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  • A Spectral Approach to the Shortest Path Problem
    arXiv.cs.DM Pub Date : 2020-04-02
    Stefan Steinerberger

    Let $G=(V,E)$ be a simple, connected graph. One is often interested in a short path between two vertices $u,v$. We propose a spectral algorithm: construct the function $\phi:V \rightarrow \mathbb{R}_{\geq 0}$ $$ \phi = \arg\min_{f:V \rightarrow \mathbb{R} \atop f(u) = 0, f \not\equiv 0} \frac{\sum_{(w_1, w_2) \in E}{(f(w_1)-f(w_2))^2}}{\sum_{w \in V}{f(w)^2}}.$$ $\phi$ can also be understood as the

    更新日期:2020-04-03
  • Scheduling Parallel-Task Jobs Subject to Packing and Placement Constraints
    arXiv.cs.DM Pub Date : 2020-04-01
    Mehrnoosh Shafiee; Javad Ghaderi

    Motivated by modern parallel computing applications, we consider the problem of scheduling parallel-task jobs with heterogeneous resource requirements in a cluster of machines. Each job consists of a set of tasks that can be processed in parallel, however, the job is considered completed only when all its tasks finish their processing, which we refer to as "synchronization" constraint. Further, assignment

    更新日期:2020-04-03
  • Subgraph densities in a surface
    arXiv.cs.DM Pub Date : 2020-03-30
    Tony Huynh; Gwenaël Joret; David R. Wood

    Given a fixed graph $H$ that embeds in a surface $\Sigma$, what is the maximum number of copies of $H$ in an $n$-vertex graph $G$ that embeds in $\Sigma$? We show that the answer is $\Theta(n^{f(H)})$, where $f(H)$ is a graph invariant called the `flap-number' of $H$, which is independent of $\Sigma$. This simultaneously answers two open problems posed by Eppstein (1993). When $H$ is a complete graph

    更新日期:2020-04-01
  • A formula on the weight distribution of linear codes with applications to AMDS codes
    arXiv.cs.DM Pub Date : 2020-03-31
    Alessio Meneghetti; Marco Pellegrini; Massimiliano Sala

    The determination of the weight distribution of linear codes has been a fascinating problem since the very beginning of coding theory. There has been a lot of research on weight enumerators of special cases, such as self-dual codes and codes with small Singleton's defect. We propose a new set of linear relations that must be satisfied by the coefficients of the weight distribution. From these relations

    更新日期:2020-04-01
  • VC density of set systems defnable in tree-like graphs
    arXiv.cs.DM Pub Date : 2020-03-31
    Adam Paszke; Michał Pilipczuk

    We study set systems definable in graphs using variants of logic with different expressive power. Our focus is on the notion of Vapnik-Chervonenkis density: the smallest possible degree of a polynomial bounding the cardinalities of restrictions of such set systems. On one hand, we prove that if $\varphi(\bar x,\bar y)$ is a fixed CMSO$_1$ formula and $\cal C$ is a class of graphs with uniformly bounded

    更新日期:2020-04-01
  • DAG-width is PSPACE-complete
    arXiv.cs.DM Pub Date : 2014-11-10
    Saeed Akhoondian Amiri; Stephan Kreutzer; Roman Rabinovich

    Berwanger et al. show that for every graph $G$ of size $n$ and DAG-width $k$ there is a DAG decomposition of width $k$ and size $n^{O(k)}$. This gives a polynomial time algorithm for determining the DAG-width of a graph for any fixed $k$. However, if the DAG-width of the graphs from a class is not bounded, such algorithms become exponential. This raises the question whether we can always find a DAG

    更新日期:2020-04-01
  • Algorithms for weighted independent transversals and strong colouring
    arXiv.cs.DM Pub Date : 2019-06-28
    Alessandra Graf; David G. Harris; Penny Haxell

    An independent transversal (IT) in a graph with a given vertex partition is an independent set consisting of one vertex in each partition class. Several sufficient conditions are known for the existence of an IT in a given graph with a given vertex partition, which have been used over the years to solve many combinatorial problems. Some of these IT existence theorems have algorithmic proofs, but there

    更新日期:2020-04-01
  • Duality pairs and homomorphisms to oriented and unoriented cycles
    arXiv.cs.DM Pub Date : 2020-03-12
    Santiago Guzmán-Pro; César Hernández-Cruz

    In the homomorphism order of digraphs, a duality pair is an ordered pair of digraphs $(G,H)$ such that for any digraph, $D$, $G\to D$ if and only if $D\not\to H$. The directed path on $k+1$ vertices together with the transitive tournament on $k$ vertices is a classic example of a duality pair. This relation between paths and tournaments implies that a graph is $k$-colourable if and only if it admits

    更新日期:2020-04-01
  • word2vec, node2vec, graph2vec, X2vec: Towards a Theory of Vector Embeddings of Structured Data
    arXiv.cs.DM Pub Date : 2020-03-27
    Martin Grohe

    Vector representations of graphs and relational structures, whether hand-crafted feature vectors or learned representations, enable us to apply standard data analysis and machine learning techniques to the structures. A wide range of methods for generating such embeddings have been studied in the machine learning and knowledge representation literature. However, vector embeddings have received relatively

    更新日期:2020-03-31
  • The impossibility of low rank representations for triangle-rich complex networks
    arXiv.cs.DM Pub Date : 2020-03-27
    C. Seshadhri; Aneesh Sharma; Andrew Stolman; Ashish Goel

    The study of complex networks is a significant development in modern science, and has enriched the social sciences, biology, physics, and computer science. Models and algorithms for such networks are pervasive in our society, and impact human behavior via social networks, search engines, and recommender systems to name a few. A widely used algorithmic technique for modeling such complex networks is

    更新日期:2020-03-31
  • Algorithmic Counting of Zero-Dimensional Finite Topological Spaces With Respect to the Covering Dimension
    arXiv.cs.DM Pub Date : 2020-03-28
    Rudolf Berghammer; Steffen Börm; Michael Winter

    Taking the covering dimension dim as notion for the dimension of a topological space, we first specify thenumber zdim_{T_0}(n) of zero-dimensional T_0-spaces on {1,...,n}$ and the number zdim(n) of zero-dimensional arbitrary topological spaces on {1,\ldots,n} by means oftwo mappings po and P that yieldthe number po(n) of partial orders on {1,...,n} and the set P(n) of partitions of {1,...,n}, respectively

    更新日期:2020-03-31
  • Periodicity of lively quantum walks on cycles with generalized Grover coin
    arXiv.cs.DM Pub Date : 2020-03-29
    Rohit Sarma Sarkar; Amrita Mandal; Bibhas Adhikari

    In this paper we extend the study of three state lively quantum walks on cycles by considering the coin operator as a linear sum of permutation matrices, which is a generalization of the Grover matrix. First we provide a complete characterization of orthogonal matrices of order $3\times 3$ which are linear sum of permutation matrices. Consequently, we determine several groups of complex, real and rational

    更新日期:2020-03-31
  • Graphical Games and Decomposition
    arXiv.cs.DM Pub Date : 2020-03-29
    Laura Arditti; Giacomo Como; Fabio Fagnani

    We consider graphical games as introduced by Kearns et al. (2001). First we analyse the interaction of graphicality with a notion of strategic equivalence of games, providing a minimal complexity graphical description for games. Then we study the interplay between graphicality and the classical decomposition of games proposed by Candogan et al. (2011), characterizing the graphical properties of each

    更新日期:2020-03-31
  • Long Alternating Paths Exist
    arXiv.cs.DM Pub Date : 2020-03-30
    Wolfgang Mulzer; Pavel Valtr

    Let $P$ be a set of $2n$ points in convex position, such that $n$ points are colored red and $n$ points are colored blue. A non-crossing alternating path on $P$ of length $\ell$ is a sequence $p_1, \dots, p_\ell$ of $\ell$ points from $P$ so that (i) all points are pairwise distinct; (ii) any two consecutive points $p_i$, $p_{i+1}$ have different colors; and (iii) any two segments $p_i p_{i+1}$ and

    更新日期:2020-03-31
  • Separable games
    arXiv.cs.DM Pub Date : 2020-03-29
    Laura Arditti; Giacomo Como; Fabio Fagnani

    We introduce the notion of separable games, which refines and generalizes that of graphical games. We prove that there exists a minimal splitting with respect to which a game is separable. Moreover we prove that in every strategic equivalence class, there is a game separable with respect to the minimal splitting in the class. This game is also graphical with respect to the smallest graph in the class

    更新日期:2020-03-31
  • The One-way Communication Complexity of Submodular Maximization with Applications to Streaming and Robustness
    arXiv.cs.DM Pub Date : 2020-03-30
    Moran Feldman; Ashkan Norouzi-Fard; Ola Svensson; Rico Zenklusen

    We consider the classical problem of maximizing a monotone submodular function subject to a cardinality constraint, which, due to its numerous applications, has recently been studied in various computational models. We consider a clean multi-player model that lies between the offline and streaming model, and study it under the aspect of one-way communication complexity. Our model captures the streaming

    更新日期:2020-03-31
  • Hyperplane arrangements in polymake
    arXiv.cs.DM Pub Date : 2020-03-30
    Lars Kastner; Marta Panizzut

    Hyperplane arrangements form the latest addition to the zoo of combinatorial objects dealt with by polymake. We report on their implementation and on a algorithm to compute the associated cell decomposition. The implemented algorithm performs significantly better than brute force alternatives, as it requires less convex hulls computations.

    更新日期:2020-03-31
  • On different Versions of the Exact Subgraph Hierarchy for the Stable Set Problem
    arXiv.cs.DM Pub Date : 2020-03-30
    Elisabeth Gaar

    One of many different hierarchies towards the stability number of a graph is the exact subgraph hierarchy (ESH). On the first level it starts to compute the Lov\'{a}sz theta function as a semidefinite program (SDP) with a matrix variable of order $n+1$ and $n+m+1$ constraints, where $n$ is the number of vertices and $m$ is the number of edges of a graph $G$. On the $k$-th level of the ESH it adds all

    更新日期:2020-03-31
  • On the probability of being synchronizable
    arXiv.cs.DM Pub Date : 2013-04-21
    Mikhail V. Berlinkov

    We prove that a random automaton with $n$ states and any fixed non-singleton alphabet is synchronizing with high probability. Moreover, we also prove that the convergence rate is exactly $1-\Theta(\frac{1}{n})$ as conjectured by Cameron \cite{CamConj} for the most interesting binary alphabet case. Finally, we describe a deterministic algorithm which decides whether a given random automaton is synchronizing

    更新日期:2020-03-31
  • Ramsey properties and extending partial automorphisms for classes of finite structures
    arXiv.cs.DM Pub Date : 2017-05-05
    David M. Evans; Jan Hubička; Jaroslav Nešetřil

    We show that every free amalgamation class of finite structures with relations and (symmetric) partial functions is a Ramsey class when enriched by a free linear ordering of vertices. This is a common strengthening of the Ne\v{s}et\v{r}il-R\"odl Theorem and the second and third authors' Ramsey theorem for finite models (that is, structures with both relations and functions). We also find subclasses

    更新日期:2020-03-31
  • A Polyhedral Approach to Bisubmodular Function Minimization
    arXiv.cs.DM Pub Date : 2020-03-12
    Qimeng Yu; Simge Kucukyavuz

    We consider minimization problems with bisubmodular objective functions. We propose a class of valid inequalities, which we call the poly-bimatroid inequalities and prove that these inequalities, along with trivial bound constraints, fully describe the convex hull of the epigraph of a bisubmodular function. We develop a cutting plane algorithm for general bisubmodular minimization problems using the

    更新日期:2020-03-31
  • A squarefree term not occurring in the Leech sequence
    arXiv.cs.DM Pub Date : 2020-03-27
    Benjamin Wells

    Let \[ \begin{array}{c}\overline{A} = ABCBA\ CBC\ ABCBA,\\ \overline{B} = BCACB\ ACA\ BCACB,\\ \overline{C} = CABAC\ BAB\ CABAC. \end{array} \] The Leech sequence $L$ is the squarefree sequence obtained as the limit of the palindromes \[ A, \overline{A}, \overline{\overline{A}}, \ldots . \] In order to specify a certain class of pseudorecursive varieties of semigroups, it is helpful to have a squarefree

    更新日期:2020-03-30
  • Covering minimal separators and potential maximal cliques in $P_t$-free graphs
    arXiv.cs.DM Pub Date : 2020-03-27
    Andrzej Grzesik; Tereza Klimošová; Marcin Pilipczuk; Michał Pilipczuk

    A graph is called $P_t$-free} if it does not contain a $t$-vertex path as an induced subgraph. While $P_4$-free graphs are exactly cographs, the structure of $P_t$-free graphs for $t \geq 5$ remains little understood. On one hand, classic computational problems such as Maximum Weight Independent Set (MWIS) and $3$-Coloring are not known to be NP-hard on $P_t$-free graphs for any fixed $t$. On the other

    更新日期:2020-03-30
  • Tight Bounds on the Coeffcients of Consecutive $k$-out-of-$n$:$F$ Systems
    arXiv.cs.DM Pub Date : 2020-03-27
    Vlad-Florin Drăgoi; Simon R. Cowell; Valeriu Beiu

    In this paper we compute the coefficients of the reliability polynomial of a consecutive-$k$-out-of-$n$:$F$ system, in Bernstein basis, using the generalized Pascal coefficients. Based on well-known combinatorial properties of the generalized Pascal triangle we determine simple closed formulae for the reliability polynomial of a consecutive system for particular ranges of $k$. Moreover, for the remaining

    更新日期:2020-03-30
  • On $\underline{12}0$-avoiding inversion and ascent sequences
    arXiv.cs.DM Pub Date : 2020-03-26
    Zhicong Lin; Shishuo Fu

    Recently, Yan and the first named author investigated systematically the enumeration of inversion or ascent sequences avoiding vincular patterns of length $3$, where two of the three letters are required to be adjacent. They established many connections with familiar combinatorial families and proposed several interesting conjectures. The objective of this paper is to address two of their conjectures

    更新日期:2020-03-30
  • From Information Theory Puzzles in Deletion Channels to Deniability in Quantum Cryptography
    arXiv.cs.DM Pub Date : 2020-03-25
    Arash Atashpendar

    From the output produced by a memoryless deletion channel with a uniformly random input of known length $n$, one obtains a posterior distribution on the channel input. The difference between the Shannon entropy of this distribution and that of the uniform prior measures the amount of information about the channel input which is conveyed by the output of length $m$. We first conjecture on the basis

    更新日期:2020-03-27
  • Explicit expanders of every degree and size
    arXiv.cs.DM Pub Date : 2020-03-25
    Noga Alon

    An $(n,d,\lambda)$-graph is a $d$ regular graph on $n$ vertices in which the absolute value of any nontrivial eigenvalue is at most $\lambda$. For any constant $d \geq 3$, $\epsilon>0$ and all sufficiently large $n$ we show that there is a deterministic poly(n) time algorithm that outputs an $(n,d, \lambda)$-graph (on exactly $n$ vertices) with $\lambda \leq 2 \sqrt{d-1}+\epsilon$. For any $d=p+2$

    更新日期:2020-03-27
  • Regular partitions of gentle graphs
    arXiv.cs.DM Pub Date : 2020-03-26
    Yiting Jiang; Jaroslav Nesetril; Patrice Ossona de Mendez; Sebastian Siebertz

    Szemeredi's Regularity Lemma is a very useful tool of extremal combinatorics. Recently, several refinements of this seminal result were obtained for special, more structured classes of graphs. We survey these results in their rich combinatorial context. In particular, we stress the link to the theory of (structural) sparsity, which leads to alternative proofs, refinements and solutions of open problems

    更新日期:2020-03-27
  • Hypernetwork Science: From Multidimensional Networks to Computational Topology
    arXiv.cs.DM Pub Date : 2020-03-26
    Cliff A. Joslyn; Sinan Aksoy; Tiffany J. Callahan; Lawrence E. Hunter; Brett Jefferson; Brenda Praggastis; Emilie A. H. Purvine; Ignacio J. Tripodi

    As data structures and mathematical objects used for complex systems modeling, hypergraphs sit nicely poised between on the one hand the world of network models, and on the other that of higher-order mathematical abstractions from algebra, lattice theory, and topology. They are able to represent complex systems interactions more faithfully than graphs and networks, while also being some of the simplest

    更新日期:2020-03-27
  • A Topological Characterization of Modulo-p Arguments and Implications for Necklace Splitting
    arXiv.cs.DM Pub Date : 2020-03-26
    Aris Filos-Ratsikas; Alexandros Hollender; Katerina Sotiraki; Manolis Zampetakis

    The classes PPA-$p$ have attracted attention lately, because they are the main candidates for capturing the complexity of Necklace Splitting with $p$ thieves, for prime $p$. However, these classes are not known to have complete problems of a topological nature, which impedes any progress towards settling the complexity of the problem. On the contrary, such problems have been pivotal in obtaining completeness

    更新日期:2020-03-27
  • A Nonexistence Certificate for Projective Planes of Order Ten with Weight 15 Codewords
    arXiv.cs.DM Pub Date : 2019-11-11
    Curtis Bright; Kevin Cheung; Brett Stevens; Dominique Roy; Ilias Kotsireas; Vijay Ganesh

    Using techniques from the fields of symbolic computation and satisfiability checking we verify one of the cases used in the landmark result that projective planes of order ten do not exist. In particular, we show that there exist no projective planes of order ten that generate codewords of weight fifteen, a result first shown in 1973 via an exhaustive computer search. We provide a simple satisfiability

    更新日期:2020-03-27
  • Fair packing of independent sets
    arXiv.cs.DM Pub Date : 2020-03-25
    Nina Chiarelli; Matjaž Krnc; Martin Milanič; Ulrich Pferschy; Nevena Pivač; Joachim Schauer

    In this work we add a graph theoretical perspective to a classical problem of fairly allocating indivisible items to several agents. Agents have different profit valuations of items and we allow an incompatibility relation between pairs of items described in terms of a conflict graph. Hence, every feasible allocation of items to the agents corresponds to a partial coloring, that is, a collection of

    更新日期:2020-03-26
  • Topology and adjunction in promise constraint satisfaction
    arXiv.cs.DM Pub Date : 2020-03-25
    Andrei Krokhin; Jakub Opršal; Marcin Wrochna; Stanislav Živný

    The approximate graph colouring problem concerns colouring a $k$-colourable graph with $c$ colours, where $c\geq k$. This problem naturally generalises to promise graph homomorphism and further to promise constraint satisfaction problems. Complexity analysis of all these problems is notoriously difficult. In this paper, we introduce two new techniques to analyse the complexity of promise CSPs: one

    更新日期:2020-03-26
  • Improved lower bound on the dimension of the EU council's voting rules
    arXiv.cs.DM Pub Date : 2020-03-25
    Stefan Kober; Stefan Weltge

    Kurz and Napel (2015) proved that the voting system of the EU council (based on the 2014 population data) cannot be represented as the intersection of six weighted games, i.e., its dimension is at least 7. This set a new record for real-world voting rules and the authors posed the exact determination as a challenge. Recently, Chen, Cheung, and Ng (2019) showed that the dimension is at most 24. We provide

    更新日期:2020-03-26
  • Diseño e implementación de una meta-heurística multi-poblacional de optimización combinatoria enfocada a la resolución de problemas de asignación de rutas a vehículos
    arXiv.cs.DM Pub Date : 2020-03-24
    Eneko Osaba

    Transportation is an essential area in the nowadays society, both for business sector and citizenry. There are different kinds of transportation systems, each one with its own characteristics. In the same way, various areas of knowledge can deal efficiently with the transport planning. The majority of the problems related with the transport and logistics have common characteristics, so they can be

    更新日期:2020-03-26
  • Fast and resilient manipulation planning for target retrieval in clutter
    arXiv.cs.DM Pub Date : 2020-03-24
    Changjoo Nam; Jinhwi Lee; Sang Hun Cheong; Brian Y. Cho; ChangHwan Kim

    This paper presents a task and motion planning (TAMP) framework for a robotic manipulator in order to retrieve a target object from clutter. We consider a configuration of objects in a confined space with a high density so no collision-free path to the target exists. The robot must relocate some objects to retrieve the target without collisions. For fast completion of object rearrangement, the robot

    更新日期:2020-03-26
  • Cantor-solus and Cantor-multus Distributions
    arXiv.cs.DM Pub Date : 2020-03-20
    Steven Finch

    The Cantor distribution is obtained from bitstrings; the Cantor-solus distribution (a new name) admits only strings without adjacent 1 bits. We review moments and order statistics associated with these. The Cantor-multus distribution is introduced -- which instead admits only strings without isolated 1 bits -- and more complicated formulas emerge.

    更新日期:2020-03-24
  • An Online Framework to Interact and Efficiently Compute Linear Layouts of Graphs
    arXiv.cs.DM Pub Date : 2020-03-21
    Michael A. Bekos; Mirco Haug; Michael Kaufmann; Julia Männecke

    We present a prototype online system to automate the procedure of computing different types of linear layouts of graphs under different user-specific constraints. The system consists of two main components; the client and the server sides. The client side is built upon an easy-to-use editor, which supports basic interaction with graphs, enriched with several additional features to allow the user to

    更新日期:2020-03-24
  • A proof of the Total Coloring Conjecture
    arXiv.cs.DM Pub Date : 2020-03-21
    T Srinivasa Murthy

    A $total\ coloring$ of a graph G is a map $f:V(G) \cup E(G) \rightarrow \mathcal{K}$, where $\mathcal{K}$ is a set of colors, satisfying the following three conditions: 1. $f(u) \neq f(v)$ for any two adjacent vertices $u, v \in V(G)$; 2. $f(e) \neq f(e')$ for any two adjacent edges $e, e' \in E(G)$; and 3. $f(v) \neq f(e)$ for any vertex $v \in V(G)$ and any edge $e \in E(G)$ that is incident to same

    更新日期:2020-03-24
  • Spectral Clustering Revisited: Information Hidden in the Fiedler Vector
    arXiv.cs.DM Pub Date : 2020-03-22
    Adela DePavia; Stefan Steinerberger

    We are interested in the clustering problem on graphs: it is known that if there are two underlying clusters, then the signs of the eigenvector corresponding to the second largest eigenvalue of the adjacency matrix can reliably reconstruct the two clusters. We argue that the vertices for which the eigenvector has the largest and the smallest entries, respectively, are unusually strongly connected to

    更新日期:2020-03-24
  • On the diameter of the polytope of the stable marriage with ties
    arXiv.cs.DM Pub Date : 2020-03-23
    Felix Bauckholt; Laura Sanità

    The stable marriage problem with ties is a well-studied and interesting problem in game theory. We are given a set of men and a set of women. Each individual has a preference ordering on the opposite group, which can possibly contain ties. A stable marriage is given by a matching between men and women for which there is no blocking pair, i.e., a men and a women who strictly prefer each other to their

    更新日期:2020-03-24
  • Think Globally, Act Locally: On the Optimal Seeding for Nonsubmodular Influence Maximization
    arXiv.cs.DM Pub Date : 2020-03-23
    Grant Schoenebeck; Biaoshuai Tao; Fang-Yi Yu

    We study the $r$-complex contagion influence maximization problem. In the influence maximization problem, one chooses a fixed number of initial seeds in a social network to maximize the spread of their influence. In the $r$-complex contagion model, each uninfected vertex in the network becomes infected if it has at least $r$ infected neighbors. In this paper, we focus on a random graph model named

    更新日期:2020-03-24
  • Polynomial-time algorithm for Maximum Weight Independent Set on $P_6$-free graphs
    arXiv.cs.DM Pub Date : 2017-07-18
    Andrzej Grzesik; Tereza Klimošová; Marcin Pilipczuk; Michał Pilipczuk

    In the classic Maximum Weight Independent Set problem we are given a graph $G$ with a nonnegative weight function on vertices, and the goal is to find an independent set in $G$ of maximum possible weight. While the problem is NP-hard in general, we give a polynomial-time algorithm working on any $P_6$-free graph, that is, a graph that has no path on $6$ vertices as an induced subgraph. This improves

    更新日期:2020-03-24
  • Treewidth-Pliability and PTAS for Max-CSPs
    arXiv.cs.DM Pub Date : 2019-11-08
    Miguel Romero; Marcin Wrochna; Stanislav Živný

    We identify a sufficient condition, treewidth-pliability, that gives a polynomial-time approximation scheme (PTAS) for a large class of Max-2-CSPs parametrised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main

    更新日期:2020-03-24
  • Decreasing the maximum average degree by deleting an independent set or a d-degenerate subgraph
    arXiv.cs.DM Pub Date : 2019-09-24
    Wojciech Nadara; Marcin Smulewicz

    The maximum average degree $\mathrm{mad}(G)$ of a graph $G$ is the maximum average degree over all subgraphs of $G$. In this paper we prove that for every $G$ and positive integer $k$ such that $\mathrm{mad}(G) \ge k$ there exists $S \subseteq V(G)$ such that $\mathrm{mad}(G - S) \le \mathrm{mad}(G) - k$ and $G[S]$ is $(k-1)$-degenerate. Moreover, such $S$ can be computed in polynomial time. In particular

    更新日期:2020-03-24
  • Globally Optimal Selection of Ground Stations in Satellite Systems with Site Diversity
    arXiv.cs.DM Pub Date : 2019-12-30
    Christos N. Efrem; Athanasios D. Panagopoulos

    The availability of satellite communication systems is extremely limited by atmospheric impairments, such as rain (for radio frequencies) and cloud coverage (for optical frequencies). A solution to this problem is the site diversity technique, where a network of geographically distributed ground stations (GSs) can ensure, with high probability, that at least one GS is available for connection to the

    更新日期:2020-03-24
  • Mutants and Residents with Different Connection Graphs in the Moran Process
    arXiv.cs.DM Pub Date : 2017-10-19
    Themistoklis Melissourgos; Sotiris Nikoletseas; Christoforos Raptopoulos; Paul Spirakis

    The Moran process, as studied by Lieberman et al. [L05], is a stochastic process modeling the spread of genetic mutations in populations. In this process, agents of a two-type population (i.e. mutants and residents) are associated with the vertices of a graph. Initially, only one vertex chosen u.a.r. is a mutant, with fitness $r > 0$, while all other individuals are residents, with fitness $1$. In

    更新日期:2020-03-24
  • Fan-Crossing Free Graphs
    arXiv.cs.DM Pub Date : 2020-03-18
    Franz J. Brandenburg

    A graph is fan-crossing free if it admits a drawing in the plane so that each edge can be crossed by independent edges. Then the crossing edges have distinct vertices. In complement, a graph is fan-crossing if each edge can be crossed by edges of a fan. Then the crossing edges are incident to a common vertex. Graphs are k-planar if each edge is crossed by at most k edges, and k-gap-planar if each crossing

    更新日期:2020-03-20
  • Independent sets in ($P_4+P_4$,Triangle)-free graphs
    arXiv.cs.DM Pub Date : 2020-03-19
    Raffaele Mosca

    The Maximum Weight Independent Set Problem (WIS) is a well-known NP-hard problem. A popular way to study WIS is to detect graph classes for which WIS can be solved in polynomial time, with particular reference to hereditary graph classes, i.e., defined by a hereditary graph property or equivalently by forbidding one or more induced subgraphs. Given two graphs $G$ and $H$, $G+H$ denotes the disjoint

    更新日期:2020-03-20
  • Minimum Scan Cover with Angular Transition Costs
    arXiv.cs.DM Pub Date : 2020-03-19
    Sándor P. Fekete; Linda Kleist; Dominik Krupke

    We provide a comprehensive study of a natural geometric optimization problem motivated by questions in the context of satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs (MSC), we are given a graph $G$ that is embedded in Euclidean space. The edges of $G$ need to be scanned, i.e., probed from both of their vertices. In order to scan their edge, two vertices

    更新日期:2020-03-20
  • A Real Polynomial for Bipartite Graph Minimum Weight Perfect Matchings
    arXiv.cs.DM Pub Date : 2020-03-19
    Karthik Gajulapalli; Thorben Tröbst; Vijay V. Vazirani

    In a recent paper, Beniamini and Nisan \cite{Beniamini2020bipartite} gave a closed-form formula for the unique multilinear polynomial for the Boolean function determining whether a given bipartite graph $G \subseteq K_{n,n}$ has a perfect matching, together with an efficient algorithm for computing its terms. We give the following generalization: Given an arbitrary non-negative weight function $w$

    更新日期:2020-03-20
  • Computing Maximum Matchings in Temporal Graphs
    arXiv.cs.DM Pub Date : 2019-05-13
    George B. Mertzios; Hendrik Molter; Rolf Niedermeier; Viktor Zamaraev; Philipp Zschoche

    Temporal graphs are graphs whose topology is subject to discrete changes over time. Given a static underlying graph $G$, a temporal graph is represented by assigning a set of integer time-labels to every edge $e$ of $G$, indicating the discrete time steps at which $e$ is active. We introduce and study the complexity of a natural temporal extension of the classical graph problem Maximum Matching, taking

    更新日期:2020-03-20
  • Finding Dominating Induced Matchings in $S_{1,1,5}$-Free Graphs in Polynomial Time
    arXiv.cs.DM Pub Date : 2019-05-14
    Andreas Brandstädt; Raffaele Mosca

    Let $G=(V,E)$ be a finite undirected graph. An edge set $E' \subseteq E$ is a {\em dominating induced matching} ({\em d.i.m.}) in $G$ if every edge in $E$ is intersected by exactly one edge of $E'$. The \emph{Dominating Induced Matching} (\emph{DIM}) problem asks for the existence of a d.i.m.\ in $G$; this problem is also known as the \emph{Efficient Edge Domination} problem; it is the Efficient Domination

    更新日期:2020-03-20
  • Enumeration of Unordered Forests
    arXiv.cs.DM Pub Date : 2020-03-18
    Florian Ingels; Romain Azaïs

    Reverse search is a convenient method for enumerating structured objects, that can be used both to address theoretical issues and to solve data mining problems. This method has already been successfully developed to handle unordered trees. If the literature proposes solutions to enumerate singletons of trees, we study in this article a more general, higher combinatorial problem, the enumeration of

    更新日期:2020-03-19
  • Counting Bounded Tree Depth Homomorphisms
    arXiv.cs.DM Pub Date : 2020-03-18
    Martin Grohe

    We prove that graphs G, G' satisfy the same sentences of first-order logic with counting of quantifier rank at most k if and only if they are homomorphism-indistinguishable over the class of all graphs of tree depth at most k. Here G, G' are homomorphism-indistinguishable over a class C of graphs if for each graph F in C, the number of homomorphisms from F to G equals the number of homomorphisms from

    更新日期:2020-03-19
  • Colorings of complements of line graphs
    arXiv.cs.DM Pub Date : 2020-03-18
    Hamid Reza Daneshpajouh; Frédéric Meunier; Guilhem Mizrahi

    Our purpose is to show that complements of line graphs enjoy nice coloring properties. We show that for all graphs in this class the local and usual chromatic numbers are equal. We also prove a sufficient condition for the chromatic number to be equal with a natural upper bound. A consequence of this latter condition is a complete characterization of all induced subgraphs of the Kneser graph $\operatorname{KG}(n

    更新日期:2020-03-19
  • Probabilistic Tools for the Analysis of Randomized Optimization Heuristics
    arXiv.cs.DM Pub Date : 2018-01-20
    Benjamin Doerr

    This chapter collects several probabilistic tools that proved to be useful in the analysis of randomized search heuristics. This includes classic material like Markov, Chebyshev and Chernoff inequalities, but also lesser known topics like stochastic domination and coupling or Chernoff bounds for geometrically distributed random variables and for negatively correlated random variables. Most of the results

    更新日期:2020-03-19
  • Maximum Bipartite Subgraph of Geometric Intersection Graphs
    arXiv.cs.DM Pub Date : 2019-09-09
    Satyabrata Jana; Anil Maheshwari; Saeed Mehrabi; Sasanka Roy

    We study the Maximum Bipartite Subgraph (MBS) problem, which is defined as follows. Given a set $S$ of $n$ geometric objects in the plane, we want to compute a maximum-size subset $S'\subseteq S$ such that the intersection graph of the objects in $S'$ is bipartite. We first give a simple $O(n)$-time algorithm that solves the MBS problem on a set of $n$ intervals. We also give an $O(n^2)$-time algorithm

    更新日期:2020-03-19
  • Hard to Solve Instances of the Euclidean Traveling Salesman Problem
    arXiv.cs.DM Pub Date : 2018-08-08
    Stefan Hougardy; Xianghui Zhong

    The well known $4/3$ conjecture states that the integrality ratio of the subtour LP is at most $4/3$ for metric Traveling Salesman instances. We present a family of Euclidean Traveling Salesman instances for which we prove that the integrality ratio of the subtour LP converges to $4/3$. These instances (using the rounded Euclidean norm) turn out to be hard to solve exactly with Concorde, the fastest

    更新日期:2020-03-18
  • A randomly weighted minimum arborescence with a random cost constraint
    arXiv.cs.DM Pub Date : 2019-07-08
    Alan Frieze; Tomasz Tkocz

    We study the minimum spanning arborescence problem on the complete digraph $\vec{K}_n$ where an edge $e$ has a weight $W_e$ and a cost $C_e$, each of which is an independent uniform random variable $U^\alpha$ where $\alpha\leq 1$ and $U$ is uniform $[0,1]$. There is also a constraint that the spanning arborescence $T$ must satisfy $C(T)\leq c_0$. We establish, for a range of values for $c_0,\alpha$

    更新日期:2020-03-18
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