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  • Kirchhoff's Circuit Law Applications to Graph Simplification in Search Problems
    arXiv.cs.DM Pub Date : 2020-09-23
    Jaeho Choi; Joongheon Kim

    This paper proposes a new analysis of graph using the concept of electric potential, and also proposes a graph simplification method based on this analysis. Suppose that each node in the weighted-graph has its respective potential value. Furthermore, suppose that the start and terminal nodes in graphs have maximum and zero potentials, respectively. When we let the level of each node be defined as the

    更新日期:2020-09-25
  • Counting five-node subgraphs
    arXiv.cs.DM Pub Date : 2020-09-23
    Steve Lawford

    We propose exact count formulae for the 21 topologically distinct non-induced connected subgraphs on five nodes, in simple, unweighted and undirected graphs. We prove the main result using short and purely combinatorial arguments that can be adapted to derive count formulae for larger subgraphs. To illustrate, we give analytic results for some regular graphs, and present a short empirical application

    更新日期:2020-09-25
  • The Quotient in Preorder Theories
    arXiv.cs.DM Pub Date : 2020-09-23
    Íñigo X. Íncer RomeoUniversity of California, Berkeley; Leonardo MangerucaRaytheon Technologies Research Center, Rome, Italy; Tiziano VillaUniversità di Verona, Italy; Alberto Sangiovanni-VincentelliUniversity of California, Berkeley

    Seeking the largest solution to an expression of the form A x <= B is a common task in several domains of engineering and computer science. This largest solution is commonly called quotient. Across domains, the meanings of the binary operation and the preorder are quite different, yet the syntax for computing the largest solution is remarkably similar. This paper is about finding a common framework

    更新日期:2020-09-24
  • Bounds on the Spectral Sparsification of Symmetric and Off-Diagonal Nonnegative Real Matrices
    arXiv.cs.DM Pub Date : 2020-09-23
    Sergio Mercado; Marcos Villagra

    We say that a square real matrix $M$ is \emph{off-diagonal nonnegative} if and only if all entries outside its diagonal are nonnegative real numbers. In this note we show that for any off-diagonal nonnegative symmetric matrix $M$, there exists a nonnegative symmetric matrix $\widehat{M}$ which is sparse and close in spectrum to $M$.

    更新日期:2020-09-24
  • A simple $(2+ε)$-approximation algorithm for Split Vertex Deletion
    arXiv.cs.DM Pub Date : 2020-09-23
    Matthew Drescher; Samuel Fiorini; Tony Huynh

    A split graph is a graph whose vertex set can be partitioned into a clique and a stable set. Given a graph $G$ and weight function $w: V(G) \to \mathbb{Q}_{\geq 0}$, the Split Vertex Deletion (SVD) problem asks to find a minimum weight set of vertices $X$ such that $G-X$ is a split graph. It is easy to show that a graph is a split graph if and only it it does not contain a $4$-cycle, $5$-cycle, or

    更新日期:2020-09-24
  • On uncertainty inequalities related to subcube partitions and additive energy
    arXiv.cs.DM Pub Date : 2020-09-21
    Norbert Hegyvari

    The additive energy plays a central role in combinatorial number theory. We show an uncertainty inequality which indicates how the additive energy of support of a Boolean function, its degree and subcube partition are related.

    更新日期:2020-09-23
  • On Regular Set Systems Containing Regular Subsystems
    arXiv.cs.DM Pub Date : 2020-09-21
    Amin Bahmanian; Sadegheh Haghshenas

    Let $X,Y$ be finite sets, $r,s,h, \lambda \in \mathbb{N}$ with $s\geq r, X\subsetneq Y$. By $\lambda \binom{X}{h}$ we mean the collection of all $h$-subsets of $X$ where each subset occurs $\lambda$ times. A coloring of $\lambda\binom{X}{h}$ is {\it $r$-regular} if in every color class each element of $X$ occurs $r$ times. A one-regular color class is a {\it perfect matching}. We are interested in

    更新日期:2020-09-23
  • The Ising antiferromagnet and max cut on random regular graphs
    arXiv.cs.DM Pub Date : 2020-09-22
    Amin Coja-Oghlan; Philipp Loick; Balázs F. Mezei; Gregory B. Sorkin

    The Ising antiferromagnet is an important statistical physics model with close connections to the {\sc Max Cut} problem. Combining spatial mixing arguments with the method of moments and the interpolation method, we pinpoint the replica symmetry breaking phase transition predicted by physicists. Additionally, we rigorously establish upper bounds on the {\sc Max Cut} of random regular graphs predicted

    更新日期:2020-09-23
  • Improving on Best-of-Many-Christofides for $T$-tours
    arXiv.cs.DM Pub Date : 2020-09-21
    Vera Traub

    The $T$-tour problem is a natural generalization of TSP and Path TSP. Given a graph $G=(V,E)$, edge cost $c: E \to \mathbb{R}_{\ge 0}$, and an even cardinality set $T\subseteq V$, we want to compute a minimum-cost $T$-join connecting all vertices of $G$ (and possibly containing parallel edges). In this paper we give an $\frac{11}{7}$-approximation for the $T$-tour problem and show that the integrality

    更新日期:2020-09-22
  • Connected Fair Detachments of Hypergraphs
    arXiv.cs.DM Pub Date : 2020-09-21
    Amin Bahmanian

    Let $\mathcal G$ be a hypergraph whose edges are colored. An {\it $(\alpha,n)$-detachment} of $\mathcal G$ is a hypergraph obtained by splitting a vertex $\alpha$ into $n$ vertices, say $\alpha_1,\dots,\alpha_n$, and sharing the incident hinges and edges among the subvertices. A detachment is {\it fair} if the degree of vertices and multiplicity of edges are shared as evenly as possible among the subvertices

    更新日期:2020-09-22
  • Note on Sunflowers
    arXiv.cs.DM Pub Date : 2020-09-20
    Tolson Bell; Suchakree Chueluecha; Lutz Warnke

    A sunflower with p petals consists of p sets whose pairwise intersections are identical. Building upon a breakthrough of Alweiss, Lovett, Wu, and Zhang from 2019, Rao proved that any family of (Cp\log(pk))^k distinct k-element sets contains a sunflower with p petals, where C>0 is a constant; this bound was reproved by Tao. In this note we record that, by a minor variant of their probabilistic arguments

    更新日期:2020-09-22
  • Blocking total dominating sets via edge contractions
    arXiv.cs.DM Pub Date : 2020-09-18
    Esther Galby; Felix Mann; Bernard Ries

    In this paper, we study the problem of deciding whether the total domination number of a given graph $G$ can be reduced using exactly one edge contraction (called 1-Edge Contraction($\gamma_t$)). We focus on several graph classes and determine the computational complexity of this problem. By putting together these results, we manage to obtain a complete dichotomy for $H$-free graphs.

    更新日期:2020-09-21
  • Directed branch-width: A directed analogue of tree-width
    arXiv.cs.DM Pub Date : 2020-09-18
    Benjamin Merlin Bumpus; Kitty Meeks; William Pettersson

    We introduce a new digraph width measure called directed branch-width. To do this, we generalize a characterization of graph classes of bounded tree-width in terms of their line graphs to digraphs. Under parameterizations by directed branch-width we obtain linear time algorithms for many problems, such as directed Hamilton path and Max-Cut, which are hard when parameterized by other known directed

    更新日期:2020-09-21
  • A polynomial size model with implicit SWAP gate counting for exact qubit reordering
    arXiv.cs.DM Pub Date : 2020-09-18
    Jesse Mulderij; Karen I. Aardal; Irina Chiscop; Frank Phillipson

    Due to the physics behind quantum computing, quantum circuit designers must adhere to the constraints posed by the limited interaction distance of qubits. Existing circuits need therefore to be modified via the insertion of SWAP gates, which alter the qubit order by interchanging the location of two qubits' quantum states. We consider the Nearest Neighbor Compliance problem on a linear array, where

    更新日期:2020-09-21
  • Prisoners, Rooms, and Lightswitches
    arXiv.cs.DM Pub Date : 2020-09-18
    Daniel M. Kane; Scott Duke Kominers

    We examine a new variant of the classic prisoners and lightswitches puzzle: A warden leads his $n$ prisoners in and out of $r$ rooms, one at a time, in some order, with each prisoner eventually visiting every room an arbitrarily large number of times. The rooms are indistinguishable, except that each one has $s$ lightswitches; the prisoners win their freedom if at some point a prisoner can correctly

    更新日期:2020-09-21
  • On the Enumeration of Sweep-Covers and Their Relation to Raney Numbers
    arXiv.cs.DM Pub Date : 2020-09-17
    Blake Wilson

    We define a new structure for collections of nodes in trees which are called "Sweep-Covers" for their 'covering' of all the nodes in the tree by some ancestor-descendent relationship. Then, we analyze an algorithm for finding all sweep covers of a given size in any tree. The complexity of the algorithm is analyzed on a class of infinite $\Delta$-ary trees with constant path lengths between the $\Delta$-star

    更新日期:2020-09-21
  • Metrical Service Systems with Transformations
    arXiv.cs.DM Pub Date : 2020-09-17
    Sébastien Bubeck; Niv Buchbinder; Christian Coester; Mark Sellke

    We consider a generalization of the fundamental online metrical service systems (MSS) problem where the feasible region can be transformed between requests. In this problem, which we call T-MSS, an algorithm maintains a point in a metric space and has to serve a sequence of requests. Each request is a map (transformation) $f_t\colon A_t\to B_t$ between subsets $A_t$ and $B_t$ of the metric space. To

    更新日期:2020-09-20
  • Algorithms and Complexity for Variants of Covariates Fine Balance
    arXiv.cs.DM Pub Date : 2020-09-17
    Dorit S. Hochbaum; Asaf Levin; Xu Rao

    We study here several variants of the covariates fine balance problem where we generalize some of these problems and introduce a number of others. We present here a comprehensive complexity study of the covariates problems providing polynomial time algorithms, or a proof of NP-hardness. The polynomial time algorithms described are mostly combinatorial and rely on network flow techniques. In addition

    更新日期:2020-09-20
  • On Weak Flexibility in Planar Graphs
    arXiv.cs.DM Pub Date : 2020-09-16
    Bernard Lidický; Tomáš Masařík; Kyle Murphy; Shira Zerbib

    Recently, Dvo\v{r}\'ak, Norin, and Postle introduced flexibility as an extension of list coloring on graphs [JGT 19']. In this new setting, each vertex $v$ in some subset of $V(G)$ has a request for a certain color $r(v)$ in its list of colors $L(v)$. The goal is to find an $L$ coloring satisfying many, but not necessarily all, of the requests. The main studied question is whether there exists a universal

    更新日期:2020-09-20
  • On path ranking in time-dependent graphs
    arXiv.cs.DM Pub Date : 2020-09-16
    Tommaso Adamo; Gianpaolo Ghiani; Emanuela Guerriero

    In this paper we study a property of time-dependent graphs, dubbed path ranking invariance. Broadly speaking, a time-dependent graph is path ranking invariant if the ordering of its paths (w.r.t. travel time) is independent of the start time. In this paper we show that, if a graph is path ranking invariant, the solution of a large class of time-dependent vehicle routing problems can be obtained by

    更新日期:2020-09-18
  • Typical and Extremal Aspects of Friends-and-Strangers Graphs
    arXiv.cs.DM Pub Date : 2020-09-16
    Noga Alon; Colin Defant; Noah Kravitz

    Given graphs $X$ and $Y$ with vertex sets $V(X)$ and $V(Y)$ of the same cardinality, the friends-and-strangers graph $\mathsf{FS}(X,Y)$ is the graph whose vertex set consists of all bijections $\sigma:V(X)\to V(Y)$, where two bijections $\sigma$ and $\sigma'$ are adjacent if they agree everywhere except for two adjacent vertices $a,b \in V(X)$ such that $\sigma(a)$ and $\sigma(b)$ are adjacent in $Y$

    更新日期:2020-09-18
  • Accelerating Domain Propagation: an Efficient GPU-Parallel Algorithm over Sparse Matrices
    arXiv.cs.DM Pub Date : 2020-09-16
    Boro Sofranac; Ambros Gleixner; Sebastian Pokutta

    Fast domain propagation of linear constraints has become a crucial component of today's best algorithms and solvers for mixed integer programming and pseudo-boolean optimization to achieve peak solving performance. Irregularities in the form of dynamic algorithmic behaviour, dependency structures, and sparsity patterns in the input data make efficient implementations of domain propagation on GPUs and

    更新日期:2020-09-18
  • MIP and Set Covering approaches for Sparse Approximation
    arXiv.cs.DM Pub Date : 2020-09-14
    Diego Delle Donne; Matthieu Kowalski; Leo Liberti

    The Sparse Approximation problem asks to find a solution $x$ such that $||y - Hx|| < \alpha$, for a given norm $||\cdot||$, minimizing the size of the support $||x||_0 := \#\{j \ |\ x_j \neq 0 \}$. We present valid inequalities for Mixed Integer Programming (MIP) formulations for this problem and we show that these families are sufficient to describe the set of feasible supports. This leads to a reformulation

    更新日期:2020-09-15
  • Sparsity of weighted networks: measures and applications
    arXiv.cs.DM Pub Date : 2020-09-14
    Swati Goswami; Asit K. Das; Subhas C. Nandy

    A majority of real life networks are weighted and sparse. The present article aims at characterization of weighted networks based on sparsity, as a measure of inherent diversity, of different network parameters. It utilizes sparsity index defined on ordered degree sequence of simple networks and derives further properties of this index. The range of possible values of sparsity index of any connected

    更新日期:2020-09-15
  • An Algorithmic Information Distortion in Multidimensional Networks
    arXiv.cs.DM Pub Date : 2020-09-12
    Felipe S. Abrahão; Klaus Wehmuth; Hector Zenil; Artur Ziviani

    Network complexity, network information content analysis, and lossless compressibility of graph representations have been played an important role in network analysis and network modeling. As multidimensional networks, such as time-varying, multilayer, or dynamic multilayer networks, gain more relevancy in network science, it becomes crucial to investigate in which situations universal algorithmic

    更新日期:2020-09-15
  • Terminating cases of flooding
    arXiv.cs.DM Pub Date : 2020-09-12
    Walter Hussak; Amitabh Trehan

    Basic synchronous flooding proceeds in rounds. Given a finite undirected (network) graph $G$, a set of sources $I \subseteq G$ initiate flooding in the first round by every node in $I$ sending the same message to all of its neighbours. In each subsequent round, nodes send the message to all of their neighbours from which they did not receive the message in the previous round. Flooding terminates when

    更新日期:2020-09-15
  • Finite Horn Monoids and Near-Semirings
    arXiv.cs.DM Pub Date : 2020-09-12
    Christian Antic

    Describing complex objects as the composition of elementary ones is a common strategy in computer science and science in general. This paper contributes to the foundations of knowledge representation and database theory by introducing and studying the sequential composition of propositional Horn theories. Specifically, we show that the notion of composition gives rise to a family of monoids and near-semirings

    更新日期:2020-09-15
  • Balancing Polynomials in the Chebyshev Norm
    arXiv.cs.DM Pub Date : 2020-09-12
    Victor Reis

    Given $n$ polynomials $p_1, \dots, p_n$ of degree at most $n$ with $\|p_i\|_\infty \le 1$ for $i \in [n]$, we show there exist signs $x_1, \dots, x_n \in \{-1,1\}$ so that \[\Big\|\sum_{i=1}^n x_i p_i\Big\|_\infty < 30\sqrt{n}, \] where $\|p\|_\infty := \sup_{|x| \le 1} |p(x)|$. This result extends the Rudin-Shapiro sequence, which gives an upper bound of $O(\sqrt{n})$ for the Chebyshev polynomials

    更新日期:2020-09-15
  • Graphs and matrices: A translation of "Graphok és matrixok" by Dénes Kőnig (1931)
    arXiv.cs.DM Pub Date : 2020-09-05
    Gábor Szárnyas

    This paper, originally written in Hungarian by D\'{e}nes K\H{o}nig in 1931, proves that in a bipartite graph, the minimum vertex cover and the maximum matching have the same size. This statement is now known as K\H{o}nig's theorem. The paper also discusses the connection of graphs and matrices, then makes some observations about the combinatorial properties of the latter.

    更新日期:2020-09-15
  • Hypercontractivity on the symmetric group
    arXiv.cs.DM Pub Date : 2020-09-11
    Yuval Filmus; Guy Kindler; Noam Lifshitz; Dor Minzer

    The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly for product spaces, which raises the question of whether analogous results hold over non-product domains. We consider the symmetric group, $S_n$, one of the most

    更新日期:2020-09-14
  • The chromatic number of 2-edge-colored and signed graphs of bounded maximum degree
    arXiv.cs.DM Pub Date : 2020-09-11
    Christopher Duffy; Fabien Jacques; Mickael Montassier; Alexandre Pinlou

    A 2-edge-colored graph or a signed graph is a simple graph with two types of edges. A homomorphism from a 2-edge-colored graph $G$ to a 2-edge-colored graph $H$ is a mapping $\varphi: V(G) \rightarrow V(H)$ that maps every edge in $G$ to an edge of the same type in $H$. Switching a vertex $v$ of a 2-edge-colored or signed graph corresponds to changing the type of each edge incident to $v$. There is

    更新日期:2020-09-14
  • A new algorithm for the $^K$DMDGP subclass of Distance Geometry Problems
    arXiv.cs.DM Pub Date : 2020-09-11
    Douglas S. Goncalves; Carlile Lavor; Leo Liberti; Michael Souza

    The fundamental inverse problem in distance geometry is the one of finding positions from inter-point distances. The Discretizable Molecular Distance Geometry Problem (DMDGP) is a subclass of the Distance Geometry Problem (DGP) whose search space can be discretized and represented by a binary tree, which can be explored by a Branch-and-Prune (BP) algorithm. It turns out that this combinatorial search

    更新日期:2020-09-14
  • Sharp threshold rates for random codes
    arXiv.cs.DM Pub Date : 2020-09-09
    Venkatesan Guruswami; Jonathan Mosheiff; Nicolas Resch; Shashwat Silas; Mary Wootters

    Suppose that $\mathcal{P}$ is a property that may be satisfied by a random code $C \subset \Sigma^n$. For example, for some $p \in (0,1)$, $\mathcal{P}$ might be the property that there exist three elements of $C$ that lie in some Hamming ball of radius $pn$. We say that $R^*$ is the threshold rate for $\mathcal{P}$ if a random code of rate $R^{*} + \varepsilon$ is very likely to satisfy $\mathcal{P}$

    更新日期:2020-09-11
  • Refined Wilf-equivalences by Comtet statistics
    arXiv.cs.DM Pub Date : 2020-09-09
    Shishuo Fu; Zhicong Lin; Yaling Wang

    We launch a systematic study of the refined Wilf-equivalences by the statistics $\mathsf{comp}$ and $\mathsf{iar}$, where $\mathsf{comp}(\pi)$ and $\mathsf{iar}(\pi)$ are the number of components and the length of the initial ascending run of a permutation $\pi$, respectively. As Comtet was the first one to consider the statistic $\mathsf{comp}$ in his book {\em Analyse combinatoire}, any statistic

    更新日期:2020-09-10
  • Edge Degeneracy: Algorithmic and Structural Results
    arXiv.cs.DM Pub Date : 2020-09-08
    Stratis Limnios; Christophe Paul; Joanny Perret; Dimitrios M. Thilikos

    We consider a cops and robber game where the cops are blocking edges of a graph, while the robber occupies its vertices. At each round of the game, the cops choose some set of edges to block and right after the robber is obliged to move to another vertex traversing at most $s$ unblocked edges ($s$ can be seen as the speed of the robber). Both parts have complete knowledge of the opponent's moves and

    更新日期:2020-09-10
  • Tight List-Sizes for Oblivious AVCs under Constraints
    arXiv.cs.DM Pub Date : 2020-09-08
    Yihan Zhang; Sidharth Jaggi; Amitalok J. Budkuley

    We study list-decoding over adversarial channels governed by oblivious adversaries (a.k.a. oblivious Arbitrarily Varying Channels (AVCs)). This type of adversaries aims to maliciously corrupt the communication without knowing the actual transmission from the sender. For any oblivious AVCs potentially with constraints on the sender's transmitted sequence and the adversary's noise sequence, we determine

    更新日期:2020-09-10
  • Connectivity matters: Construction and exact random sampling of connected graphs
    arXiv.cs.DM Pub Date : 2020-09-08
    Szabolcs Horvát; Carl D. Modes

    We describe a new method for the random sampling of connected graphs with a specified degree sequence. We consider both the case of simple graphs and that of loopless multigraphs. Our method builds on a recently introduced novel sampling approach that constructs graphs independently (unlike edge-switching Markov Chain Monte Carlo methods) and efficiently (unlike the configuration model), and extends

    更新日期:2020-09-10
  • Probabilistic analysis of algorithms for cost constrained minimum weighted combinatorial objects
    arXiv.cs.DM Pub Date : 2020-09-07
    Alan Frieze; Tomasz Tkocz

    We consider cost constrained versions of the minimum spanning tree problem and the assignment problem. We assume edge weights are independent copies of a continuous random variable $Z$ that satisfies $F(x)=\Pr(Z\leq x)\approx x^\alpha$ as $x\to0$, where $\alpha\geq 1$. Also, there are $r=O(1)$ budget constraints with edge costs chosen from the same distribution. We use Lagrangean duality to construct

    更新日期:2020-09-10
  • On the spectrum and linear programming bound for hypergraphs
    arXiv.cs.DM Pub Date : 2020-09-07
    Sebastian M. Cioabă; Jack H. Koolen; Masato Mimura; Hiroshi Nozaki; Takayuki Okuda

    The spectrum of a graph is closely related to many graph parameters. In particular, the spectral gap of a regular graph which is the difference between its valency and second eigenvalue, is widely seen an algebraic measure of connectivity and plays a key role in the theory of expander graphs. In this paper, we extend previous work done for graphs and bipartite graphs and present a linear programming

    更新日期:2020-09-08
  • On prefix palindromic length of automatic words
    arXiv.cs.DM Pub Date : 2020-09-07
    Anna E. Frid; Enzo Laborde; Jarkko Peltomäki

    The prefix palindromic length $\mathrm{PPL}_{\mathbf{u}}(n)$ of an infinite word $\mathbf{u}$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $\mathbf{u}$. Since 2013, it is still unknown if $\mathrm{PPL}_{\mathbf{u}}(n)$ is unbounded for every aperiodic infinite word $\mathbf{u}$, even though this has been proven for almost all aperiodic words. At the

    更新日期:2020-09-08
  • Vertex-Domatic, Edge-Domatic and Total Domatic Number of Uniform Hypergraphs
    arXiv.cs.DM Pub Date : 2020-09-06
    Smruti Prava Dash

    E. J. Cockayne and S. T. Hedetniemi introduced the concept of domatic number of a graph. B. Zelinka extended the concept to the uniform hypergraphs. Further, B. Zelinka defined the concept of edge-domatic number and total edge-domatic number of a graph. In this paper, we investigate and prove some assertions in connection with vertex domatic number, edge-domatic number and total domatic number of some

    更新日期:2020-09-08
  • On the implementation of a global optimization method for mixed-variable problems
    arXiv.cs.DM Pub Date : 2020-09-04
    Giacomo Nannicini

    We describe the optimization algorithm implemented in the open-source derivative-free solver RBFOpt. The algorithm is based on the radial basis function method of Gutmann and the metric stochastic response surface method of Regis and Shoemaker. We propose several modifications aimed at generalizing and improving these two algorithms: (i) the use of an extended space to represent categorical variables

    更新日期:2020-09-08
  • Nonrepetitive graph colouring
    arXiv.cs.DM Pub Date : 2020-09-04
    David R. Wood

    A vertex colouring of a graph $G$ is "nonrepetitive" if $G$ contains no path for which the first half of the path is assigned the same sequence of colours as the second half. Thue's famous theorem says that every path is nonrepetitively 3-colourable. This paper surveys results about nonrepetitive colourings of graphs. The goal is to give a unified and comprehensive presentation of the major results

    更新日期:2020-09-08
  • MAX CUT in Weighted Random Intersection Graphs and Discrepancy of Sparse Random Set Systems
    arXiv.cs.DM Pub Date : 2020-09-03
    Sotiris Nikoletseas; Christoforos Raptopoulos; Paul Spirakis

    Let $V$ be a set of $n$ vertices, ${\cal M}$ a set of $m$ labels, and let $\mathbf{R}$ be an $m \times n$ matrix of independent Bernoulli random variables with success probability $p$. A random instance $G(V,E,\mathbf{R}^T\mathbf{R})$ of the weighted random intersection graph model is constructed by drawing an edge with weight $[\mathbf{R}^T\mathbf{R}]_{v,u}$ between any two vertices $u,v$ for which

    更新日期:2020-09-05
  • Strong Hanani-Tutte for the Torus
    arXiv.cs.DM Pub Date : 2020-09-01
    Radoslav Fulek; Michael J. Pelsmajer; Marcus Schaefer

    If a graph can be drawn on the torus so that every two independent edges cross an even number of times, then the graph can be embedded on the torus.

    更新日期:2020-09-05
  • Automated Market Makers for Decentralized Finance (DeFi)
    arXiv.cs.DM Pub Date : 2020-09-03
    Yongge Wang

    This paper compares mathematical models for automated market makers including logarithmic market scoring rule (LMSR), liquidity sensitive LMSR (LS-LMSR), constant product/mean/sum, and others. It is shown that though LMSR may not be a good model for Decentralized Finance (DeFi) applications, LS-LMSR has several advantages over constant product/mean based automated market makers. However, LS-LMSR requires

    更新日期:2020-09-05
  • Service Rate Region: A New Aspect of Coded Distributed System Design
    arXiv.cs.DM Pub Date : 2020-09-03
    Mehmet Aktas; Gauri Joshi; Swanand Kadhe; Fatemeh Kazemi; Emina Soljanin

    Erasure coding has been recently employed as a powerful method to mitigate delays due to slow or straggling nodes in distributed systems. In this work, we show that erasure coding of data objects can flexibly handle skews in the request rates. Coding can help boost the service rate region, that is, increase the overall volume of data access requests that can be handled by the system. The goal of this

    更新日期:2020-09-05
  • Exact square coloring of subcubic planar graphs *
    arXiv.cs.DM Pub Date : 2020-09-02
    Florent FoucaudLaBRI; Hervé HocquardLaBRI; Suchismita MishraIIT Madras; Narayanan NarayananIIT Madras; Reza NaserasrLaBRI; Éric SopenaLaBRI; Petru ValicovLIS, LIRMM

    We study the exact square chromatic number of subcubic planar graphs. An exact square coloring of a graph G is a vertex-coloring in which any two vertices at distance exactly 2 receive distinct colors. The smallest number of colors used in such a coloring of G is its exact square chromatic number, denoted $\chi^{\sharp 2}(G)$. This notion is related to other types of distance-based colorings, as well

    更新日期:2020-09-03
  • Big Ramsey degrees using parameter spaces
    arXiv.cs.DM Pub Date : 2020-09-02
    Jan Hubička

    We show that the universal homogeneous partial order has finite big Ramsey degrees and discuss several corollaries. Our proof uses parameter spaces and the Carlson-Simpson theorem rather than (a strengthening of) the Halpern-L\"auchli theorem and the Milliken tree theorem, which are the primary tools used to give bounds on big Ramsey degrees elsewhere (originating from work of Laver and Milliken).

    更新日期:2020-09-03
  • Towards constant-factor approximation for chordal / distance-hereditary vertex deletion
    arXiv.cs.DM Pub Date : 2020-09-02
    Jungho Ahn; Eun Jung Kim; Euiwoong Lee

    For a family of graphs $\mathcal{F}$, Weighted $\mathcal{F}$-Deletion is the problem for which the input is a vertex weighted graph $G=(V,E)$ and the goal is to delete $S\subseteq V$ with minimum weight such that $G\setminus S\in\mathcal{F}$. Designing a constant-factor approximation algorithm for large subclasses of perfect graphs has been an interesting research direction. Block graphs, 3-leaf power

    更新日期:2020-09-03
  • On sensitivity in bipartite Cayley graphs
    arXiv.cs.DM Pub Date : 2020-09-01
    Ignacio García-Marco; Kolja Knauer

    Huang proved that every set of more than half the vertices of the $d$-dimensional hypercube $Q_d$ induces a subgraph of maximum degree at least $\sqrt{d}$, which is tight by a result of Chung, F\"uredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs. First, we present three infinite families of Cayley graphs of unbounded degree that contain

    更新日期:2020-09-02
  • Choosability with Separation of Cycles and Outerplanar Graphs
    arXiv.cs.DM Pub Date : 2020-09-01
    Jean-Christophe Godin; Olivier Togni

    We consider the following list coloring with separation problem of graphs: Given a graph $G$ and integers $a,b$, find the largest integer $c$ such that for any list assignment $L$ of $G$ with $|L(v)|\le a$ for any vertex $v$ and $|L(u)\cap L(v)|\le c$ for any edge $uv$ of $G$, there exists an assignment $\varphi$ of sets of integers to the vertices of $G$ such that $\varphi(u)\subset L(u)$ and $|\varphi(v)|=b$

    更新日期:2020-09-02
  • Reconfiguration graphs of zero forcing sets
    arXiv.cs.DM Pub Date : 2020-09-01
    Jesse Geneson; Ruth Haas; Leslie Hogben

    This paper begins the study of reconfiguration of zero forcing sets, and more specifically, the zero forcing graph. Given a base graph $G$, its zero forcing graph, $\mathscr{Z}(G)$, is the graph whose vertices are the minimum zero forcing sets of $G$ with an edge between vertices $B$ and $B'$ of $\mathscr{Z}(G)$ if and only if $B$ can be obtained from $B'$ by changing a single vertex of $G$. It is

    更新日期:2020-09-02
  • Sorting an Array Using the Topological Sort of a Corresponding Comparison Graph
    arXiv.cs.DM Pub Date : 2020-08-31
    Balaram Behera

    The quest for efficient sorting is ongoing, and we will explore a graph-based stable sorting strategy, in particular employing comparison graphs. We use the topological sort to map the comparison graph to a linear domain, and we can manipulate our graph such that the resulting topological sort is the sorted array. By taking advantage of the many relations between Hamiltonian paths and topological sorts

    更新日期:2020-09-02
  • Collectively canalizing Boolean functions
    arXiv.cs.DM Pub Date : 2020-08-31
    Claus Kadelka; Benjamin Keilty; Reinhard Laubenbacher

    Boolean networks are frequently used to model biological systems, such as gene regulatory networks. Canalizing functions have been suggested as suitable update rules for genetic networks and have been studied rigorously. Mathematically, the biology-inspired concept of canalization means that a single input can determine the output of a function, irrespective of all the other inputs. Here, we relax

    更新日期:2020-09-01
  • Semipaired Domination in Some Subclasses of Chordal Graphs
    arXiv.cs.DM Pub Date : 2020-08-31
    Michael A. Henning; Arti Pandey; Vikash Tripathi

    A dominating set $D$ of a graph $G$ without isolated vertices is called semipaired dominating set if $D$ can be partitioned into $2$-element subsets such that the vertices in each set are at distance at most $2$. The semipaired domination number, denoted by $\gamma_{pr2}(G)$ is the minimum cardinality of a semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices, the \textsc{Minimum

    更新日期:2020-09-01
  • Extremal results for graphs of bounded metric dimension
    arXiv.cs.DM Pub Date : 2020-08-31
    Jesse Geneson; Suchir Kaustav; Antoine Labelle

    Metric dimension is a graph parameter motivated by problems in robot navigation, drug design, and image processing. In this paper, we answer several open extremal problems on metric dimension and pattern avoidance in graphs from (Geneson, Metric dimension and pattern avoidance, Discrete Appl. Math. 284, 2020, 1-7). Specifically, we construct a new family of graphs that allows us to determine the maximum

    更新日期:2020-09-01
  • Maximizing the algebraic connectivity in multilayer networks with arbitrary interconnections
    arXiv.cs.DM Pub Date : 2020-08-29
    Ali Tavasoli; Ehsan Ardjmand; Heman Shakeri

    The second smallest eigenvalue of the Laplacian matrix is determinative in characterizing many network properties and is known as algebraic connectivity. In this paper, we investigate the problem of maximizing algebraic connectivity in multilayer networks by allocating interlink weights subject to a budget while allowing arbitrary interconnections. For budgets below a threshold, we identify an upper-bound

    更新日期:2020-09-01
  • The Double Exponential Runtime is Tight for 2-Stage Stochastic ILPs
    arXiv.cs.DM Pub Date : 2020-08-29
    Klaus Jansen; Kim-Manuel Klein; Alexandra Lassota

    We consider fundamental algorithmic number theoretic problems and their relation to a class of block structured Integer Linear Programs (ILPs) called $2$-stage stochastic. A $2$-stage stochastic ILP is an integer program of the form $\min \{w^T x \mid \mathcal{A} x = b, L \leq x \leq U, x \in \mathbb{Z}^{s + nt} \}$ where the constraint matrix $\mathcal{A} \in \mathbb{Z}^{r n \times s +nt}$ consists

    更新日期:2020-09-01
  • The complexity of L(p,q)-Edge-Labelling
    arXiv.cs.DM Pub Date : 2020-08-27
    Gaetan Berthe; Barnaby Martin; Daniel Paulusma; Siani Smith

    We classify the complexity of L(p,q)-Edge-k-Labelling in the sense that for all positive integers p and q we exhibit k so that we can show L(p,q)-Edge-k-Labelling is NP-complete.

    更新日期:2020-08-28
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