 2019-10-01
    Matthias Bentert; René van Bevern; Till Fluschnik; André Nichterlein; Rolf Niedermeier

    Kernelization is the fundamental notion for polynomial-time data reduction with performance guarantees. Kernelization for weighted problems particularly requires to also shrink weights. Marx and V\'egh [ACM Trans. Algorithms 2015] and Etscheid et al. [J. Comput. Syst. Sci. 2017] used a technique of Frank and Tardos [Combinatorica 1987] to obtain polynomial-size kernels for weighted problems, mostly with additive goal functions. We lift the technique to linearizable functions, a function type that we introduce and that also contains non-additive functions. Using the lifted technique, we obtain kernelization results for natural problems in graph partitioning, network design, facility location, scheduling, vehicle routing, and computational social choice, thereby improving and generalizing results from the literature.

    更新日期:2020-02-19
  • Symmetric Arithmetic Circuits
    arXiv.cs.DM Pub Date : 2020-02-15
    Anuj Dawar; Gregory Wilsenach

    We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry restriction. In the context of circuits computing polynomials defined on a matrix of variables, such as the determinant or the permanent, the restriction amounts to requiring that the shape of the circuit is invariant under row and column permutations of the matrix. We establish unconditional, nearly exponential, lower bounds on the size of any symmetric circuit for computing the permanent over any field of characteristic other than 2. In contrast, we show that there are polynomial-size symmetric circuits for computing the determinant over fields of characterisitic zero.

    更新日期:2020-02-18
  • Directed Graph Hashing
    arXiv.cs.DM Pub Date : 2020-02-16
    Caleb Helbling

    This paper presents an algorithm for structurally hashing directed graphs. The algorithm seeks to fulfill the recursive principle that a hash of a node should depend only on the hash of its neighbors. The algorithm works even in the presence of cycles, which prevents a naive recursive algorithm from functioning. We also discuss the implications of the recursive principle, limitations of the algorithm, and potential use cases.

    更新日期:2020-02-18
  • Generating clause sequences of a CNF formula
    arXiv.cs.DM Pub Date : 2020-02-17
    Kristóf Bérczi; Endre Boros; Ondřej Čepek; Khaled Elbassioni; Petr Kučera; Kazuhisa Makino

    Given a CNF formula $\Phi$ with clauses $C_1,\ldots,C_m$ and variables $V=\{x_1,\ldots,x_n\}$, a truth assignment $a:V\rightarrow\{0,1\}$ of $\Phi$ leads to a clause sequence $\sigma_\Phi(a)=(C_1(a),\ldots,C_m(a))\in\{0,1\}^m$ where $C_i(a) = 1$ if clause $C_i$ evaluates to $1$ under assignment $a$, otherwise $C_i(a) = 0$. The set of all possible clause sequences carries a lot of information on the formula, e.g. SAT, MAX-SAT and MIN-SAT can be encoded in terms of finding a clause sequence with extremal properties. We consider a problem posed at Dagstuhl Seminar 19211 "Enumeration in Data Management" (2019) about the generation of all possible clause sequences of a given CNF with bounded dimension. We prove that the problem can be solved in incremental polynomial time. We further give an algorithm with polynomial delay for the class of tractable CNF formulas. We also consider the generation of maximal and minimal clause sequences, and show that generating maximal clause sequences is NP-hard, while minimal clause sequences can be generated with polynomial delay.

    更新日期:2020-02-18
  • A two-stage algorithm for aircraft conflict resolution with trajectory recovery
    arXiv.cs.DM Pub Date : 2020-02-17
    Fernando H. C. Dias; Stephanie Rahme; David Rey

    As air traffic volume is continuously increasing, it has become a priority to improve traffic control algorithms to handle future air travel demand and improve airspace capacity. We address the conflict resolution problem in air traffic control using a novel approach for aircraft collision avoidance with trajectory recovery. We present a two-stage algorithm that first solves all initial conflicts by adjusting aircraft headings and speeds, before identifying the optimal time for aircraft to recover towards their target destination. The collision avoidance stage extends an existing mixed-integer programming formulation to heading control. For the trajectory recovery stage, we introduce a novel exact mixed-integer programming formulation as well as a greedy heuristic algorithm. The proposed two-stage approach guarantees that all trajectories during both the collision avoidance and recovery stages are conflict-free. Numerical results on benchmark problems show that the proposed heuristic for trajectory recovery is competitive while also emphasizing the difficulty of this optimization problem. The proposed approach can be used as a decision-support tool for introducing automation in air traffic control.

    更新日期:2020-02-18
  • A Unifying Framework for Spectrum-Preserving Graph Sparsification and Coarsening
    arXiv.cs.DM Pub Date : 2019-02-26
    Gecia Bravo-Hermsdorff; Lee M. Gunderson

    How might one "reduce" a graph? That is, generate a smaller graph that preserves the global structure at the expense of discarding local details? There has been extensive work on both graph sparsification (removing edges) and graph coarsening (merging nodes, often by edge contraction); however, these operations are currently treated separately. Interestingly, for a planar graph, edge deletion corresponds to edge contraction in its planar dual (and more generally, for a graphical matroid and its dual). Moreover, with respect to the dynamics induced by the graph Laplacian (e.g., diffusion), deletion and contraction are physical manifestations of two reciprocal limits: edge weights of $0$ and $\infty$, respectively. In this work, we provide a unifying framework that captures both of these operations, allowing one to simultaneously sparsify and coarsen a graph while preserving its large-scale structure. The limit of infinite edge weight is rarely considered, as many classical notions of graph similarity diverge. However, its algebraic, geometric, and physical interpretations are reflected in the Laplacian pseudoinverse $\mathbf{\mathit{L}}^{\dagger}$, which remains finite in this limit. Motivated by this insight, we provide a probabilistic algorithm that reduces graphs while preserving $\mathbf{\mathit{L}}^{\dagger}$, using an unbiased procedure that minimizes its variance. We compare our algorithm with several existing sparsification and coarsening algorithms using real-world datasets, and demonstrate that it more accurately preserves the large-scale structure.

    更新日期:2020-02-18
  • Two-dimensional partial cubes
    arXiv.cs.DM Pub Date : 2019-06-11
    Victor Chepoi; Kolja Knauer; Manon Philibert

    We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube $Q_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variety of results. In particular, we prove that all partial cubes of VC-dimension 2 can be extended to ample aka lopsided partial cubes of VC-dimension 2, yielding that the set families defined by such graphs satisfy the sample compression conjecture by Littlestone and Warmuth (1986). Furthermore we point out relations to tope graphs of COMs of low rank and region graphs of pseudoline arrangements.

    更新日期:2020-02-18
  • On the minimum value of the Colless index and the bifurcating trees that achieve it
    arXiv.cs.DM Pub Date : 2019-07-11
    Tomás M. Coronado; Mareike Fischer; Lina Herbst; Francesc Rosselló; Kristina Wicke

    Measures of tree balance play an important role in the analysis of phylogenetic trees. One of the oldest and most popular indices in this regard is the Colless index for rooted bifurcating trees, introduced by Colless (1982). While many of its statistical properties under different probabilistic models for phylogenetic trees have already been established, little is known about its minimum value and the trees that achieve it. In this manuscript, we fill this gap in the literature. To begin with, we derive both recursive and closed expressions for the minimum Colless index of a tree with $n$ leaves. Surprisingly, these expressions show a connection between the minimum Colless index and the so-called Blancmange curve, a fractal curve. We then fully characterize the tree shapes that achieve this minimum value and we introduce both an algorithm to generate them and a recurrence to count them. After focusing on two extremal classes of trees with minimum Colless index (the maximally balanced trees and the greedy from the bottom trees), we conclude by showing that all trees with minimum Colless index also have minimum Sackin index, another popular balance index.

    更新日期:2020-02-18
  • Equitable vertex arboricity of $d$-degenerate graphs
    arXiv.cs.DM Pub Date : 2019-08-14
    Xin Zhang; Bei Niu; Yan Li; Bi Li

    A minimization problem in graph theory so-called the equitable tree-coloring problem can be used to formulate a structure decomposition problem on the communication network with some security considerations. Precisely, an equitable tree-$k$-coloring of a graph is a vertex coloring using $k$ distinct colors such that every color class induces a forest and the sizes of any two color classes differ by at most one. In this paper, we establish some theoretical results on the equitable tree-colorings of graphs by showing that every $d$-degenerate graph with maximum degree at most $\Delta$ is equitably tree-$k$-colorable for every integer $k\geq (\Delta+1)/2$ provided that $\Delta\geq 10d$. This generalises the result of Chen et al.[J. Comb. Optim. 34(2) (2017) 426--432] which states that every $5$-degenerate graph with maximum degree at most $\Delta$ is equitably tree-$k$-colorable for every integer $k\geq (\Delta+1)/2$.

    更新日期:2020-02-18
  • On the Approximation Ratio of the $k$-Opt and Lin-Kernighan Algorithm for Metric and Graph TSP
    arXiv.cs.DM Pub Date : 2019-09-27
    Xianghui Zhong

    The $k$-Opt and Lin-Kernighan algorithm are two of the most important local search approaches for the Metric TSP. Both start with an arbitrary tour and make local improvements in each step to get a shorter tour. We show that, for any fixed $k\geq 3$, the approximation ratio of the $k$-Opt algorithm for Metric TSP is $O(\sqrt[k]{n})$. Assuming the Erd\H{o}s girth conjecture, we prove a matching lower bound of $\Omega(\sqrt[k]{n})$. Unconditionally, we obtain matching bounds for $k=3,4,6$ and a lower bound of $\Omega(n^{\frac{2}{3k-3}})$. Our most general bounds depend on the values of a function from extremal graph theory and are tight up to a factor logarithmic in the number of vertices unconditionally. Moreover, all the upper bounds also apply to a parameterized version of the Lin-Kernighan algorithm with appropriate parameter. In the end, we show that the approximation ratio of $k$-Opt for Graph TSP is $\Omega\left(\frac{\log(n)}{\log\log(n)}\right)$ and $O\left(\left(\frac{\log(n)}{\log\log(n)}\right)^{\log_2(9)+\epsilon}\right)$ for all $\epsilon>0$.

    更新日期:2020-02-18
  • A polynomial kernel for $3$-leaf power deletion
    arXiv.cs.DM Pub Date : 2019-11-11
    Jungho Ahn; Eduard Eiben; O-joung Kwon; Sang-il Oum

    A graph $G$ is an $\ell$-leaf power of a tree $T$ if $V(G)$ is equal to the set of leaves of $T$, and distinct vertices $v$ and $w$ of $G$ are adjacent if and only if the distance between $v$ and $w$ in $T$ is at most $\ell$. Given a graph $G$, the $3$-Leaf Power Deletion problem asks whether there is a set $S\subseteq V(G)$ of size at most $k$ such that $G\setminus S$ is a $3$-leaf power of some tree $T$. We provide a polynomial kernel for this problem. More specifically, we present a polynomial-time algorithm for an input instance $(G,k)$ to output an equivalent instance $(G',k')$ such that $k'\leq k$ and $G'$ has at most $O(k^{14})$ vertices.

    更新日期:2020-02-18
  • Introducing Graph Cumulants: What is the Kurtosis of your Social Network?
    arXiv.cs.DM Pub Date : 2020-02-10
    Lee M. Gunderson; Gecia Bravo-Hermsdorff

    In an increasingly interconnected world, understanding and summarizing the structure of these networks becomes increasingly relevant. However, this task is nontrivial; proposed summary statistics are as diverse as the networks they describe, and a standardized hierarchy has not yet been established. In contrast, vector-valued random variables admit such a description in terms of their cumulants (e.g., mean, (co)variance, skew, kurtosis). Here, we introduce the natural analogue of cumulants for networks, building a hierarchical description based on correlations between an increasing number of connections, seamlessly incorporating additional information, such as directed edges, node attributes, and edge weights. These graph cumulants provide a principled and unifying framework for quantifying the propensity of a network to display any substructure of interest (such as cliques to measure clustering). Moreover, they give rise to a natural hierarchical family of maximum entropy models for networks (i.e., ERGMs) that do not suffer from the "degeneracy problem", a common practical pitfall of other ERGMs.

    更新日期:2020-02-18
  • Relationship of $k$-Bend and Monotonic $\ell$-Bend Edge Intersection Graphs of Paths on a Grid
    arXiv.cs.DM Pub Date : 2020-02-14
    Eranda Cela; Elisabeth Gaar

    If a graph $G$ can be represented by means of paths on a grid, such that each vertex of $G$ corresponds to one path on the grid and two vertices of $G$ are adjacent if and only if the corresponding paths share a grid edge, then this graph is called EPG and the representation is called EPG representation. A $k$-bend EPG representation is an EPG representation in which each path has at most $k$ bends. The class of all graphs that have a $k$-bend EPG representation is denoted by $B_k$. $B_\ell^m$ is the class of all graphs that have a monotonic (each path is ascending in both columns and rows) $\ell$-bend EPG representation. It is known that $B_k^m \subsetneqq B_k$ holds for $k=1$. We prove that $B_k^m \subsetneqq B_k$ holds also for $k \in \{2, 3, 5\}$ and for $k \geqslant 7$ by investigating the $B_k$-membership and $B_k^m$-membership of complete bipartite graphs. In particular we derive necessary conditions for this membership that have to be fulfilled by $m$, $n$ and $k$, where $m$ and $n$ are the number of vertices on the two partition classes of the bipartite graph. We conjecture that $B_{k}^{m} \subsetneqq B_{k}$ holds also for $k\in \{4,6\}$. Furthermore we show that $B_k \not\subseteq B_{2k-9}^m$ holds for all $k\geqslant 5$. This implies that restricting the shape of the paths can lead to a significant increase of the number of bends needed in an EPG representation. So far no bounds on the amount of that increase were known. We prove that $B_1 \subseteq B_3^m$ holds, providing the first result of this kind.

    更新日期:2020-02-17
  • Abstract rewriting internalized
    arXiv.cs.DM Pub Date : 2020-02-14
    Maxime Lucas

    In traditional rewriting theory, one studies a set of terms up to a set of rewriting relations. In algebraic rewriting, one instead studies a vector space of terms, up to a vector space of relations. Strikingly, although both theories are very similar, most results (such as Newman's Lemma) require different proofs in these two settings. In this paper, we develop rewriting theory internally to a category $\mathcal C$ satisfying some mild properties. In this general setting, we define the notions of termination, local confluence and confluence using the notion of reduction strategy, and prove an analogue of Newman's Lemma. In the case of $\mathcal C= \operatorname{Set}$ or $\mathcal C = \operatorname{Vect}$ we recover classical results of abstract and algebraic rewriting in a slightly more general form, closer to von Oostrom's notion of decreasing diagrams.

    更新日期:2020-02-17
  • A Breezing Proof of the KMW Bound
    arXiv.cs.DM Pub Date : 2020-02-14
    Corinna Coupette; Christoph Lenzen

    In their seminal paper from 2004, Kuhn, Moscibroda, and Wattenhofer (KMW) proved a hardness result for several fundamental graph problems in the LOCAL model: For any (randomized) algorithm, there are input graphs with $n$ nodes and maximum degree $\Delta$ on which $\Omega(\min\{\sqrt{\log n/\log \log n},\log \Delta/\log \log \Delta\})$ (expected) communication rounds are required to obtain polylogarithmic approximations to a minimum vertex cover, minimum dominating set, or maximum matching, respectively. Via reduction, this hardness extends to symmetry breaking tasks like finding maximal independent sets or maximal matchings. Today, more than $15$ years later, there is still no proof of this result that is easy on the reader. Setting out to change this, in this work, we provide a fully self-contained and \emph{simple} proof of the KMW lower bound. The key argument is algorithmic, and it relies on an invariant that can be readily verified from the generation rules of the lower bound graphs.

    更新日期:2020-02-17
  • Some Problems and Algorithms Related to the Weight Order Relation on the $n$-dimensional Boolean Cube
    arXiv.cs.DM Pub Date : 2018-11-11
    Valentin Bakoev

    The problem "Given a Boolean function $f$ of $n$ variables by its truth table vector. Find (if exists) a vector $\alpha \in \{0,1\}^n$ of maximal (or minimal) weight, such that $f(\alpha)= 1$." is considered here. It is closely related to the problem of fast computing the algebraic degree of Boolean functions. It is an important cryptographic parameter used in the design of S-boxes in modern block ciphers, PRNGs in stream ciphers, at Reed-Muller codes, etc. To find effective solutions to this problem we explore the orders of the vectors of the $n$-dimensional Boolean cube $\{0,1\}^n$ in accordance with their weights. The notion of "$k$-th layer" of $\{0,1\}^n$ is involved in the definition and examination of the "weight order" relation. It is compared with the known relation "precedes". Several enumeration problems concerning these relations are solved and the corresponding comments were added to 3 sequences in the On-line Encyclopedia of Integer Sequences (OEIS). One special order (among the numerous weight orders) is defined and examined in detail. The lexicographic order is a second criterion for an ordinance of the vectors of equal weights. So a total order called Weight-Lexicographic Order (WLO) is obtained. Two algorithms for generating the WLO sequence and two algorithms for generating the characteristic vectors of the layers are proposed. Their results were used in creating 2 new sequences: A294648 and A305860 in the OEIS. Two algorithms for solving the problem considered are developed--the first one works in a byte-wise manner and uses the WLO sequence, and the second one works in a bitwise manner and uses the characteristic vector as masks. The experimental results after many tests confirm the efficiency of these algorithms. Some other applications of the obtained algorithms are also discussed--for example, when representing, generating and ranking other combinatorial objects.

    更新日期:2020-02-17
  • Two-sided matching markets with correlated random preferences have few stable pairs
    arXiv.cs.DM Pub Date : 2019-04-08
    Hugo Gimbert; Claire Mathieu; Simon Mauras

    Stable matching in a community consisting of $N$ men and $N$ women is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal paper by Gale and Shapley. In this paper, we study the number of stable pairs, that is, the man/woman pairs that appear in some stable matching. We prove that if the preference lists on one side are generated at random using the popularity model of Immorlica and Mahdian, the expected number of stable edges is bounded by $N \ln N + N$, matching the asymptotic value for uniform preference lists. If in addition that popularity model is a geometric distribution, then the number of stable edges is $\mathcal O(N)$ and the incentive to manipulate is limited. If in addition the preference lists on the other side are uniform, then the number of stable edges is asymptotically $N$ up to lower order terms: most participants have a unique stable partner, hence non-manipulability.

    更新日期:2020-02-17
  • The structure of multigranular rough sets
    arXiv.cs.DM Pub Date : 2020-02-12
    Jouni Järvinen; Sándor Radeleczki

    We study multigranulation spaces of two equivalences. The lattice-theoretical properties of so-called "optimistic" and "pessimistic" multigranular approximation systems are given. We also consider the ordered sets of rough sets determined by these approximation pairs.

    更新日期:2020-02-14
  • Breaking limitation of quantum annealer in solving optimization problems under constraints
    arXiv.cs.DM Pub Date : 2020-02-13
    Masayuki Ohzeki

    Quantum annealing is a generic solver for optimization problems that uses fictitious quantum fluctuation. The most groundbreaking progress in the research field of quantum annealing is its hardware implementation, i.e., the so-called quantum annealer, using artificial spins. However, the connectivity between the artificial spins is sparse and limited on a special network known as the chimera graph. Several embedding techniques have been proposed, but the number of logical spins, which represents the optimization problems to be solved, is drastically reduced. In particular, an optimization problem including fully or even partly connected spins suffers from low embeddable size on the chimera graph. In the present study, we propose an alternative approach to solve a large-scale optimization problem on the chimera graph via a well-known method in statistical mechanics called the Hubbard-Stratonovich transformation or its variants. The proposed method can be used to deal with a fully connected Ising model without embedding on the chimera graph and leads to nontrivial results of the optimization problem. We tested the proposed method with a number of partition problems involving solving linear equations and the traffic flow optimization problem in Sendai and Kyoto cities in Japan.

    更新日期:2020-02-14
  • A generalization of the Kővári-Sós-Turán theorem
    arXiv.cs.DM Pub Date : 2020-02-13
    Jesse Geneson

    We present a new proof of the K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n theorem that $ex(n, K_{s,t}) = O(n^{2-1/t})$. The new proof is elementary, avoiding the use of convexity. For any $d$-uniform hypergraph $H$, let $ex_d(n,H)$ be the maximum possible number of edges in an $H$-free $d$-uniform hypergraph on $n$ vertices. Let $K_{H, t}$ be the $(d+1)$-uniform hypergraph obtained from $H$ by adding $t$ new vertices $v_1, \dots, v_t$ and replacing every edge $e$ in $E(H)$ with $t$ edges $e \cup \left\{v_1\right\},\dots, e \cup \left\{v_t\right\}$ in $E(K_{H, t})$. If $H$ is the $1$-uniform hypergraph on $s$ vertices with $s$ edges, then $K_{H, t} = K_{s, t}$. We prove that $ex_{d+1}(n,K_{H,t}) = O(ex_d(n, H)^{1/t} n^{d+1-d/t})$. Thus $ex_{d+1}(n,K_{H,t}) = O(n^{d+1-1/t})$ for any $d$-uniform hypergraph $H$ with $ex_d(n, H) = \Theta(n^{d-1})$, which implies the K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n theorem in the $d = 1$ case. As a corollary, this implies that $ex_{d+1}(n, K_{H,t}) = O(n^{d+1-1/t})$ when $H$ is a $d$-uniform hypergraph in which all edges are pairwise disjoint, which generalizes an upper bound proved by Mubayi and Verstra\"{e}te (JCTA, 2004). We also obtain analogous bounds for 0-1 matrix Tur\'{a}n problems.

    更新日期:2020-02-14
  • Notes on Tree- and Path-chromatic Number
    arXiv.cs.DM Pub Date : 2020-02-13
    Tony Huynh; Bruce Reed; David R. Wood; Liana Yepremyan

    Tree-chromatic number is a chromatic version of treewidth, where the cost of a bag in a tree-decomposition is measured by its chromatic number rather than its size. Path-chromatic number is defined analogously. These parameters were introduced by Seymour (JCTB 2016). In this paper, we survey all the known results on tree- and path-chromatic number and then present some new results and conjectures. In particular, we propose a version of Hadwiger's Conjecture for tree-chromatic number. As evidence that our conjecture may be more tractable than Hadwiger's Conjecture, we give a short proof that every $K_5$-minor-free graph has tree-chromatic number at most $4$, which avoids the Four Colour Theorem. We also present some hardness results and conjectures for computing tree- and path-chromatic number.

    更新日期:2020-02-14
  • Genetic Algorithms for Redundancy in Interaction Testing
    arXiv.cs.DM Pub Date : 2020-02-13
    Ryan E. Dougherty

    It is imperative for testing to determine if the components within large-scale software systems operate functionally. Interaction testing involves designing a suite of tests, which guarantees to detect a fault if one exists among a small number of components interacting together. The cost of this testing is typically modeled by the number of tests, and thus much effort has been taken in reducing this number. Here, we incorporate redundancy into the model, which allows for testing in non-deterministic environments. Existing algorithms for constructing these test suites usually involve one "fast" algorithm for generating most of the tests, and another "slower" algorithm to "complete" the test suite. We employ a genetic algorithm that generalizes these approaches that also incorporates redundancy by increasing the number of algorithms chosen, which we call "stages." By increasing the number of stages, we show that not only can the number of tests be reduced compared to existing techniques, but the computational time in generating them is also greatly reduced.

    更新日期:2020-02-14
  • Tight Bounds for the Cover Times of Random Walks with Heterogeneous Step Lengths
    arXiv.cs.DM Pub Date : 2020-02-13
    Brieuc GuinardIRIF, CNRS, UPD7; Amos KormanIRIF, CNRS, UPD7

    Search patterns of randomly oriented steps of different lengths have been observed on all scales of the biological world, ranging from the microscopic to the ecological, including in protein motors, bacteria, T-cells, honeybees, marine predators, and more. Through different models, it has been demonstrated that adopting a variety in the magnitude of the step lengths can greatly improve the search efficiency. However, the precise connection between the search efficiency and the number of step lengths in the repertoire of the searcher has not been identified. Motivated by biological examples in one-dimensional terrains, a recent paper studied the best cover time on an n-node cycle that can be achieved by a random walk process that uses k step lengths. By tuning the lengths and corresponding probabilities the authors therein showed that the best cover time is roughly n 1+$\Theta$(1/k). While this bound is useful for large values of k, it is hardly informative for small k values, which are of interest in biology. In this paper, we provide a tight bound for the cover time of such a walk, for every integer k > 1. Specifically, up to lower order polylogarithmic factors, the upper bound on the cover time is a polynomial in n of exponent 1+ 1/(2k--1). For k = 2, 3, 4 and 5 the exponent is thus 4/3 , 6/5 , 8/7 , and 10/9 , respectively. Informally, our result implies that, as long as the number of step lengths k is not too large, incorporating an additional step length to the repertoire of the process enables to improve the cover time by a polynomial factor, but the extent of the improvement gradually decreases with k.

    更新日期:2020-02-14
  • Algorithmic Complexity of Isolate Secure Domination in Graphs
    arXiv.cs.DM Pub Date : 2020-02-12
    Jakkepalli Pavan Kumar; P. Venkata Subba Reddy

    A dominating set $S$ is an Isolate Dominating Set (IDS) if the induced subgraph $G[S]$ has at least one isolated vertex. In this paper, we initiate the study of new domination parameter called, isolate secure domination. An isolate dominating set $S\subseteq V$ is an isolate secure dominating set (ISDS), if for each vertex $u \in V \setminus S$, there exists a neighboring vertex $v$ of $u$ in $S$ such that $(S \setminus \{v\}) \cup \{u\}$ is an IDS of $G$. The minimum cardinality of an ISDS of $G$ is called as an isolate secure domination number, and is denoted by $\gamma_{0s}(G)$. Given a graph $ G=(V,E)$ and a positive integer $ k,$ the ISDM problem is to check whether $ G $ has an isolate secure dominating set of size at most $ k.$ We prove that ISDM is NP-complete even when restricted to bipartite graphs and split graphs. We also show that ISDM can be solved in linear time for graphs of bounded tree-width.

    更新日期:2020-02-14
  • Drawing Graphs as Spanners
    arXiv.cs.DM Pub Date : 2020-02-13
    Oswin Aichholzer; Manuel Borrazzo; Prosenjit Bose; Jean Cardinal; Fabrizio Frati; Pat Morin; Birgit Vogtenhuber

    We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph $G$, the goal is to construct a straight-line drawing $\Gamma$ of $G$ in the plane such that, for any two vertices $u$ and $v$ of $G$, the ratio between the minimum length of any path from $u$ to $v$ and the Euclidean distance between $u$ and $v$ is small. The maximum such ratio, over all pairs of vertices of $G$, is the spanning ratio of $\Gamma$. First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio $1$, a proper straight-line drawing with spanning ratio $1$, and a planar straight-line drawing with spanning ratio $1$ are NP-complete, $\exists \mathbb R$-complete, and linear-time solvable problems, respectively, where a drawing is proper if no two vertices overlap and no edge overlaps a vertex. Second, we show that moving from spanning ratio $1$ to spanning ratio $1+\epsilon$ allows us to draw every graph. Namely, we prove that, for every $\epsilon>0$, every (planar) graph admits a proper (resp. planar) straight-line drawing with spanning ratio smaller than $1+\epsilon$. Third, our drawings with spanning ratio smaller than $1+\epsilon$ have large edge-length ratio, that is, the ratio between the length of the longest edge and the length of the shortest edge is exponential. We show that this is sometimes unavoidable. More generally, we identify having bounded toughness as the criterion that distinguishes graphs that admit straight-line drawings with constant spanning ratio and polynomial edge-length ratio from graphs that require exponential edge-length ratio in any straight-line drawing with constant spanning ratio.

    更新日期:2020-02-14
  • A Simpler and Faster Strongly Polynomial Algorithm for Generalized Flow Maximization
    arXiv.cs.DM Pub Date : 2016-11-06
    Neil Olver; László A. Végh

    We present a new strongly polynomial algorithm for generalized flow maximization that is significantly simpler and faster than the previous strongly polynomial algorithm [V\'egh16]. For the uncapacitated problem formulation, the complexity bound $O(mn(m+n\log n)\log (n^2/m))$ improves on the previous estimate by almost a factor $O(n^2)$. Even for small numerical parameter values, our running time bound is comparable to the best weakly polynomial algorithms. The key new technical idea is relaxing the primal feasibility conditions. This allows us to work almost exclusively with integral flows, in contrast to all previous algorithms for the problem.

    更新日期:2020-02-14
  • Cumulative subtraction games
    arXiv.cs.DM Pub Date : 2018-05-23
    Gal Cohensius; Urban Larsson; Reshef Meir; David Wahlstedt

    We study zero-sum games, a variant of the classical combinatorial Subtraction games (studied for example in the monumental work "Winning Ways", by Berlekamp, Conway and Guy), called Cumulative Subtraction (CS). Two players alternate in moving, and get points for taking pebbles out of a joint pile. We prove that the outcome in optimal play (game value) of a CS with a finite number of possible actions is eventually periodic, with period $2s$, where $s$ is the size of the largest available action. This settles a conjecture by Stewart in his Ph.D. thesis (2011). Specifically, we find a quadratic bound, in the size of $s$, on when the outcome function must have become periodic. In case of two possible actions, we give an explicit description of optimal play. We generalize the periodicity result to games with a so-called reward function, where at each stage of game, the change of `score' does not necessarily equal the number of pebbles you collect.

    更新日期:2020-02-14
  • Graph multicoloring reduction methods and application to McDiarmid-Reed's Conjecture
    arXiv.cs.DM Pub Date : 2018-12-05
    Jean-Christophe Godin; Olivier Togni

    A $(a,b)$-coloring of a graph $G$ associates to each vertex a set of $b$ colors from a set of $a$ colors in such a way that the color-sets of adjacent vertices are disjoints. We define general reduction tools for $(a,b)$-coloring of graphs for $2\le a/b\le 3$. In particular, we prove necessary and sufficient conditions for the existence of a $(a,b)$-coloring of a path with prescribed color-sets on its end-vertices. Other more complex $(a,b)$-colorability reductions are presented. The utility of these tools is exemplified on finite triangle-free induced subgraphs of the triangular lattice. Computations on millions of such graphs generated randomly show that our tools allow to find (in linear time) a $(9,4)$-coloring for each of them. Although there remain few graphs for which our tools are not sufficient for finding a $(9,4)$-coloring, we believe that pursuing our method can lead to a solution of the conjecture of McDiarmid-Reed.

    更新日期:2020-02-14
  • Freezing, Bounded-Change and Convergent Cellular Automata
    arXiv.cs.DM Pub Date : 2019-08-19
    Nicolas OllingerLIFO; Guillaume TheyssierI2M

    This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension , and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-Kurka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension 1, but also dimension 1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension.

    更新日期:2020-02-14
  • The Minimization of Random Hypergraphs
    arXiv.cs.DM Pub Date : 2019-10-01
    Thomas Bläsius; Tobias Friedrich; Martin Schirneck

    We investigate the maximum-entropy model $\mathcal{B}_{n,m,p}$ for random $n$-vertex, $m$-edge multi-hypergraphs with expected edge size $pn$. We show that the expected size of the minimization $\min(\mathcal{B}_{n,m,p})$, i.e., the number of inclusion-wise minimal edges of $\mathcal{B}_{n,m,p}$, undergoes a phase transition with respect to $m$. If $m$ is at most $1/(1-p)^{(1-p)n}$, then $\mathrm{E}[|\min(\mathcal{B}_{n,m,p})|]$ is of order $\Theta(m)$, while for $m \ge 1/(1-p)^{(1-p+\varepsilon)n}$ for any $\varepsilon > 0$, it is $\Theta( 2^{(\mathrm{H}(\alpha) + (1-\alpha) \log_2 p) n}/ \sqrt{n})$. Here, $\mathrm{H}$ denotes the binary entropy function and $\alpha = - (\log_{1-p} m)/n$. The result implies that the maximum expected number of minimal edges over all $m$ is $\Theta((1+p)^n/\sqrt{n})$. Our structural findings have algorithmic implications for minimizing an input hypergraph, which has applications in the profiling of relational databases as well as for the Orthogonal Vectors problem studied in fine-grained complexity. We make several technical contributions that are of independent interest in probability. First, we improve the Chernoff--Hoeffding theorem on the tail of the binomial distribution. In detail, we show that for a binomial variable $Y \sim \operatorname{Bin}(n,p)$ and any $0 < x < p$, it holds that $\mathrm{P}[Y \le xn] = \Theta( 2^{-\!\mathrm{D}(x \,{\|}\, p) n}/\sqrt{n})$, where $\mathrm{D}$ is the binary Kullback--Leibler divergence between Bernoulli distributions. We give explicit upper and lower bounds on the constants hidden in the big-O notation that hold for all $n$. Secondly, we establish the fact that the probability of a set of cardinality $i$ being minimal after $m$ i.i.d. maximum-entropy trials exhibits a sharp threshold behavior at $i^* = n + \log_{1-p} m$.

    更新日期:2020-02-14
  • Analyzing, Exploring, and Visualizing Complex Networks via Hypergraphs using SimpleHypergraphs.jl
    arXiv.cs.DM Pub Date : 2020-02-10
    Alessia Antelmi; Gennaro Cordasco; Bogumił Kamiński; Paweł Prałat; Vittorio Scarano; Carmine Spagnuolo; Przemyslaw Szufel

    Real-world complex networks are usually being modeled as graphs. The concept of graphs assumes that the relations within the network are binary (for instance, between pairs of nodes); however, this is not always true for many real-life scenarios, such as peer-to-peer communication schemes, paper co-authorship, or social network interactions. For such scenarios, it is often the case that the underlying network is better and more naturally modeled by hypergraphs. A hypergraph is a generalization of a graph in which a single (hyper)edge can connect any number of vertices. Hypergraphs allow modelers to have a complete representation of multi-relational (many-to-many) networks; hence, they are extremely suitable for analyzing and discovering more subtle dependencies in such data structures. Working with hypergraphs requires new software libraries that make it possible to perform operations on them, from basic algorithms (searching or traversing the network) to computing important hypergraph measures, to including more challenging algorithms (community detection). In this paper, we present a new software library, SimpleHypergraphs.jl, written in the Julia language and designed for high-performance computing on hypergraphs. We also present various approaches for hypergraph visualization that have been integrated into our tool. To demonstrate how the library can be exploited in practice, we discuss two case studies based on the 2019 Yelp Challenge dataset and the collaboration network built upon the Game of Thrones TV series. Results are promising and confirm the ability of hypergraphs to provide more insight than standard graph-based approaches.

    更新日期:2020-02-13
  • Bi-objective Optimization of Biclustering with Binary Data
    arXiv.cs.DM Pub Date : 2020-02-09
    Fred Glover; Said Hanafi; Gintaras Palubeckis

    Clustering consists of partitioning data objects into subsets called clusters according to some similarity criteria. This paper addresses a generalization called quasi-clustering that allows overlapping of clusters, and which we link to biclustering. Biclustering simultaneously groups the objects and features so that a specific group of objects has a special group of features. In recent years, biclustering has received a lot of attention in several practical applications. In this paper we consider a bi-objective optimization of biclustering problem with binary data. First we present an integer programing formulations for the bi-objective optimization biclustering. Next we propose a constructive heuristic based on the set intersection operation and its efficient implementation for solving a series of mono-objective problems used inside the Epsilon-constraint method (obtained by keeping only one objective function and the other objective function is integrated into constraints). Finally, our experimental results show that using CPLEX solver as an exact algorithm for finding an optimal solution drastically increases the computational cost for large instances, while our proposed heuristic provides very good results and significantly reduces the computational expense.

    更新日期:2020-02-13
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