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  • Efficient enumeration of maximal split subgraphs and sub-cographs and related classes
    arXiv.cs.DM Pub Date : 2020-07-02
    Caroline Brosse; Aurélie Lagoutte; Vincent Limouzy; Arnaud Mary; Lucas Pastor

    In this paper, we are interested in algorithms that take in input an arbitrary graph $G$, and that enumerate in output all the (inclusion-wise) maximal "subgraphs" of $G$ which fulfil a given property $\Pi$. All over this paper, we study several different properties $\Pi$, and the notion of subgraph under consideration (induced or not) will vary from a result to another. More precisely, we present

    更新日期:2020-07-03
  • Multicommodity Flows in Planar Graphs with Demands on Faces
    arXiv.cs.DM Pub Date : 2020-07-02
    Nikhil Kumar

    We consider the problem of multicommodity flows in planar graphs. Seymour showed that if the union of supply and demand graphs is planar, then the cut condition is sufficient for routing demands. Okamura-Seymour showed that if all demands are incident on one face, then again cut condition is sufficient for routing demands. We consider a common generalization of these settings where the end points of

    更新日期:2020-07-03
  • A Method for Fast Computing the Algebraic Degree of Boolean Functions
    arXiv.cs.DM Pub Date : 2020-07-01
    Valentin Bakoev

    The algebraic degree of Boolean functions (or vectorial Boolean functions) is an important cryptographic parameter that should be computed by fast algorithms. They work in two main ways: (1) by computing the algebraic normal form and then searching the monomial of the highest degree in it, or (2) by examination the algebraic properties of the true table vector of a given function. We have already done

    更新日期:2020-07-03
  • A simple bipartite graph projection model for clustering in networks
    arXiv.cs.DM Pub Date : 2020-07-01
    Austin R. Benson; Paul Liu; Hao Yin

    Graph datasets are frequently constructed by a projection of a bipartite graph, where two nodes are connected in the projection if they share a common neighbor in the bipartite graph; for example, a coauthorship graph is a projection of an author-publication bipartite graph. Analyzing the structure of the projected graph is common, but we do not have a good understanding of the consequences of the

    更新日期:2020-07-03
  • Build2Vec: Building Representation in Vector Space
    arXiv.cs.DM Pub Date : 2020-07-01
    Mahmoud Abdelrahman; Adrian Chong; Clayton Miller

    In this paper, we represent a methodology of a graph embeddings algorithm that is used to transform labeled property graphs obtained from a Building Information Model (BIM). Industrial Foundation Classes (IFC) is a standard schema for BIM, which is utilized to convert the building data into a graph representation. We used node2Vec with biased random walks to extract semantic similarities between different

    更新日期:2020-07-03
  • Some i-Mark games
    arXiv.cs.DM Pub Date : 2020-07-01
    Oren Friman; Gabriel Nivasch

    Let $S$ be a set of positive integers, and let $D$ be a set of integers larger than $1$. The game $i$-Mark$(S,D)$ is an impartial combinatorial game introduced by Sopena (2016), which is played with a single pile of tokens. In each turn, a player can subtract $s \in S$ from the pile, or divide the size of the pile by $d \in D$, if the pile size is divisible by $d$. Sopena partially analyzed the games

    更新日期:2020-07-03
  • Forbidden induced subgraph characterization of circle graphs within split graphs
    arXiv.cs.DM Pub Date : 2020-07-01
    Flavia Bonomo-Braberman; Guillermo A. Durán; Nina Pardal; Martín D. Safe

    A graph is circle if its vertices are in correspondence with a family of chords in a circle in such a way that every two distinct vertices are adjacent if and only if the corresponding chords have nonempty intersection. Even though there are diverse characterizations of circle graphs, a structural characterization by minimal forbidden induced subgraphs for the entire class of circle graphs is not known

    更新日期:2020-07-02
  • Dichotomizing $k$-vertex-critical $H$-free graphs for $H$ of order four
    arXiv.cs.DM Pub Date : 2020-06-30
    Ben Cameron; Chính T. Hoàng; Joe Sawada

    For $k \geq 3$, we prove (i) there is a finite number of $k$-vertex-critical $(P_2+\ell P_1)$-free graphs and (ii) $k$-vertex-critical $(P_3+P_1)$-free graphs have at most $2k-1$ vertices. Together with previous research, these results imply the following characterization where $H$ is a graph of order four: There is a finite number of $k$-vertex-critical $H$-free graphs for fixed $k \geq 5$ if and

    更新日期:2020-07-02
  • Precedence thinness in graphs
    arXiv.cs.DM Pub Date : 2020-06-30
    Flavia Bonomo-Braberman; Fabiano S. Oliveira; Moysés S. Sampaio Jr.; Jayme L. Szwarcfiter

    Interval and proper interval graphs are very well-known graph classes, for which there is a wide literature. As a consequence, some generalizations of interval graphs have been proposed, in which graphs in general are expressed in terms of $k$ interval graphs, by splitting the graph in some special way. As a recent example of such an approach, the classes of $k$-thin and proper $k$-thin graphs have

    更新日期:2020-07-01
  • Thinness of product graphs
    arXiv.cs.DM Pub Date : 2020-06-30
    Flavia Bonomo-Braberman; Carolina L. Gonzalez; Fabiano S. Oliveira; Moysés S. Sampaio Jr.; Jayme L. Szwarcfiter

    The thinness of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Many NP-complete problems can be solved in polynomial time for graphs with bounded thinness, given a suitable representation of the graph. In this paper we study the thinness and its variations of graph products. We show that the thinness behaves "well" in

    更新日期:2020-07-01
  • Concave Aspects of Submodular Functions
    arXiv.cs.DM Pub Date : 2020-06-27
    Rishabh Iyer; Jeff Bilmes

    Submodular Functions are a special class of set functions, which generalize several information-theoretic quantities such as entropy and mutual information [1]. Submodular functions have subgradients and subdifferentials [2] and admit polynomial-time algorithms for minimization, both of which are fundamental characteristics of convex functions. Submodular functions also show signs similar to concavity

    更新日期:2020-07-01
  • Linear transformations between dominating sets in the TAR-model
    arXiv.cs.DM Pub Date : 2020-06-30
    Nicolas Bousquet; Alice Joffard; Paul Ouvrard

    Given a graph $G$ and an integer $k$, a token addition and removal ({\sf TAR} for short) reconfiguration sequence between two dominating sets $D_{\sf s}$ and $D_{\sf t}$ of size at most $k$ is a sequence $S= \langle D_0 = D_{\sf s}, D_1 \ldots, D_\ell = D_{\sf t} \rangle$ of dominating sets of $G$ such that any two consecutive dominating sets differ by the addition or deletion of one vertex, and no

    更新日期:2020-07-01
  • A quantum algorithm to estimate the Gowers $U_2$ norm and linearity testing of Boolean functions
    arXiv.cs.DM Pub Date : 2020-06-30
    C. A. Jothishwaran; Anton Tkachenko; Sugata Gangopadhyay; Constanza Riera; Pantelimon Stanica

    We propose a quantum algorithm to estimate the Gowers $U_2$ norm of a Boolean function, and extend it into a second algorithm to distinguish between linear Boolean functions and Boolean functions that are $\epsilon$-far from the set of linear Boolean functions, which seems to perform better than the classical BLR algorithm. Finally, we outline an algorithm to estimate Gowers $U_3$ norms of Boolean

    更新日期:2020-07-01
  • Algorithms and complexity for geodetic sets on planar and chordal graphs
    arXiv.cs.DM Pub Date : 2020-06-30
    Dibyayan Chakraborty; Sandip Das; Florent Foucaud; Harmender Gahlawat; Dimitri Lajou; Bodhayan Roy

    We study the complexity of finding the \emph{geodetic number} on subclasses of planar graphs and chordal graphs. A set $S$ of vertices of a graph $G$ is a \emph{geodetic set} if every vertex of $G$ lies in a shortest path between some pair of vertices of $S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality of a given graph. The problem is known to

    更新日期:2020-07-01
  • Complexity of Modification Problems for Best Match Graphs
    arXiv.cs.DM Pub Date : 2020-06-30
    David Schaller; Peter F. Stadler; Marc Hellmuth

    Best match graphs (BMGs) are vertex-colored directed graphs that were introduced to model the relationships of genes (vertices) from different species (colors) given an underlying evolutionary tree that is assumed to be unknown. In real-life applications, BMGs are estimated from sequence similarity data. Measurement noise and approximation errors therefore result in empirically determined graphs that

    更新日期:2020-07-01
  • Counting Homomorphisms to $K_4$-minor-free Graphs, modulo 2
    arXiv.cs.DM Pub Date : 2020-06-30
    Jacob Focke; Leslie Ann Goldberg; Marc Roth; Stanislav Živný

    We study the problem of computing the parity of the number of homomorphisms from an input graph $G$ to a fixed graph $H$. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph $H$ and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class $\oplus\mathrm{P}$ of parity problems. We verify their conjecture

    更新日期:2020-07-01
  • On Dualization over Distributive Lattices
    arXiv.cs.DM Pub Date : 2020-06-27
    Khaled Elbassioni

    Given a partially order set (poset) $P$, and a pair of families of ideals $\cI$ and filters $\cF$ in $P$ such that each pair $(I,F)\in \cI\times\cF$ has a non-empty intersection, the dualization problem over $P$ is to check whether there is an ideal $X$ in $P$ which intersects every member of $\cF$ and does not contain any member of $\cI$. Equivalently, the problem is to check for a distributive lattice

    更新日期:2020-06-30
  • The Number of Repetitions in 2D-Strings
    arXiv.cs.DM Pub Date : 2020-06-29
    Panagiotis Charalampopoulos; Jakub Radoszewski; Wojciech Rytter; Tomasz Waleń; Wiktor Zuba

    The notions of periodicity and repetitions in strings, and hence these of runs and squares, naturally extend to two-dimensional strings. We consider two types of repetitions in 2D-strings: 2D-runs and quartics (quartics are a 2D-version of squares in standard strings). Amir et al. introduced 2D-runs, showed that there are $O(n^3)$ of them in an $n \times n$ 2D-string and presented a simple construction

    更新日期:2020-06-30
  • Flexible List Colorings in Graphs with Special Degeneracy Conditions
    arXiv.cs.DM Pub Date : 2020-06-29
    Peter Bradshaw; Tomáš Masařík; Ladislav Stacho

    For a given $\varepsilon > 0$, we say that a graph $G$ is $\epsilon$-flexibly $k$-choosable if the following holds: for any assignment $L$ of lists of size $k$ on $V(G)$, if a preferred color is requested at any set $R$ of vertices, then at least $\epsilon |R|$ of these requests may be satisfied by some $L$-coloring. We consider flexible list colorings in several graph classes with certain special

    更新日期:2020-06-30
  • Asymptotic enumeration of digraphs and bipartite graphs by degree sequence
    arXiv.cs.DM Pub Date : 2020-06-29
    Anita Liebenau; Nick Wormald

    We provide asymptotic formulae for the numbers of bipartite graphs with given degree sequence, and of loopless digraphs with given in- and out-degree sequences, for a wide range of parameters. Our results cover medium range densities and close the gaps between the results known for the sparse and dense ranges. In the case of bipartite graphs, these results were proved by Greenhill, McKay and Wang in

    更新日期:2020-06-30
  • Fast and Private Submodular and $k$-Submodular Functions Maximization with Matroid Constraints
    arXiv.cs.DM Pub Date : 2020-06-28
    Akbar Rafiey; Yuichi Yoshida

    The problem of maximizing nonnegative monotone submodular functions under a certain constraint has been intensively studied in the last decade, and a wide range of efficient approximation algorithms have been developed for this problem. Many machine learning problems, including data summarization and influence maximization, can be naturally modeled as the problem of maximizing monotone submodular functions

    更新日期:2020-06-30
  • Characterizing the Expressive Power of Invariant and Equivariant Graph Neural Networks
    arXiv.cs.DM Pub Date : 2020-06-28
    Waïss Azizian; Marc Lelarge

    Various classes of Graph Neural Networks (GNN) have been proposed and shown to be successful in a wide range of applications with graph structured data. In this paper, we propose a theoretical framework able to compare the expressive power of these GNN architectures. The current universality theorems only apply to intractable classes of GNNs. Here, we prove the first approximation guarantees for practical

    更新日期:2020-06-30
  • Reconstructing Biological and Digital Phylogenetic Trees in Parallel
    arXiv.cs.DM Pub Date : 2020-06-27
    Ramtin Afshar; Michael T. Goodrich; Pedro Matias; Martha C. Osegueda

    In this paper, we study the parallel query complexity of reconstructing biological and digital phylogenetic trees from simple queries involving their nodes. This is motivated from computational biology, data protection, and computer security settings, which can be abstracted in terms of two parties, a \emph{responder}, Alice, who must correctly answer queries of a given type regarding a degree-$d$

    更新日期:2020-06-30
  • Dissecting Power of a Finite Intersection of Deterministic Context Free Languages
    arXiv.cs.DM Pub Date : 2020-06-26
    Josef Rukavicka

    Let $\exp^{k,\alpha}$ denote a tetration function defined as follows: $\exp^{1,\alpha}=2^{\alpha}$ and $\exp^{k+1,\alpha}=2^{\exp^{k,\alpha}}$, where $k,\alpha$ are positive integers. Let $\Delta_n$ denote an alphabet with $n$ letters. If $L\subseteq\Delta_n^*$ is an infinite language such that for each $u\in L$ there is $v\in L$ with $\vert v\vert\leq \exp^{k,\alpha}\vert u\vert$ then we call $L$

    更新日期:2020-06-30
  • Cutting Polygons into Small Pieces with Chords: Laser-Based Localization
    arXiv.cs.DM Pub Date : 2020-06-26
    Esther M. Arkin; Rathish Das; Jie Gao; Mayank Goswami; Joseph S. B. Mitchell; Valentin Polishchuk; Csaba D. Toth

    Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Several versions are considered, depending on the definition of the "size" of a piece. In particular, we consider the area, the diameter, and the radius of the largest inscribed circle as a measure of the size of a piece. We also consider different

    更新日期:2020-06-29
  • An even better Density Increment Theorem and its application to Hadwiger's Conjecture
    arXiv.cs.DM Pub Date : 2020-06-25
    Luke Postle

    In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Recently, Norin, Song and the author showed that every graph with no $K_t$ minor is $O(t(\log t)^{\beta})$-colorable

    更新日期:2020-06-29
  • First-Order Model-Checking in Random Graphs and Complex Networks
    arXiv.cs.DM Pub Date : 2020-06-25
    Jan Dreier; Philipp Kuinke; Peter Rossmanith

    Complex networks are everywhere. They appear for example in the form of biological networks, social networks, or computer networks and have been studied extensively. Efficient algorithms to solve problems on complex networks play a central role in today's society. Algorithmic meta-theorems show that many problems can be solved efficiently. Since logic is a powerful tool to model problems, it has been

    更新日期:2020-06-26
  • Density of Binary Disc Packings: Playing with Stoichiometry
    arXiv.cs.DM Pub Date : 2020-06-25
    Thomas Fernique

    We consider the packings in the plane of discs of radius $1$ and $\sqrt{2}-1$ when the proportions of each type of disc are fixed. The maximal density is determined and the densest packings are described. In particular, it is shown that a phase separation phenomenon appears when there is an excess of small discs.

    更新日期:2020-06-26
  • Lifted Disjoint Paths with Application in Multiple Object Tracking
    arXiv.cs.DM Pub Date : 2020-06-25
    Andrea Hornakova; Roberto Henschel; Bodo Rosenhahn; Paul Swoboda

    We present an extension to the disjoint paths problem in which additional \emph{lifted} edges are introduced to provide path connectivity priors. We call the resulting optimization problem the lifted disjoint paths problem. We show that this problem is NP-hard by reduction from integer multicommodity flow and 3-SAT. To enable practical global optimization, we propose several classes of linear inequalities

    更新日期:2020-06-26
  • The variation of the sum of edge lengths in linear arrangements of trees
    arXiv.cs.DM Pub Date : 2020-06-24
    Ramon Ferrer-i-Cancho; Carlos Gómez-Rodríguez; Juan Luis Esteban

    A fundamental problem in network science is the normalization of the topological or physical distance between vertices, that requires understanding the range of variation of the unnormalized distances. Here we investigate the limits of the variation of the physical distance in linear arrangements of the vertices of trees. In particular, we investigate various problems on the sum of edge lengths in

    更新日期:2020-06-26
  • Reconfiguration of Spanning Trees with Many or Few Leaves
    arXiv.cs.DM Pub Date : 2020-06-25
    Nicolas Bousquet; Takehiro Ito; Yusuke Kobayashi; Haruka Mizuta; Paul Ouvrard; Akira Suzuki; Kunihiro Wasa

    Let $G$ be a graph and $T_1,T_2$ be two spanning trees of $G$. We say that $T_1$ can be transformed into $T_2$ via an edge flip if there exist two edges $e \in T_1$ and $f$ in $T_2$ such that $T_2= (T_1 \setminus e) \cup f$. Since spanning trees form a matroid, one can indeed transform a spanning tree into any other via a sequence of edge flips, as observed by Ito et al. We investigate the problem

    更新日期:2020-06-26
  • The Spectrum of the Singular Values of Z-Shaped Graph Matrices
    arXiv.cs.DM Pub Date : 2020-06-25
    Wenjun Cai; Aaron Potechin

    Graph matrices are a type of matrix which appears when analyzing the sum of squares hierarchy and other methods using higher moments. However, except for rough norm bounds, little is known about graph matrices. In this paper, we take a step towards better understanding graph matrices by determining the spectrum of the singular values of Z-shaped graph matrices.

    更新日期:2020-06-26
  • Discrepancy Minimization via a Self-Balancing Walk
    arXiv.cs.DM Pub Date : 2020-06-24
    Ryan Alweiss; Yang P. Liu; Mehtaab Sawhney

    We study discrepancy minimization for vectors in $\mathbb{R}^n$ under various settings. The main result is the analysis of a new simple random process in multiple dimensions through a comparison argument. As corollaries, we obtain bounds which are tight up to logarithmic factors for several problems in online vector balancing posed by Bansal, Jiang, Singla, and Sinha (STOC 2020), as well as linear

    更新日期:2020-06-26
  • Guarding Quadrangulations and Stacked Triangulations with Edges
    arXiv.cs.DM Pub Date : 2020-06-24
    Paul Jungeblut; Torsten Ueckerdt

    Let $G = (V,E)$ be a plane graph. A face $f$ of $G$ is guarded by an edge $vw \in E$ if at least one vertex from $\{v,w\}$ is on the boundary of $f$. For a planar graph class $\mathcal{G}$ we ask for the minimal number of edges needed to guard all faces of any $n$-vertex graph in $\mathcal{G}$. We prove that $\lfloor n/3 \rfloor$ edges are always sufficient for quadrangulations and give a construction

    更新日期:2020-06-25
  • Lyndon Words, the Three Squares Lemma, and Primitive Squares
    arXiv.cs.DM Pub Date : 2020-06-24
    Hideo Bannai; Takuya Mieno; Yuto Nakashima

    We revisit the so-called "Three Squares Lemma" by Crochemore and Rytter [Algorithmica 1995] and, using arguments based on Lyndon words, derive a more general variant which considers three overlapping squares which do not necessarily share a common prefix. We also give an improved upper bound of $n\log_2 n$ on the maximum number of (occurrences of) primitively rooted squares in a string of length $n$

    更新日期:2020-06-25
  • A Parameterized Family of Meta-Submodular Functions
    arXiv.cs.DM Pub Date : 2020-06-23
    Mehrdad Ghadiri; Richard Santiago; Bruce Shepherd

    Submodular function maximization has found a wealth of new applications in machine learning models during the past years. The related supermodular maximization models (submodular minimization) also offer an abundance of applications, but they appeared to be highly intractable even under simple cardinality constraints. Hence, while there are well-developed tools for maximizing a submodular function

    更新日期:2020-06-25
  • Relationship of Two Discrete Dynamical Models: One-dimensional Cellular Automata and Integral Value Transformations
    arXiv.cs.DM Pub Date : 2020-06-24
    Sreeya Ghosh; Sudhakar Sahoo; Sk. Sarif Hassan; Jayanta Kumar Das; Pabitra Pal Choudhury

    Cellular Automaton (CA) and an Integral Value Transformation (IVT) are two well established mathematical models which evolve in discrete time steps. Theoretically, studies on CA suggest that CA is capable of producing a great variety of evolution patterns. However computation of non-linear CA or higher dimensional CA maybe complex, whereas IVTs can be manipulated easily. The main purpose of this paper

    更新日期:2020-06-25
  • Discrete correlations of order 2 of generalised Rudin-Shapiro sequences: a combinatorial approach
    arXiv.cs.DM Pub Date : 2020-06-23
    Irène Marcovici; Thomas Stoll; Pierre-Adrien Tahay

    We introduce a family of block-additive automatic sequences, that are obtained by allocating a weight to each couple of digits, and defining the $n$th term of the sequence as being the total weight of the integer $n$ written in base $k$. Under an additional difference condition on the weight function, these sequences can be interpreted as generalised Rudin-Shapiro sequences, and we prove that they

    更新日期:2020-06-24
  • On the flip graphs on perfect matchings of complete graphs and signed reversal graphs
    arXiv.cs.DM Pub Date : 2020-06-23
    Sebastian M. Cioabă; Gordon Royle; Zhao Kuang Tan

    In this paper, we study the flip graph on the perfect matchings of a complete graph of even order. We investigate its combinatorial and spectral properties including connections to the signed reversal graph and we improve a previous upper bound on its chromatic number.

    更新日期:2020-06-24
  • A survey of repositories in graph theory
    arXiv.cs.DM Pub Date : 2020-06-23
    Srinibas Swain; C. Paul Bonnington; Graham Farr; Kerri Morgan

    Since the pioneering work of R. M. Foster in the 1930s, many graph repositories have been created to support research in graph theory. This survey reviews many of these graph repositories and summarises the scope and contents of each repository. We identify opportunities for the development of repositories that can be queried in more flexible ways.

    更新日期:2020-06-24
  • Better approximation algorithms for maximum weight internal spanning trees in cubic graphs and claw-free graphs
    arXiv.cs.DM Pub Date : 2020-06-22
    Ahmad Biniaz

    Given a connected vertex-weighted graph $G$, the maximum weight internal spanning tree (MaxwIST) problem asks for a spanning tree of $G$ that maximizes the total weight of internal nodes. This problem is NP-hard and APX-hard, with the currently best known approximation factor $1/2$ (Chen et al., Algorithmica 2019). For the case of claw-free graphs, Chen et al. present an involved approximation algorithm

    更新日期:2020-06-24
  • Sampling hypergraphs with given degrees
    arXiv.cs.DM Pub Date : 2020-06-22
    Martin Dyer; Catherine Greenhill; Pieter Kleer; James Ross; Leen Stougie

    There is a well-known connection between hypergraphs and bipartite graphs, obtained by treating the incidence matrix of the hypergraph as the biadjacency matrix of a bipartite graph. We use this connection to describe and analyse a rejection sampling algorithm for sampling simple uniform hypergraphs with a given degree sequence. Our algorithm uses, as a black box, an algorithm $\mathcal{A}$ for sampling

    更新日期:2020-06-23
  • Hierarchical Decompositions of dihypergraphs
    arXiv.cs.DM Pub Date : 2020-06-21
    Lhouari Nourine; Simon Vilmin

    In this paper we are interested in decomposing a dihypergraph $\mathcal{H} = (V, \mathcal{E})$ into simpler dihypergraphs, that can be handled more efficiently. We study the properties of dihypergraphs that can be hierarchically decomposed into trivial dihypergraphs, \ie vertex hypergraph. The hierarchical decomposition is represented by a full labelled binary tree called $\mathcal{H}$-tree, in the

    更新日期:2020-06-23
  • Improved Bounds for Metric Capacitated Covering Problems
    arXiv.cs.DM Pub Date : 2020-06-22
    Sayan Bandyapadhyay

    In the Metric Capacitated Covering (MCC) problem, given a set of balls $\mathcal{B}$ in a metric space $P$ with metric $d$ and a capacity parameter $U$, the goal is to find a minimum sized subset $\mathcal{B}'\subseteq \mathcal{B}$ and an assignment of the points in $P$ to the balls in $\mathcal{B}'$ such that each point is assigned to a ball that contains it and each ball is assigned with at most

    更新日期:2020-06-23
  • On morphisms preserving palindromic richness
    arXiv.cs.DM Pub Date : 2020-06-22
    Francesco Dolce; Edita Pelantová

    Droubay, Justin and Pirillo that each word of length $n$ contains at most $n+1$ distinct palindromes. A finite "rich word" is a word with maximal number of palindromic factors. The definition of palindromic richness can be naturally extended to infinite words. Sturmian words and Rote complementary symmetric sequences form two classes of binary rich words, while episturmian words and words coding $d$-interval

    更新日期:2020-06-23
  • Further progress towards Hadwiger's conjecture
    arXiv.cs.DM Pub Date : 2020-06-21
    Luke Postle

    In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Recently, Norin, Song and the author showed that every graph with no $K_t$ minor is $O(t(\log t)^{\beta})$-colorable

    更新日期:2020-06-23
  • Multicritera Cuts and Size-Constrained $k$-cuts in Hypergraphs
    arXiv.cs.DM Pub Date : 2020-06-20
    Calvin Beideman; Karthekeyan Chandrasekaran; Chao Xu

    We address counting and optimization variants of multicriteria global min-cut and size-constrained min-$k$-cut in hypergraphs. 1. For an $r$-rank $n$-vertex hypergraph endowed with $t$ hyperedge-cost functions, we show that the number of multiobjective min-cuts is $O(r2^{tr}n^{3t-1})$. In particular, this shows that the number of parametric min-cuts in constant rank hypergraphs for a constant number

    更新日期:2020-06-23
  • Tight Bounds on The Clique Chromatic Number
    arXiv.cs.DM Pub Date : 2020-06-19
    Gwenaël Joret; Piotr Micek; Bruce Reed; Michiel Smid

    The clique chromatic number of a graph is the minimum number of colours needed to colour its vertices so that no inclusion-wise maximal clique which is not an isolated vertex is monochromatic. We show that every graph of maximum degree $\Delta$ has clique chromatic number $O\left(\frac{\Delta}{\log~\Delta}\right)$. We obtain as a corollary that every $n$-vertex graph has clique chromatic number $O

    更新日期:2020-06-23
  • An Alon-Boppana theorem for powered graphs and generalized Ramanujan graphs
    arXiv.cs.DM Pub Date : 2020-06-18
    Emmanuel Abbe; Peter Ralli

    The r-th power of a graph modifies a graph by connecting every vertex pair within distance r. This paper gives a generalization of the Alon-Boppana Theorem for the r-th power of graphs, including irregular graphs. This leads to a generalized notion of Ramanujan graphs, those for which the powered graph has a spectral gap matching the derived Alon-Boppana bound. In particular, we show that certain graphs

    更新日期:2020-06-22
  • Robustness of Pisot-regular sequences
    arXiv.cs.DM Pub Date : 2020-06-19
    Émilie Charlier; Célia Cisternino; Manon Stipulanti

    We consider numeration systems based on a $d$-tuple $\mathbf{U}=(U_1,\ldots,U_d)$ of sequences of integers and we define $(\mathbf{U},\mathbb{K})$-regular sequences through $\mathbb{K}$-recognizable formal series, where $\mathbb{K}$ is any semiring. We show that, for any $d$-tuple $\mathbf{U}$ of Pisot numeration systems and any commutative semiring $\mathbb{K}$, this definition does not depend on

    更新日期:2020-06-22
  • A unified framework for equivalences in social networks
    arXiv.cs.DM Pub Date : 2020-06-18
    Nina Otter; Mason A. Porter

    A key concern in network analysis is the study of social positions and roles of actors in a network. The notion of "position" refers to an equivalence class of nodes that have similar ties to other nodes, whereas a "role" is an equivalence class of compound relations that connect the same pairs of nodes. An open question in network science is whether it is possible to simultaneously perform role and

    更新日期:2020-06-19
  • Random walks on randomly evolving graphs
    arXiv.cs.DM Pub Date : 2020-06-18
    Leran Cai; Thomas Sauerwald; Luca Zanetti

    A random walk is a basic stochastic process on graphs and a key primitive in the design of distributed algorithms. One of the most important features of random walks is that, under mild conditions, they converge to a stationary distribution in time that is at most polynomial in the size of the graph. This fundamental property, however, only holds if the graph does not change over time, while on the

    更新日期:2020-06-19
  • Palette Sparsification Beyond $(Δ+1)$ Vertex Coloring
    arXiv.cs.DM Pub Date : 2020-06-18
    Noga Alon; Sepehr Assadi

    A recent palette sparsification theorem of Assadi, Chen, and Khanna [SODA'19] states that in every $n$-vertex graph $G$ with maximum degree $\Delta$, sampling $O(\log{n})$ colors per each vertex independently from $\Delta+1$ colors almost certainly allows for proper coloring of $G$ from the sampled colors. Besides being a combinatorial statement of its own independent interest, this theorem was shown

    更新日期:2020-06-19
  • Train Unit Shunting and Servicing: a Real-Life Application of Multi-Agent Path Finding
    arXiv.cs.DM Pub Date : 2020-06-18
    Jesse MulderijDelft University of Technology; Bob HuismanNederlandse Spoorwegen; Denise TönissenVrije Universiteit Amsterdam; Koos van der LindenDelft University of Technology; Mathijs de WeerdtDelft University of Technology

    In between transportation services, trains are parked and maintained at shunting yards. The conflict-free routing of trains to and on these yards and the scheduling of service and maintenance tasks is known as the train unit shunting and service problem. Efficient use of the capacity of these yards is becoming increasingly important, because of increasing numbers of trains without proportional extensions

    更新日期:2020-06-19
  • A 3/2--approximation for big two-bar charts packing
    arXiv.cs.DM Pub Date : 2020-06-18
    Adil Erzin; Stepan Nazarenko; Gregory Melidi; Roman Plotnikov

    We consider a Two-Bar Charts Packing Problem (2-BCPP), in which it is necessary to pack two-bar charts (2-BCs) in a unit-height strip of minimum length. The problem is a generalization of the Bin Packing Problem (BPP). Earlier, we proposed an $O(n^2)$-time algorithm that constructs the packing which length at most $2\cdot OPT+1$, where $OPT$ is the minimum length of the packing of $n$ 2-BCs. In this

    更新日期:2020-06-19
  • Extremal overlap-free and extremal $β$-free binary words
    arXiv.cs.DM Pub Date : 2020-06-17
    Lucas Mol; Narad Rampersad; Jeffrey Shallit

    An overlap-free (or $\beta$-free) word $w$ over a fixed alphabet $\Sigma$ is extremal if every word obtained from $w$ by inserting a single letter from $\Sigma$ at any position contains an overlap (or a factor of exponent at least $\beta$, respectively). We find all lengths which admit an extremal overlap-free binary word. For every extended real number $\beta$ such that $2^+\leq\beta\leq 8/3$, we

    更新日期:2020-06-19
  • Twin-width II: small classes
    arXiv.cs.DM Pub Date : 2020-06-17
    Édouard Bonnet; Colin Geniet; Eun Jung Kim; Stéphan Thomassé; Rémi Watrigant

    The twin-width of a graph $G$ is the minimum integer $d$ such that $G$ has a $d$-contraction sequence, that is, a sequence of $|V(G)|-1$ iterated vertex identifications for which the overall maximum number of red edges incident to a single vertex is at most $d$, where a red edge appears between two sets of identified vertices if they are not homogeneous in $G$. We show that if a graph admits a $d$-contraction

    更新日期:2020-06-18
  • Bute: A Bottom-Up Exact Solver for Treedepth (Submitted to PACE 2020 under username peaty)
    arXiv.cs.DM Pub Date : 2020-06-17
    James Trimble

    This note introduces the exact solver Bute for the exact treedepth problem, along with two variants of the solver. Each of these solvers computes an elimination tree in a bottom-up fashion by finding sets of vertices that induce subgraphs of small treedepth, then combining sets of vertices together with a root vertex to produce larger sets. The algorithms make use of a trie data structure to reduce

    更新日期:2020-06-18
  • Scheduling a Proportionate Flow Shop of Batching Machines
    arXiv.cs.DM Pub Date : 2020-06-17
    Christoph Hertrich; Christian Weiß; Heiner Ackermann; Sandy Heydrich; Sven O. Krumke

    In this paper we study a proportionate flow shop of batching machines with release dates and a fixed number $m \geq 2$ of machines. The scheduling problem has so far barely received any attention in the literature, but recently its importance has increased significantly, due to applications in the industrial scaling of modern bio-medicine production processes. We show that for any fixed number of machines

    更新日期:2020-06-18
  • The No-Flippancy Game
    arXiv.cs.DM Pub Date : 2020-06-17
    Isha Agarwal; Matvey Borodin; Aidan Duncan; Kaylee Ji; Tanya Khovanova; Shane Lee; Boyan Litchev; Anshul Rastogi; Garima Rastogi; Andrew Zhao

    We analyze a coin-based game with two players where, before starting the game, each player selects a string of length $n$ comprised of coin tosses. They alternate turns, choosing the outcome of a coin toss according to specific rules. As a result, the game is deterministic. The player whose string appears first wins. If neither player's string occurs, then the game must be infinite. We study several

    更新日期:2020-06-18
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