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  • 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
  • Proper Orientation Number of Triangle-free Bridgeless Outerplanar Graphs
    arXiv.cs.DM Pub Date : 2019-07-15
    J. Ai; S. Gerke; G. Gutin; Y. Shi; Z. Taoqiu

    An orientation of $G$ is a digraph obtained from $G$ by replacing each edge by exactly one of two possible arcs with the same endpoints. We call an orientation \emph{proper} if neighbouring vertices have different in-degrees. The proper orientation number of a graph $G$, denoted by $\vec{\chi}(G)$, is the minimum maximum in-degree of a proper orientation of G. Araujo et al. (Theor. Comput. Sci. 639

    更新日期:2020-03-18
  • Improved bounds for the sunflower lemma
    arXiv.cs.DM Pub Date : 2019-08-22
    Ryan Alweiss; Shachar Lovett; Kewen Wu; Jiapeng Zhang

    A sunflower with $r$ petals is a collection of $r$ sets so that the intersection of each pair is equal to the intersection of all. Erd\H{o}s and Rado proved the sunflower lemma: for any fixed $r$, any family of sets of size $w$, with at least about $w^w$ sets, must contain a sunflower. The famous sunflower conjecture is that the bound on the number of sets can be improved to $c^w$ for some constant

    更新日期:2020-03-18
  • Harary polynomials
    arXiv.cs.DM Pub Date : 2020-03-13
    Orli Herscovici; Johann A. Makowsky; Vsevolod Rakita

    Given a graph property $\mathcal{P}$, F. Harary introduced in 1985 $\mathcal{P}$-colorings, graph colorings where each colorclass induces a graph in $\mathcal{P}$. Let $\chi_{\mathcal{P}}(G;k)$ counts the number of $\mathcal{P}$-colorings of $G$ with at most $k$ colors. It turns out that $\chi_{\mathcal{P}}(G;k)$ is a polynomial in $\mathbb{Z}[k]$ for each graph $G$. Graph polynomials of this form

    更新日期:2020-03-16
  • The mathematics of Septoku
    arXiv.cs.DM Pub Date : 2008-01-24
    George I. Bell

    Septoku is a Sudoku variant invented by Bruce Oberg, played on a hexagonal grid of 37 cells. We show that up to rotations, reflections, and symbol permutations, there are only six valid Septoku boards. In order to have a unique solution, we show that the minimum number of given values is six. We generalize the puzzle to other board shapes, and devise a puzzle on a star-shaped board with 73 cells with

    更新日期:2020-03-16
  • The Approximation Ratio of the 2-Opt Heuristic for the Metric Traveling Salesman Problem
    arXiv.cs.DM Pub Date : 2019-09-26
    Stefan Hougardy; Fabian Zaiser; Xianghui Zhong

    The 2-Opt heuristic is one of the simplest algorithms for finding good solutions to the metric Traveling Salesman Problem. It is the key ingredient to the well-known Lin-Kernighan algorithm and often used in practice. So far, only upper and lower bounds on the approximation ratio of the 2-Opt heuristic for the metric TSP were known. We prove that for the metric TSP with $n$ cities, the approximation

    更新日期:2020-03-16
  • The diameter of the directed configuration model
    arXiv.cs.DM Pub Date : 2020-03-10
    Xing Shi Cai; Guillem Perarnau

    We show that the diameter of the directed configuration model with $n$ vertices rescaled by $\log n$ converges in probability to a constant. Our assumptions are the convergence of the in- and out-degree of a uniform random vertex in distribution, first and second moment. Our result extends previous results on the diameter of the model and applies to many other random directed graphs.

    更新日期:2020-03-12
  • On Degree Sequence Optimization
    arXiv.cs.DM Pub Date : 2020-03-11
    Shmuel Onn

    We consider the problem of finding a subgraph of a given graph which maximizes a given function evaluated at its degree sequence. While the problem is intractable already for convex functions, we show that it can be solved in polynomial time for convex multi-criteria objectives. We next consider the problem with separable objectives, which is NP-hard already when all vertex functions are the square

    更新日期:2020-03-12
  • Induced subgraphs of bounded treewidth and the container method
    arXiv.cs.DM Pub Date : 2020-03-11
    Tara Abrishami; Maria Chudnovsky; Marcin Pilipczuk; Paweł Rzążewski; Paul Seymour

    A hole in a graph is an induced cycle of length at least 4. A hole is long if its length is at least 5. By $P_t$ we denote a path on $t$ vertices. In this paper we give polynomial-time algorithms for the following problems: the Maximum Weight Independent Set problem in long-hole-free graphs, and the Feedback Vertex Set problem in $P_5$-free graphs. Each of the above results resolves a corresponding

    更新日期:2020-03-12
  • Monotone Arc Diagrams with few Biarcs
    arXiv.cs.DM Pub Date : 2020-03-11
    Steven Chaplick; Henry Förster; Michael Hoffmann; Michael Kaufmann

    We show that every planar graph can be represented by a monotone topological 2-page book embedding where at most 15n/16 (of potentially 3n-6) edges cross the spine exactly once.

    更新日期:2020-03-12
  • Beyond the Guruswami-Sudan (and Parvaresh-Vardy) Radii: Folded Reed-Solomon, Multiplicity and Derivative Codes
    arXiv.cs.DM Pub Date : 2020-03-11
    Neophytos Charalambides

    The classical family of Reed-Solomon codes consist of evaluations of polynomials over the finite field $\mathbb{F}_q$ of degree less than $k$, at $n$ distinct field elements. These are arguably the most widely used and studied codes, as they have both erasure and error-correction capabilities, among many others nice properties. In this survey we study closely related codes, folded Reed-Solomon codes

    更新日期:2020-03-12
  • General three person two color Hat Game
    arXiv.cs.DM Pub Date : 2017-04-13
    Theo van Uem

    Three distinguishable players are randomly fitted with a white or black hat, where the probabilities of getting a white or black hat may be different for each player, but known to all the players. All players guess simultaneously the color of their own hat observing only the hat colors of the other two players. It is also allowed for each player to pass: no color is guessed. The team wins if at least

    更新日期:2020-03-12
  • Clustering powers of sparse graphs
    arXiv.cs.DM Pub Date : 2020-03-07
    Jaroslav Nešetřil; Patrice Ossona de Mendez; Michał Pilipczuk; Xuding Zhu

    We prove that if $G$ is a sparse graph --- it belongs to a fixed class of bounded expansion $\mathcal{C}$ --- and $d\in \mathbb{N}$ is fixed, then the $d$th power of $G$ can be partitioned into cliques so that contracting each of these clique to a single vertex again yields a sparse graph. This result has several graph-theoretic and algorithmic consequences for powers of sparse graphs, including bounds

    更新日期:2020-03-10
  • Classification of minimally unsatisfiable 2-CNFs
    arXiv.cs.DM Pub Date : 2020-03-07
    Hoda Abbasizanjani; Oliver Kullmann

    We consider minimally unsatisfiable 2-CNFs, i.e., minimally unsatisfiable conjunctive normal forms, where each clause contains at most 2 literals (short 2-MUs). Characterisations of 2-MUs in the literature have been restricted to the nonsingular case (where every variable occurs positively and negatively at least twice), and those with a unit-clause. We provide the full classification of 2-MUs, and

    更新日期:2020-03-10
  • The Game of Cops and Eternal Robbers
    arXiv.cs.DM Pub Date : 2020-03-08
    Anthony Bonato; Melissa Hugan; Trent Marbach; Fionn Mc Inerney

    We introduce the game of Cops and Eternal Robbers played on graphs, where there are infinitely many robbers that appear sequentially over distinct plays of the game. A positive integer $t$ is fixed, and the cops are required to capture the robber in at most $t$ time-steps in each play. The associated optimization parameter is the eternal cop number, denoted by $c_t^{\infty},$ which equals the eternal

    更新日期:2020-03-10
  • Quasi-random words and limits of word sequences
    arXiv.cs.DM Pub Date : 2020-03-07
    Hiêp Hàn; Marcos Kiwi; Matías Pavez-Signé

    Words are sequences of letters over a finite alphabet. We study two intimately related topics for this object: quasi-randomness and limit theory. With respect to the first topic we investigate the notion of uniform distribution of letters over intervals, and in the spirit of the famous Chung-Graham-Wilson theorem for graphs we provide a list of word properties which are equivalent to uniformity. In

    更新日期:2020-03-10
  • Graph Clustering Via QUBO and Digital Annealing
    arXiv.cs.DM Pub Date : 2020-03-09
    Pierre Miasnikof; Seo Hong; Yuri Lawryshyn

    This article empirically examines the computational cost of solving a known hard problem, graph clustering, using novel purpose-built computer hardware. We express the graph clustering problem as an intra-cluster distance or dissimilarity minimization problem. We formulate our poblem as a quadratic unconstrained binary optimization problem and employ a novel computer architecture to obtain a numerical

    更新日期:2020-03-10
  • The Fast Loaded Dice Roller: A Near-Optimal Exact Sampler for Discrete Probability Distributions
    arXiv.cs.DM Pub Date : 2020-03-08
    Feras A. Saad; Cameron E. Freer; Martin C. Rinard; Vikash K. Mansinghka

    This paper introduces a new algorithm for the fundamental problem of generating a random integer from a discrete probability distribution using a source of independent and unbiased random coin flips. We prove that this algorithm, which we call the Fast Loaded Dice Roller (FLDR), is highly efficient in both space and time: (i) the size of the sampler is guaranteed to be linear in the number of bits

    更新日期:2020-03-10
  • Complexity of tree-coloring interval graphs equitably
    arXiv.cs.DM Pub Date : 2020-03-09
    Bei Niu; Bi Li; Xin Zhang

    An equitable tree-$k$-coloring of a graph is a vertex $k$-coloring such that each color class induces a forest and the size of any two color classes differ by at most one. In this work, we show that every interval graph $G$ has an equitable tree-$k$-coloring for any integer $k\geq \lceil(\Delta(G)+1)/2\rceil$, solving a conjecture of Wu, Zhang and Li (2013) for interval graphs, and furthermore, give

    更新日期:2020-03-10
  • Smoothed Analysis of Trie Height by Star-like PFAs
    arXiv.cs.DM Pub Date : 2020-03-09
    Stefan Eckhardt; Sven Kosub; Johannes Nowak

    Tries are general purpose data structures for information retrieval. The most significant parameter of a trie is its height $H$ which equals the length of the longest common prefix of any two string in the set $A$ over which the trie is built. Analytical investigations of random tries suggest that ${\bf E}(H)\in O(\log(\|A\|))$, although $H$ is unbounded in the worst case. Moreover, sharp results on

    更新日期:2020-03-10
  • b-Coloring Parameterized by Clique-Width
    arXiv.cs.DM Pub Date : 2020-03-09
    Lars Jaffke; Paloma T. Lima; Daniel Lokshtanov

    We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This unifies and extends previously known polynomial-time results on several graph classes, and answers open questions posed by Campos and Silva [Algorithmica, 2018] and Bonomo et al. [Graphs Combin., 2009]. This constitutes the first result concerning structural parameterizations of this problem. We show that

    更新日期:2020-03-10
  • Opinion Dynamics under Voter and Majority Rule Models with Biased and Stubborn Agents
    arXiv.cs.DM Pub Date : 2020-03-05
    Arpan Mukhopadhyay; Ravi R. Mazumdar; Rahul Roy

    We study binary opinion dynamics in a network of social agents interacting among themselves according to one of the following rules: (1) Voter rule: An updating agent simply copies the opinion of another randomly sampled agent; (2) Majority rule: An updating agent samples multiple agents and adopts the majority opinion in the selected group. While opinion dynamics have been studied extensively under

    更新日期:2020-03-09
  • Algorithms for the rainbow vertex coloring problem on graph classes
    arXiv.cs.DM Pub Date : 2020-03-06
    Paloma T. Lima; Erik Jan van Leeuwen; Marieke van der Wegen

    Given a vertex-colored graph, we say a path is a rainbow vertex path if all its internal vertices have distinct colors. The graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. In the Rainbow Vertex Coloring (RVC) problem we want to decide whether the vertices of a given graph can be colored with at most $k$ colors so that the graph becomes rainbow

    更新日期:2020-03-09
  • Fast calculation of the variance of edge crossings in random linear arrangements
    arXiv.cs.DM Pub Date : 2020-03-06
    Lluís Alemany-Puig; Ramon Ferrer-i-Cancho

    The interest in spatial networks where vertices are embedded in a one-dimensional space is growing. Remarkable examples of these networks are syntactic dependency trees and RNA structures. In this setup, the vertices of the network are arranged linearly and then edges may cross when drawn above the sequence of vertices. Recently, two aspects of the distribution of the number of crossings in uniformly

    更新日期:2020-03-09
  • On the Collection of Fringe Subtrees in Random Binary Trees
    arXiv.cs.DM Pub Date : 2020-03-06
    Louisa Seelbach Benkner; Stephan Wagner

    A fringe subtree of a rooted tree is a subtree consisting of one of the nodes and all its descendants. In this paper, we are specifically interested in the number of non-isomorphic trees that appear in the collection of all fringe subtrees of a binary tree. This number is analysed under two different random models: uniformly random binary trees and random binary search trees. In the case of uniformly

    更新日期:2020-03-09
  • Reappraising the distribution of the number of edge crossings of graphs on a sphere
    arXiv.cs.DM Pub Date : 2020-03-06
    Lluís Alemany-Puig; Mercè Mora; Ramon Ferrer-i-Cancho

    Many real transportation and mobility networks have their vertices placed on the surface of the Earth. In such embeddings, the edges laid on that surface may cross. In his pioneering research, Moon analyzed the distribution of the number of crossings on complete graphs and complete bipartite graphs whose vertices are located uniformly at random on the surface of a sphere assuming that vertex placements

    更新日期:2020-03-09
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