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Characteristic Power Series of Graph LimitsJun 13 2019In this note, we show how to obtain a ``characteristic power series'' of graphons -- infinite limits of graphs -- as the limit of normalized reciprocal characteristic polynomials. This leads to a characterization of graph quasi-randomness and another ... More

The rank of sparse random matricesJun 13 2019Generalising prior work on the rank of random matrices over finite fields [Coja-Oghlan and Gao 2018], we determine the rank of a random matrix with prescribed numbers of non-zero entries in each row and column over any field. The rank formula turns out ... More

Decremental Optimization of Dominating Sets Under Reachability ConstraintsJun 12 2019Given a dominating set, how much smaller a dominating set can we find through elementary operations? Here, we proceed by iterative vertex addition and removal while maintaining the property that the set forms a dominating set of bounded size. This can ... More

On the Universal Near-Shortest Simple Paths ProblemJun 12 2019This article generalizes the Near-Shortest Paths Problem introduced by Byers and Waterman in 1984 using concepts of the Universal Shortest Path Problem established by Turner and Hamacher in 2011. The generalization covers a variety of shortest path problems ... More

Broadcasts on Paths and CyclesJun 12 2019A broadcast on a graph $G=(V,E)$ is a function $f: V\longrightarrow \{0,\ldots,\operatorname{diam}(G)\}$ such that $f(v)\leq e_G(v)$ for every vertex $v\in V$, where$\operatorname{diam}(G)$ denotes the diameter of $G$ and $e_G(v)$ the eccentricity of ... More

Relative Hausdorff Distance for Network AnalysisJun 12 2019Similarity measures are used extensively in machine learning and data science algorithms. The newly proposed graph Relative Hausdorff (RH) distance is a lightweight yet nuanced similarity measure for quantifying the closeness of two graphs. In this work ... More

A mixed-integer linear programming approach for soft graph clusteringJun 11 2019This paper proposes a Mixed-Integer Linear Programming approach for the Soft Graph Clustering Problem. This is the first method that simultaneously allocates membership proportion for vertices that lie in multiple clusters, and that enforces an equal ... More

A Linear Algorithm for Minimum Dominator Colorings of Orientations of PathsJun 11 2019In this paper we present an algorithm for finding a minimum dominator coloring of orientations of paths. To date this is the first algorithm for dominator colorings of digraphs in any capacity. We prove that the algorithm always provides a minimum dominator ... More

Two-dimensional partial cubesJun 11 2019Jun 12 2019We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube ... More

Two-dimensional partial cubesJun 11 2019We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube ... More

A quantum walk with both a continuous-time and a continuous-spacetime limitJun 11 2019Nowadays, quantum simulation schemes come in two flavours. Either they are continuous-time discrete-space models (a.k.a Hamiltonian-based), pertaining to non-relativistic quantum mechanics. Or they are discrete-spacetime models (a.k.a Quantum Walks or ... More

Efficient enumeration of non-isomorphic interval graphsJun 10 2019Recently, Yamazaki et al. provided an algorithm that enumerates all non-isomorphic interval graphs on $n$ vertices with an $O(n^6)$ time delay. In this paper, we improve their algorithm and achieve $O(n^3 \log n)$ time delay. We also extend the catalog ... More

Big Ramsey degrees of 3-uniform hypergraphsJun 10 2019Given a countably infinite hypergraph $\mathcal R$ and a finite hypergraph $\mathcal A$, the big Ramsey degree of $\mathcal A$ in $\mathcal R$ is the least number $L$ such that, for every finite $k$ and every $k$-colouring of the embeddings of $\mathcal ... More

Symmetry Properties of Nested Canalyzing FunctionsJun 10 2019Many researchers have studied symmetry properties of various Boolean functions. A class of Boolean functions, called nested canalyzing functions (NCFs), has been used to model certain biological phenomena. We identify some interesting relationships between ... More

Borders, Palindrome Prefixes, and Square PrefixesJun 09 2019We show that the number of length-$n$ words over a $k$-letter alphabet having no even palindromic prefix is the same as the number of length-$n$ unbordered words, by constructing an explicit bijection between the two sets. A similar result holds for those ... More

An Efficient Characterization of Submodular Spanning Tree GamesJun 06 2019Cooperative games are an important class of problems in game theory, where the goal is to distribute a value among a set of players who are allowed to cooperate by forming coalitions. An outcome of the game is given by an allocation vector that assigns ... More

Ihara Zeta EntropyJun 06 2019In this article, we introduce an entropy based on the formal power series expansion of the Ihara Zeta function. We find a number of inequalities based on the values of the Ihara zeta function. These new entropies are applicable in symbolic dynamics and ... More

On Colourability of Polygon Visibility GraphsJun 05 2019We study the problem of colouring visibility graphs of polygons. In particular, for visibility graphs of simple polygons, we provide a polynomial algorithm for 4-colouring, and prove that the 5-colourability question is already NP-complete for them. For ... More

Motivo: fast motif counting via succinct color coding and adaptive samplingJun 04 2019The randomized technique of color coding is behind state-of-the-art algorithms for estimating graph motif counts. Those algorithms, however, are not yet capable of scaling well to very large graphs with billions of edges. In this paper we develop novel ... More

Embedded hyper-parameter tuning by Simulated AnnealingJun 04 2019We propose a new metaheuristic training scheme that combines Stochastic Gradient Descent (SGD) and Discrete Optimization in an unconventional way. Our idea is to define a discrete neighborhood of the current SGD point containing a number of "potentially ... More

Characteristic Parameters and Special Trapezoidal WordsJun 04 2019Following earlier work by Aldo de Luca and others, we study trapezoidal words and their prefixes, with respect to their characteristic parameters $K$ and $R$ (length of shortest unrepeated suffix, and shortest length without right special factors, respectively), ... More

A deterministic algorithm for counting colorings with $2Δ$ colorsJun 04 2019Jun 06 2019We give a polynomial time deterministic approximation algorithm (an FPTAS) for counting the number of $q$-colorings of a graph of maximum degree $\Delta$, provided only that $q\ge 2\Delta$. This substantially improves on previous deterministic algorithms ... More

A deterministic algorithm for counting colorings with $2Δ$ colorsJun 04 2019We give a polynomial time deterministic approximation algorithm (an FPTAS) for counting the number of $q$-colorings of a graph of maximum degree $\Delta$, provided only that $q\ge 2\Delta$. This substantially improves on previous deterministic algorithms ... More

Probabilistic Existence Results for Parent-Identifying SchemesJun 03 2019Parent-identifying schemes provide a way to identify causes from effects for some information systems such as digital fingerprinting and group testing. In this paper, we consider combinatorial structures for parent-identifying schemes. First, we establish ... More

Generalizations of $k$-Weisfeiler-Leman partitions and related graph invariantsJun 03 2019The family of Weisfeiler-Leman equivalences on graphs is a widely studied approximation of graph isomorphism with many different characterizations. We give a generalization of these by adding a width parameter $r$ and show that these can be linked to ... More

Multistage Vertex CoverJun 03 2019Covering all edges of a graph by a minimum number of vertices, this is the NP-hard Vertex Cover problem, is among the most fundamental algorithmic tasks. Following a recent trend in studying dynamic and temporal graphs, we initiate the study of Multistage ... More

Circularly compatible ones, $D$-circularity, and proper circular-arc bigraphsJun 02 2019In 1969, Alan Tucker characterized proper circular-arc graphs as those graphs whose augmented adjacency matrices have the circularly compatible ones property. Moreover, he also found a polynomial-time algorithm for deciding whether any given augmented ... More

Lower Bounds for Small Ramsey Numbers on HypergraphsJun 01 2019The Ramsey number $r_k(p, q)$ is the smallest integer $N$ that satisfies for every red-blue coloring on $k$-subsets of $[N]$, there exist $p$ integers such that any $k$-subset of them is red, or $q$ integers such that any $k$-subset of them is blue. In ... More

Budget Minimization with Precedence ConstraintsMay 31 2019Budget Minimization is a scheduling problem with precedence constraints, i.e., a scheduling problem on a partially ordered set of jobs $(N, \unlhd)$. A job $j \in N$ is available for scheduling, if all jobs $i \in N$ with $i \unlhd j$ are completed. Further, ... More

The cut metric for probability distributionsMay 31 2019Guided by the theory of graph limits, we investigate a variant of the cut metric for limit objects of sequences of discrete probability distributions. Apart from establishing basic results, we introduce a natural operation called {\em pinning} on the ... More

Concurrency in Boolean networksMay 31 2019Boolean networks (BNs) are widely used to model the qualitative dynamics of biological systems. Besides the logical rules determining the evolution of each component with respect to the state of its regulators, the scheduling of component updates can ... More

More on Numbers and GraphsMay 31 2019In this note we revisit a "ring of graphs" Q in which the set of finite simple graphs N extend the role of the natural numbers and the signed graphs Z extend the role of the integers. We point out the existence of a norm which allows to complete Q to ... More

Cospectral Bipartite Graphs with the Same Degree Sequences but with Different Number of Large CyclesMay 31 2019Finding the multiplicity of cycles in bipartite graphs is a fundamental problem of interest in many fields including the analysis and design of low-density parity-check (LDPC) codes. Recently, Blake and Lin computed the number of shortest cycles ($g$-cycles, ... More

Parametrised Algorithms for Directed Modular WidthMay 30 2019Many well-known NP-hard algorithmic problems on directed graphs resist efficient parametrisations with most known width measures for directed graphs, such as directed treewidth, DAG-width, Kelly-width and many others. While these focus on measuring how ... More

Oriented coloring of graphs with low maximum degreeMay 29 2019Duffy et al. [C. Duffy, G. MacGillivray, and \'E. Sopena, Oriented colourings of graphs with maximum degree three and four, Discrete Mathematics, 342(4), p. 959--974, 2019] recently considered the oriented chromatic number of connected oriented graphs ... More

On the Clique-Width of UnigraphsMay 29 2019Clique-width is a well-studied graph parameter. For graphs of bounded clique-width, many problems that are NP-hard in general can be polynomial-time solvable. The fact motivates many studies to investigate whether the clique-width of graphs in a certain ... More

Minimizing approximately submodular functionsMay 29 2019The problem of minimizing a submodular function is well studied; several polynomial-time algorithms have been developed to solve it exactly or up to arbitrary accuracy. However, in many applications, the objective functions are not exactly submodular. ... More

Sketch-based Randomized Algorithms for Dynamic Graph RegressionMay 28 2019A well-known problem in data science and machine learning is {\em linear regression}, which is recently extended to dynamic graphs. Existing exact algorithms for updating the solution of dynamic graph regression problem require at least a linear time ... More

Sketch-based Randomized Algorithms for Dynamic Graph RegressionMay 28 2019Jun 04 2019A well-known problem in data science and machine learning is {\em linear regression}, which is recently extended to dynamic graphs. Existing exact algorithms for updating the solution of dynamic graph regression problem require at least a linear time ... More

Average Bias and Polynomial SourcesMay 28 2019We identify a new notion of pseudorandomness for randomness sources, which we call the average bias. Given a distribution $Z$ over $\{0,1\}^n$, its average bias is: $b_{\text{av}}(Z) =2^{-n} \sum_{c \in \{0,1\}^n} |\mathbb{E}_{z \sim Z}(-1)^{\langle c, ... More

Average Bias and Polynomial SourcesMay 28 2019May 30 2019We identify a new notion of pseudorandomness for randomness sources, which we call the average bias. Given a distribution $Z$ over $\{0,1\}^n$, its average bias is: $b_{\text{av}}(Z) =2^{-n} \sum_{c \in \{0,1\}^n} |\mathbb{E}_{z \sim Z}(-1)^{\langle c, ... More

On the orthogonal arrays of parameters OA(1536,13,2,7) and relatedMay 27 2019With a computer-aided approach based on the connection with equitable partitions, we establish the uniqueness of the orthogonal array OA$(1536,13,2,7)$ (or, in a different notation, OA$_{12}(7,13,2))$, constructed in [D.G.Fon-Der-Flaass. Perfect $2$-Colorings ... More

Hierarchy of Transportation Network Parameters and Hardness ResultsMay 27 2019The graph parameters highway dimension and skeleton dimension were introduced to capture the properties of transportation networks. As many important optimization problems like Travelling Salesperson, Steiner Tree or $k$-Center arise in such networks, ... More

Spanning eulerian subdigraphs in semicomplete digraphsMay 27 2019A digraph is eulerian if it is connected and every vertex has its in-degree equal to its out-degree. Having a spanning eulerian subdigraph is thus a weakening of having a hamiltonian cycle. In this paper, we first characterize the pairs $(D,a)$ of a semicomplete ... More

Taxonomization of Combinatorial Optimization Problems in Fourier SpaceMay 26 2019We propose and develop a novel framework for analyzing permutation-based combinatorial optimization problems, which could eventually be extended to other types of problems. Our approach is based on the decomposition of the objective functions via the ... More

Evacuating Two Robots from a Disk: A Second CutMay 25 2019We present an improved algorithm for the problem of evacuating two robots from the unit disk via an unknown exit on the boundary. Robots start at the center of the disk, move at unit speed, and can only communicate locally. Our algorithm improves previous ... More

Properly colored $C_{4}$'s in edge-colored graphsMay 25 2019May 29 2019When many colors appear in edge-colored graphs, it is only natural to expect rainbow subgraphs to appear. This anti-Ramsey problem has been studied thoroughly and yet there remain many gaps in the literature. Expanding upon classical and recent results ... More

Properly colored $C_{4}$'s in edge-colored graphsMay 25 2019When many colors appear in edge-colored graphs, it is only natural to expect rainbow subgraphs to appear. This anti-Ramsey problem has been studied thoroughly and yet there remain many gaps in the literature. Expanding upon classical and recent results ... More

Semi-Supervised Classification on Non-Sparse Graphs Using Low-Rank Graph Convolutional NetworksMay 24 2019Graph Convolutional Networks (GCNs) have proven to be successful tools for semi-supervised learning on graph-based datasets. For sparse graphs, linear and polynomial filter functions have yielded impressive results. For large non-sparse graphs, however, ... More

Testing Graphs against an Unknown DistributionMay 23 2019The area of graph property testing seeks to understand the relation between the global properties of a graph and its local statistics. In the classical model, the local statistics of a graph is defined relative to a uniform distribution over the graph's ... More

On the Critical Difference of Almost Bipartite GraphsMay 22 2019A set $S\subseteq V$ is \textit{independent} in a graph $G=\left( V,E\right) $ if no two vertices from $S$ are adjacent. The \textit{independence number} $\alpha(G)$ is the cardinality of a maximum independent set, while $\mu(G)$ is the size of a maximum ... More

A Hypergraph Based Approach for the 4-Constraint Satisfaction Problem TractabilityMay 22 2019Constraint Satisfaction Problem (CSP) is a framework for modeling and solving a variety of real-world problems. Once the problem is expressed as a finite set of constraints, the goal is to find the variables' values satisfying them. Even though the problem ... More

The Steiner triple systems of order 21 with a transversal subdesign TD(3,6)May 22 2019May 23 2019We prove several structural properties of Steiner triple systems (STS) of order 3w+3 that include one or more transversal subdesigns TD(3,w). Using an exhaustive search, we find that there are 2004720 isomorphism classes of STS(21) including a subdesign ... More

The Steiner triple systems of order 21 with a transversal subdesign TD(3,6)May 22 2019We prove several structural properties of Steiner triple systems (STS) of order 3w+3 that include one or more transversal subdesigns TD(3,w). Using an exhaustive search, we find that there are 2004720 isomorphism classes of STS(21) including a subdesign ... More

On Wormald's differential equation methodMay 22 2019This note contains a short and simple proof of Wormald's differential equation method (that yields slightly improved approximation guarantees and error probabilities). Intuitively, this method uses differential equations to approximate the time-evolution/dynamics ... More

An Optimal Monotone Contention Resolution Scheme for Bipartite Matchings via a Polyhedral ViewpointMay 21 2019Relaxation and rounding approaches became a standard and extremely versatile tool for constrained submodular function maximization. One of the most common rounding techniques in this context are contention resolution schemes. Such schemes round a fractional ... More

Fast Computing the Algebraic Degree of Boolean FunctionsMay 21 2019Here we consider an approach for fast computing the algebraic degree of Boolean functions. It combines fast computing the ANF (known as ANF transform) and thereafter the algebraic degree by using the weight-lexicographic order (WLO) of the vectors of ... More

Shortest-Path-Preserving RoundingMay 21 2019Various applications of graphs, in particular applications related to finding shortest paths, naturally get inputs with real weights on the edges. However, for algorithmic or visualization reasons, inputs with integer weights would often be preferable ... More

Approximation results for makespan minimization with budgeted uncertaintyMay 21 2019We study approximation algorithms for the problem of minimizing the makespan on a set of machines with uncertainty on the processing times of jobs. In the model we consider, which goes back to~\cite{BertsimasS03}, once the schedule is defined an adversary ... More

A new proof on the Ramsey number of matchingsMay 21 2019For given simple graphs $H_{1},H_{2},\ldots,H_{c}$, the Ramsey number $r(H_{1},H_{2},\ldots,H_{c})$ is the smallest positive integer $n$ such that every edge-colored $K_{n}$ with $c$ colors contains a subgraph isomorphic to $H_{i}$ in color $i$ for some ... More

Inference under Information Constraints II: Communication Constraints and Shared RandomnessMay 20 2019A central server needs to perform statistical inference based on samples that are distributed over multiple users who can each send a message of limited length to the center. We study problems of distribution learning and identity testing in this distributed ... More

Random Walks on Hypergraphs with Edge-Dependent Vertex WeightsMay 20 2019Hypergraphs are used in machine learning to model higher-order relationships in data. While spectral methods for graphs are well-established, spectral theory for hypergraphs remains an active area of research. In this paper, we use random walks to develop ... More

Sharp results concerning disjoint cross-intersecting familiesMay 20 2019For an $n$-element set $X$ let $\binom{X}{k}$ be the collection of all its $k$-subsets. Two families of sets $\mathcal A$ and $\mathcal B$ are called cross-intersecting if $A\cap B \neq \emptyset$ holds for all $A\in\mathcal A$, $B\in\mathcal B$. Let ... More

About a 'concrete' Rauszer Boolean algebra generated by a preorderMay 19 2019Inspired by the fundamental results obtained by P. Halmos and A. Monteiro, concerning equivalence relations and monadic Boolean algebras, we recall the `concrete' Rauszer Boolean algebra pointed out by C. Rauszer (1971), via un preorder R. On this algebra ... More

Rough sets and three-valued structuresMay 19 2019In recent years, many papers have been published showing relationships between rough sets and some lattice theoretical structures. We present here some strong relations between rough sets and three-valued {\L}ukasiewicz algebras.

Shortest Path Algorithms between Theory and PracticeMay 17 2019Utilizing graph algorithms is a common activity in computer science. Algorithms that perform computations on large graphs are not always efficient. This work investigates the Single-Source Shortest Path (SSSP) problem, which is considered to be one of ... More

Daisy cubes: a characterization and a generalizationMay 17 2019Daisy cubes are a recently introduced class of isometric subgraphs of hypercubes $Q_n$. They are induced with intervals between chosen vertices of $Q_n$ and the vertex $0^n\in V(Q_n)$. In this paper we characterize daisy cubes in terms of an expansion ... More

Perfect Italian domination on planar and regular graphsMay 15 2019A perfect Italian dominating function of a graph $G=(V,E)$ is a function $f : V \to \{0,1,2\}$ such that for every vertex $f(v) = 0$, it holds that $\sum_{u \in N(v)} f(u) = 2$, i.e., the weight of the labels assigned by $f$ to the neighbors of $v$ is ... More

Strong chromatic index and Hadwiger numberMay 15 2019We investigate the effect of a fixed forbidden clique minor upon the strong chromatic index, both in multigraphs and in simple graphs. We conjecture for each $k\ge 4$ that any $K_k$-minor-free multigraph of maximum degree $\Delta$ has strong chromatic ... More

Finding Dominating Induced Matchings in $S_{1,1,5}$-Free Graphs in Polynomial TimeMay 14 2019Let $G=(V,E)$ 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}) ... More

An analytical bound on the fleet size in vehicle routing problems: a dynamic programming approachMay 14 2019We present an analytical upper bound on the number of required vehicles for vehicle routing problems with split deliveries and any number of capacitated depots. We show that a fleet size greater than the proposed bound is not achievable based on a set ... More

Generating Weighted MAX-2-SAT Instances of Tunable Difficulty with Frustrated LoopsMay 14 2019Many optimization problems can be cast into the maximum satisfiability (MAX-SAT) form, and many solvers have been developed for tackling such problems. To evaluate the performance of a MAX-SAT solver, it is convenient to generate difficult MAX-SAT instances ... More

Computing Maximum Matchings in Temporal GraphsMay 13 2019We study the computational complexity of finding maximum-cardinality temporal matchings in temporal graphs (where the edge set may change over time while the vertex set remains fixed). Our model of temporal matching (which seems to be slightly more general ... More

Transtemporal edges and crosslayer edges in incompressible high-order networksMay 13 2019This work presents some outcomes of a theoretical investigation of incompressible high-order networks defined by a generalized graph representation. We study some of their network topological properties and how these may be related to real-world complex ... More

Counting and sampling gene family evolutionary histories in the duplication-loss and duplication-loss-transfer modelsMay 13 2019Given a set of species whose evolution is represented by a species tree, a gene family is a group of genes having evolved from a single ancestral gene. A gene family evolves along the branches of a species tree through various mechanisms, including - ... More

Dehn-Sommerville from Gauss-BonnetMay 13 2019We give a zero curvature proof of Dehn-Sommerville for finite simple graphs. It uses a parametrized Gauss-Bonnet formula telling that the curvature of the valuation G to f_G(t)=1+f0 t + ... + fd t^(d+1) defined by the f-vector of G is the anti-derivative ... More

Satisfiability Threshold for Power Law Random 2-SAT in Configuration ModelMay 13 2019The Random Satisfiability problem has been intensively studied for decades. For a number of reasons the focus of this study has mostly been on the model, in which instances are sampled uniformly at random from a set of formulas satisfying some clear conditions, ... More

Generalized Lyndon Factorizations of Infinite WordsMay 12 2019A generalized lexicographic order on words is a lexicographic order where the total order of the alphabet depends on the position of the comparison. A generalized Lyndon word is a finite word which is strictly smallest among its class of rotations with ... More

Routing and Scheduling of Network Flows with Deadlines and Discrete Capacity AllocationMay 12 2019Joint scheduling and routing of data flows with deadline constraints in communication networks has been attracting research interest. This type of problem distinguishes from conventional multicommodity flows due to the presence of the time dimension. ... More

Complexity of fall coloring for restricted graph classesMay 12 2019We strengthen a result by Laskar and Lyle (Discrete Appl. Math. (2009), 330-338) by proving that it is NP-complete to decide whether a bipartite planar graph can be partitioned into three independent dominating sets. In contrast, we show that this is ... More

Triangle-creation processes on cubic graphsMay 11 2019An edge switch is an operation which makes a local change in a graph while maintaining the degree of every vertex. We introduce a switch move, called a triangle switch, which creates or deletes at least one triangle. Specifically, a make move is a triangle ... More

Group Fairness in Committee SelectionMay 11 2019In this paper, we study fairness in committee selection problems. We consider a general notion of fairness via stability: A committee is stable if no coalition of voters can deviate and choose a committee of proportional size, so that all these voters ... More

Low-Complexity Tilings of the PlaneMay 10 2019A two-dimensional configuration is a coloring of the infinite grid Z^2 with finitely many colors. For a finite subset D of Z^2, the D-patterns of a configuration are the colored patterns of shape D that appear in the configuration. The number of distinct ... More

Fast delta evaluation for the Vehicle Routing Problem with Multiple Time WindowsMay 10 2019In many applications of vehicle routing, a set of time windows are feasible for each visit, giving rise to the Vehicle Routing Problem with Multiple Time Windows (VRPMTW). We argue that such disjunctions are problematic for many solution methods, and ... More

Robustness: a New Form of Heredity Motivated by Dynamic NetworksMay 10 2019We investigate a special case of hereditary property in graphs, referred to as {\em robustness}. A property (or structure) is called robust in a graph $G$ if it is inherited by all the connected spanning subgraphs of $G$. We motivate this definition using ... More

Linear Work Generation of R-MAT GraphsMay 09 2019R-MAT is a simple, widely used recursive model for generating `complex network' graphs with a power law degree distribution and community structure. We make R-MAT even more useful by reducing the required work per edge from logarithmic to constant. The ... More

Learning Erdős-Rényi Random Graphs via Edge Detecting QueriesMay 09 2019May 11 2019In this paper, we consider the problem of learning an unknown graph via queries on groups of nodes, with the result indicating whether or not at least one edge is present among those nodes. While learning arbitrary graphs with $n$ nodes and $k$ edges ... More

Learning Erdős-Rényi Random Graphs via Edge Detecting QueriesMay 09 2019In this paper, we consider the problem of learning an unknown graph via queries on groups of nodes, with the result indicating whether or not at least one edge is present among those nodes. While learning arbitrary graphs with $n$ nodes and $k$ edges ... More

On the Approximate Compressibility of Connected Vertex CoverMay 08 2019The Connected Vertex Cover problem, where the goal is to compute a minimum set of vertices in a given graph which forms a vertex cover and induces a connected subgraph, is a fundamental combinatorial problem and has received extensive attention in various ... More

Zeros and approximations of Holant polynomials on the complex planeMay 08 2019We present fully polynomial approximation schemes for general classes of Holant problems with complex edge weights, which we call Holant polynomials. Our results are based on a recent technique for approximating graph polynomials of degree $n$ by computing ... More

Switches in Eulerian graphsMay 08 2019We show that the graph transformation problem of turning a simple graph into an Eulerian one by a minimum number of single edge switches is NP-hard. Further, we show that any simple Eulerian graph can be transformed into any other such graph by a sequence ... More

Walk refinement, walk logic, and the iteration number of the Weisfeiler-Leman algorithmhttps://arxiv.org/submit/2681062/previewMay 08 2019We show that the 2-dimensional Weisfeiler-Leman algorithm stabilizes n-vertex graphs after at most O(n log n) iterations. This implies that if such graphs are distinguishable in 3-variable first order logic with counting, then they can also be distinguished ... More

Walk refinement, walk logic, and the iteration number of the Weisfeiler-Leman algorithmMay 08 2019May 09 2019We show that the 2-dimensional Weisfeiler-Leman algorithm stabilizes n-vertex graphs after at most O(n log n) iterations. This implies that if such graphs are distinguishable in 3-variable first order logic with counting, then they can also be distinguished ... More

Target Set in Threshold ModelsMay 08 2019Consider a graph $G$ and an initial coloring, where each node is blue or red. In each round, all nodes simultaneously update their color based on a predefined rule. In a threshold model, a node becomes blue if a certain number or fraction of its neighbors ... More

On the semi-proper orientations of graphsMay 08 2019A {\it semi-proper orientation} of a given graph $G$ is a function $(D,w)$ that assigns an orientation $D(e)$ and a positive integer weight $ w(e)$ to each edge $e$ such that for every two adjacent vertices $v$ and $u$, $S_{(D,w)}(v) \neq S_{(D,w)}(u) ... More

Max-Cut in Degenerate $H$-Free GraphsMay 08 2019We obtain several lower bounds on the $\textsf{Max-Cut}$ of $d$-degenerate $H$-free graphs. Let $f(m,d,H)$ denote the smallest $\textsf{Max-Cut}$ of an $H$-free $d$-degenerate graph on $m$ edges. We show that $f(m,d,K_r)\ge \left(\frac{1}{2} + d^{-1+\Omega(r^{-1})}\right)m$, ... More

Max-Cut in Degenerate $H$-Free GraphsMay 08 2019May 14 2019We obtain several lower bounds on the $\textsf{Max-Cut}$ of $d$-degenerate $H$-free graphs. Let $f(m,d,H)$ denote the smallest $\textsf{Max-Cut}$ of an $H$-free $d$-degenerate graph on $m$ edges. We show that $f(m,d,K_r)\ge \left(\frac{1}{2} + d^{-1+\Omega(r^{-1})}\right)m$, ... More

Kendall Tau Sequence Distance: Extending Kendall Tau from Ranks to SequencesMay 07 2019An edit distance is a measure of the minimum cost sequence of edit operations to transform one structure into another. Edit distance is most commonly encountered within the context of strings, where Wagner and Fischer's string edit distance is perhaps ... More

The structure of graphs with given number of blocks and the maximum Wiener indexMay 07 2019The Wiener index (the distance) of a connected graph is the sum of distances between all pairs of vertices. In this paper, we study the maximum possible value of this invariant among graphs on $n$ vertices with fixed number of blocks $p$. It is known ... More

Graphs with the second and third maximum Wiener index over the 2-vertex connected graphsMay 07 2019Wiener index, defined as the sum of distances between all unordered pairs of vertices, is one of the most popular molecular descriptors. It is well known that among 2-vertex connected graphs on $n\ge 3$ vertices, the cycle $C_n$ attains the maximum value ... More