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Ranking Bracelets in Polynomial Time arXiv.cs.DM Pub Date : 2021-04-09 Duncan Adamson, Argyrios Deligkas, Vladimir V. Gusev, Igor Potapov
The main result of the paper is the first polynomial-time algorithm for ranking bracelets. The time-complexity of the algorithm is O(k^2 n^4), where k is the size of the alphabet and n is the length of the considered bracelets. The key part of the algorithm is to compute the rank of any word with respect to the set of bracelets by finding three other ranks: the rank over all necklaces, the rank over
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Intersection models and forbidden pattern characterizations for 2-thin and proper 2-thin graphs arXiv.cs.DM Pub Date : 2021-04-08 Flavia Bonomo-Braberman, Gastón Abel Brito
The thinness of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Graphs with thinness at most two include, for example, bipartite convex graphs. Many NP-complete problems can be solved in polynomial time for graphs with bounded thinness, given a suitable representation of the graph. Proper thinness is defined analogously
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Eternal k-domination on graphs arXiv.cs.DM Pub Date : 2021-04-08 Danielle Cox, Erin Meger, M. E. Messinger
Eternal domination is a dynamic process by which a graph is protected from an infinite sequence of vertex intrusions. In eternal $k$-domination, guards initially occupy the vertices of a $k$-dominating set. After a vertex is attacked, guards "defend" by each move up to distance $k$ to form a $k$-dominating set containing the attacked vertex. The eternal $k$-domination number of a graph is the minimum
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A Cycle Joining Construction of the Prefer-Max De Bruijn Sequence arXiv.cs.DM Pub Date : 2021-04-07 Gal Amram, Amir Rubin, Gera Weiss
We propose a novel construction for the well-known prefer-max De Bruijn sequence, based on the cycle joining technique. We further show that the construction implies known results from the literature in a straightforward manner. First, it implies the correctness of the onion theorem, stating that, effectively, the reverse of prefer-max is in fact an infinite De Bruijn sequence. Second, it implies the
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Exact Algorithms for No-Rainbow Coloring and Phylogenetic Decisiveness arXiv.cs.DM Pub Date : 2021-04-05 Ghazaleh Parvini, David Fernández-Baca
The input to the no-rainbow hypergraph coloring problem is a hypergraph $H$ where every hyperedge has $r$ nodes. The question is whether there exists an $r$-coloring of the nodes of $H$ such that all $r$ colors are used and there is no rainbow hyperedge -- i.e., no hyperedge uses all $r$ colors. The no-rainbow hypergraph $r$-coloring problem is known to be NP-complete for $r \geq 3$. The special case
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Destroying Multicolored Paths and Cycles in Edge-Colored Graphs arXiv.cs.DM Pub Date : 2021-04-07 Nils Jakob Eckstein, Niels Grüttemeier, Christian Komusiewicz, Frank Sommer
We study the computational complexity of $c$-Colored $P_\ell$ Deletion and $c$-Colored $C_\ell$ Deletion. In these problems, one is given a $c$-edge-colored graph and wants to destroy all induced $c$-colored paths or cycles, respectively, on $\ell$ vertices by deleting at most $k$ edges. Herein, a path or cycle is $c$-colored if it contains edges of $c$ distinct colors. We show that $c$-Colored $P_\ell$
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Two-Stage Robust Optimization Problems with Two-Stage Uncertainty arXiv.cs.DM Pub Date : 2021-04-07 Marc Goerigk, Stefan Lendl, Lasse Wulf
We consider robust two-stage optimization problems, which can be considered as a game between the decision maker and an adversary. After the decision maker fixes part of the solution, the adversary chooses a scenario from a specified uncertainty set. Afterwards, the decision maker can react to this scenario by completing the partial first-stage solution to a full solution. We extend this classic setting
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Parameterized Complexity of Elimination Distance to First-Order Logic Properties arXiv.cs.DM Pub Date : 2021-04-07 Fedor V. Fomin, Petr A. Golovach, Dimitrios M. Thilikos
The elimination distance to some target graph property P is a general graph modification parameter introduced by Bulian and Dawar. We initiate the study of elimination distances to graph properties expressible in first-order logic. We delimit the problem's fixed-parameter tractability by identifying sufficient and necessary conditions on the structure of prefixes of first-order logic formulas. Our
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Coloring graph classes with no induced fork via perfect divisibility arXiv.cs.DM Pub Date : 2021-04-06 T. Karthick, Jenny Kaufmann, Vaidy Sivaraman
For a graph $G$, $\chi(G)$ will denote its chromatic number, and $\omega(G)$ its clique number. A graph $G$ is said to be perfectly divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A$, $B$ such that $H[A]$ is perfect and $\omega(H[B]) < \omega(H)$. An integer-valued function $f$ is called a $\chi$-binding function for a hereditary class of graphs $\cal C$
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Upper paired domination versus upper domination arXiv.cs.DM Pub Date : 2021-04-06 Hadi Alizadeh, Didem Gözüpek
A paired dominating set $P$ is a dominating set with the additional property that $P$ has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph $G$ is called the upper domination number of $G$, denoted by $\Gamma(G)$, the maximum cardinality of a minimal paired dominating set in $G$ is called the upper paired domination number of $G$, denoted by $\Gamma_{pr}(G)$
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Matrix Chain Multiplication and Polygon Triangulation Revisited and Generalized arXiv.cs.DM Pub Date : 2021-04-05 Thong Le, Dan Gusfield
The {\it matrix-chain multiplication} problem is a classic problem that is widely taught to illustrate dynamic programming. The textbook solution runs in $\theta(n^3)$ time. However, there is a complex $O(n \log n)$-time method \cite{HU82}, based on triangulating convex polygons, and a description without proofs or implementation detail, of a much simpler $O(n^2)$-time method \cite{YAO82}. There is
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A strongly universal cellular automaton in the dodecagrid with five states arXiv.cs.DM Pub Date : 2021-04-04 Maurice Margenstern
In this paper, we prove that there is a strongly universal cellular automaton in the dodecagrid, the tessellation {5,3,4} of the hyperbolic 3D-space, with five states which is rotation invariant. This improves a previous paper of the author where the automaton required ten states.
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Constraint Programming to Discover One-Flip Local Optima of Quadratic Unconstrained Binary Optimization Problems arXiv.cs.DM Pub Date : 2021-04-04 Amit Verma, Mark Lewis
The broad applicability of Quadratic Unconstrained Binary Optimization (QUBO) constitutes a general-purpose modeling framework for combinatorial optimization problems and are a required format for gate array and quantum annealing computers. QUBO annealers as well as other solution approaches benefit from starting with a diverse set of solutions with local optimality an additional benefit. This paper
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Extremal Graphs for a Spectral Inequality on Edge-Disjoint Spanning Trees arXiv.cs.DM Pub Date : 2021-04-04 Sebastian M. Cioabă, Anthony Ostuni, Davin Park, Sriya Potluri, Tanay Wakhare, Wiseley Wong
Liu, Hong, Gu, and Lai proved if the second largest eigenvalue of the adjacency matrix of graph $G$ with minimum degree $\delta \ge 2m+2 \ge 4$ satisfies $\lambda_2(G) < \delta - \frac{2m+1}{\delta+1}$, then $G$ contains at least $m+1$ edge-disjoint spanning trees, which verified a generalization of a conjecture by Cioab\u{a} and Wong. We show this bound is essentially the best possible by constructing
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A Note on Consistent Rotation Maps of Graph Cartesian Products arXiv.cs.DM Pub Date : 2021-04-03 Clark Alexander
Given two regular graphs with consistent rotation maps, we produce a constructive method for a consistent rotation map on their Cartesian product. This method will be given as a simple set of rules of addition and table look ups. We assume that the combinatorial construction of both consistent rotation maps has occurred before we construct the Cartesian product.
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Bijections from Dyck and Motzkin meanders with catastrophes to pattern avoiding Dyck paths arXiv.cs.DM Pub Date : 2021-04-02 Jean-Luc Baril, Sergey Kirgizov
In this note, we present constructive bijections from Dyck and Motzkin meanders with catastrophes to Dyck paths avoiding some patterns. As a byproduct, we deduce correspondences from Dyck and Motzkin excursions to restricted Dyck paths.
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Ultra-low memory seismic inversion with randomized trace estimation arXiv.cs.DM Pub Date : 2021-04-01 Mathias Louboutin, Felix J. Herrmann
Inspired by recent work on extended image volumes that lays the ground for randomized probing of extremely large seismic wavefield matrices, we present a memory frugal and computationally efficient inversion methodology that uses techniques from randomized linear algebra. By means of a carefully selected realistic synthetic example, we demonstrate that we are capable of achieving competitive inversion
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Using Graph Theory to Derive Inequalities for the Bell Numbers arXiv.cs.DM Pub Date : 2021-04-01 Alain Hertz, Anaelle Hertz, Hadrien Mélot
The Bell numbers count the number of different ways to partition a set of $n$ elements while the graphical Bell numbers count the number of non-equivalent partitions of the vertex set of a graph into stable sets. This relation between graph theory and integer sequences has motivated us to study properties on the average number of colors in the non-equivalent colorings of a graph to discover new non
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Pareto optimal exchange with indifferent endowments arXiv.cs.DM Pub Date : 2021-04-01 Pavlos Eirinakis, Ioannis Mourtos, Michalis Samaris
We investigate a market without money in which agents can offer certain goods (or multiple copies of an agent-specific good) in exchange for goods of other agents. The exchange must be balanced in the sense that each agent should receive a quantity of good(s) equal to the one she transfers to others. In addition, each agent has strict preferences over the agents from which she will receive goods, and
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Hereditary rigidity, separation and density In memory of Professor I.G. Rosenberg arXiv.cs.DM Pub Date : 2021-04-01 Lucien Haddad, Masahiro Miyakawa, Maurice Pouzet, Hisayuki Tatsumi
We continue the investigation of systems of hereditarily rigid relations started in Couceiro, Haddad, Pouzet and Sch\"olzel [1]. We observe that on a set $V$ with $m$ elements, there is a hereditarily rigid set $\mathcal R$ made of $n$ tournaments if and only if $m(m-1)\leq 2^n$. We ask if the same inequality holds when the tournaments are replaced by linear orders. This problem has an equivalent formulation
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Sub-GMN: The Subgraph Matching Network Model arXiv.cs.DM Pub Date : 2021-04-01 Zixun Lan, Limin Yu, Linglong Yuan, Zili Wu, Fei Ma
We propose an end-to-end learning-based approximate method for subgraph matching task, called subgraph matching network (Sub-GMN). First, Sub-GMN uses graph representation learning to map nodes to node-level embedding, and then combines metric learning and attention mechanisms to model the relationship between matched nodes in the data graph and query graph. Compared with the previous GNNs-based method
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On Deeply Critical Oriented Cliques arXiv.cs.DM Pub Date : 2021-03-31 Christopher Duffy, Pavan P D, Sandeep R. B., Sagnik Sen
In this work we consider arc criticality in colourings of oriented graphs. We study deeply critical oriented graphs, those graphs for which the removal of any arc results in a decrease of the oriented chromatic number by $2$. We prove the existence of deeply critical oriented cliques of every odd order $n\geq 9$, closing an open question posed by Borodin et al. (Journal of Combinatorial Theory, Series
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Implicit completeness criterion in three-valued logic in terms of maximal classes arXiv.cs.DM Pub Date : 2021-03-30 Mikhail Starostin
Implicit expressability was introduced by A.V. Kuznetsov in 1979 as generalization of functional expressability. Set of functions is called implicitly complete if any function has an implicit representation over this set. The system of all implicitly maximal classes in three-valued logic is described. The implicit completeness criterion is stated.
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Weak Coloring Numbers of Intersection Graphs arXiv.cs.DM Pub Date : 2021-03-31 Zdeněk Dvořák, Jakub Pekárek, Torsten Ueckerdt, Yelena Yuditsky
Weak and strong coloring numbers are generalizations of the degeneracy of a graph, where for each natural number $k$, we seek a vertex ordering such every vertex can (weakly respectively strongly) reach in $k$ steps only few vertices with lower index in the ordering. Both notions capture the sparsity of a graph or a graph class, and have interesting applications in the structural and algorithmic graph
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Revisiting regular sequences in light of rational base numeration systems arXiv.cs.DM Pub Date : 2021-03-31 Michel Rigo, Manon Stipulanti
Regular sequences generalize the extensively studied automatic sequences. Let $S$ be an abstract numeration system. When the numeration language $L$ is prefix-closed and regular, a sequence is said to be $S$-regular if the module generated by its $S$-kernel is finitely generated. In this paper, we give a new characterization of such sequences in terms of the underlying numeration tree $T(L)$ whose
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Sharp Thresholds for a SIR Model on One-Dimensional Small-World Networks arXiv.cs.DM Pub Date : 2021-03-30 Luca Becchetti, Andrea Clementi, Riccardo Denni, Francesco Pasquale, Luca Trevisan, Isabella Ziccardi
We study epidemic spreading according to a \emph{Susceptible-Infectious-Recovered} (for short, \emph{SIR}) network model known as the {\em Reed-Frost} model, and we establish sharp thresholds for two generative models of {\em one-dimensional small-world graphs}, in which graphs are obtained by adding random edges to a cycle. In $3$-regular graphs obtained as the union of a cycle and a random perfect
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Approximation algorithm for finding short synchronizing words in weighted automata arXiv.cs.DM Pub Date : 2021-03-30 Jakub Ruszil
In this paper we are dealing with the issue of finding possibly short synchronizing words in automata with weight assigned to each letter in the alphabet $\Sigma$. First we discuss some complexity problems, and then we present new approximation algorithm in four variations.
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Hybrid Power-Law Models of Network Traffic arXiv.cs.DM Pub Date : 2021-03-29 Pat Devlin, Jeremy Kepner, Ashley Luo, Erin Meger
The availability of large scale streaming network data has reinforced the ubiquity of power-law distributions in observations and enabled precision measurements of the distribution parameters. The increased accuracy of these measurements allows new underlying generative network models to be explored. The preferential attachment model is a natural starting point for these models. This work adds additional
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An upper bound on the size of Sidon sets arXiv.cs.DM Pub Date : 2021-03-29 József Balogh, Zoltán Füredi, Souktik Roy
Combining two elementary proofs we decrease the gap between the upper and lower bounds by $0.2\%$ in a classical combinatorial number theory problem. We show that the maximum size of a Sidon set of $\{ 1, 2, \ldots, n\}$ is at most $\sqrt{n}+ 0.998n^{1/4}$ for sufficiently large $n$.
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ZX-Calculus and Extended Wolfram Model Systems II: Fast Diagrammatic Reasoning with an Application to Quantum Circuit Simplification arXiv.cs.DM Pub Date : 2021-03-29 Jonathan Gorard, Manojna Namuduri, Xerxes D. Arsiwalla
This article presents a novel algorithmic methodology for performing automated diagrammatic deductions over combinatorial structures, using a combination of modified equational theorem-proving techniques and the extended Wolfram model hypergraph rewriting formalism developed by the authors in previous work. We focus especially upon the application of this new algorithm to the problem of automated circuit
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Computational Complexity of Covering Two-vertex Multigraphs with Semi-edges arXiv.cs.DM Pub Date : 2021-03-28 Jan Bok, Jiří Fiala, Petr Hliněný, Nikola Jedličková, Jan Kratochvíl
We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for {\em graphs with semi-edges}. The notion of graph covering is a discretization of coverings between surfaces or topological spaces, a notion well known and deeply studied in classical topology. Graph covers have found applications in discrete mathematics for constructing highly symmetric
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A Sharp Discrepancy Bound for Jittered Sampling arXiv.cs.DM Pub Date : 2021-03-29 Benjamin Doerr
For $m, d \in {\mathbb N}$, a jittered sampling point set $P$ having $N = m^d$ points in $[0,1)^d$ is constructed by partitioning the unit cube $[0,1)^d$ into $m^d$ axis-aligned cubes of equal size and then placing one point independently and uniformly at random in each cube. We show that there are constants $c \ge 0$ and $C$ such that for all $d$ and all $m \ge d$ the expected non-normalized star
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Mathematics of Digital Hyperspace arXiv.cs.DM Pub Date : 2021-03-28 Jeremy Kepner, Timothy Davis, Vijay Gadepally, Hayden Jananthan, Lauren Milechin
Social media, e-commerce, streaming video, e-mail, cloud documents, web pages, traffic flows, and network packets fill vast digital lakes, rivers, and oceans that we each navigate daily. This digital hyperspace is an amorphous flow of data supported by continuous streams that stretch standard concepts of type and dimension. The unstructured data of digital hyperspace can be elegantly represented, traversed
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On objects dual to tree-cut decompositions arXiv.cs.DM Pub Date : 2021-03-26 Łukasz Bożyk, Oscar Defrain, Karolina Okrasa, Michał Pilipczuk
Tree-cut width is a graph parameter introduced by Wollan that is an analogue of treewidth for the immersion order on graphs in the following sense: the tree-cut width of a graph is functionally equivalent to the largest size of a wall that can be found in it as an immersion. In this work we propose a variant of the definition of tree-cut width that is functionally equivalent to the original one, but
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Some properties of the parking function poset arXiv.cs.DM Pub Date : 2021-03-26 Bérénice Delcroix-OgerIRIF, Matthieu Josuat-VergèsIRIF, Lucas RandazzoLIGM
In 1980, Edelman defined a poset on objects called the noncrossing 2-partitions. They are closely related with noncrossing partitions and parking functions. To some extent, his definition is a precursor of the parking space theory, in the framework of finite reflection groups. We present some enumerative and topological properties of this poset. In particular, we get a formula counting certain chains
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Palindromic Length and Reduction of Powers arXiv.cs.DM Pub Date : 2021-03-26 Josef Rukavicka
Given a nonempty finite word $v$, let $PL(v)$ be the palindromic length of $v$; it means the minimal number of palindromes whose concatenation is equal to $v$. Let $v^R$ denote the reversal of $v$. Given a finite or infinite word $y$, let $Fac(y)$ denote the set of all finite factors of $y$ and let $maxPL(y)=\max\{PL(t)\mid t\in Fac(y)\}$. Let $x$ be an infinite non-ultimately periodic word with $maxPL(x)=k<\infty$
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Minimum Scan Cover and Variants -- Theory and Experiments arXiv.cs.DM Pub Date : 2021-03-26 Kevin Buchin, Sándor P. Fekete, Alexander Hill, Linda Kleist, Irina Kostitsyna, Dominik Krupke, Roel Lambers, Martijn Struijs
We consider a spectrum of geometric optimization problems motivated by contexts such as satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs, 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 need to face each other; changing
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The combinatorial game nofil played on Steiner Triple Systems arXiv.cs.DM Pub Date : 2021-03-24 Melissa A. Huggan, Svenja Huntemann, Brett Stevens
We introduce an impartial combinatorial game on Steiner triple systems called Nofil. Players move alternately, choosing points of the triple system. If a player is forced to fill a block on their turn, they lose. We explore the play of Nofil on all Steiner triple systems up to order 15 and a sampling for orders 19, 21, and 25. We determine the optimal strategies by computing the nim-values for each
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Note on the offspring distribution for group testing in the linear regime arXiv.cs.DM Pub Date : 2021-03-24 Oliver Gebhard, Philipp Loick
The group testing problem is concerned with identifying a small set of $k$ infected individuals in a large population of $n$ people. At our disposal is a testing scheme that can test groups of individuals. A test comes back positive if and only if at least one individual is infected. In this note, we lay groundwork for analysing belief propagation for group testing when $k$ scales linearly in $n$.
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The Multicolored Graph Realization Problem arXiv.cs.DM Pub Date : 2021-03-24 Josep Díaz, Öznur Yaşar Diner, Maria Serna, Oriol Serra
We introduce the Multicolored Graph Realization problem (MGRP). The input to the problem is a colored graph $(G,\varphi)$, i.e., a graph together with a coloring on its vertices. We can associate to each colored graph a cluster graph ($G_\varphi)$ in which, after collapsing to a node all vertices with the same color, we remove multiple edges and self-loops. A set of vertices $S$ is multicolored when
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Combinators: A Centennial View arXiv.cs.DM Pub Date : 2021-03-23 Stephen Wolfram
We give a modern computational introduction to the S,K combinators invented by Moses Sch\"onfinkel in 1920, and present a variety of new results and ideas about combinators. We explore the spectrum of behavior obtained with small combinator expressions, showing a variety of approaches to analysis and visualization. We discuss the implications of evaluation strategies, and of multiway systems representing
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Approximately Counting Answers to Conjunctive Queries with Disequalities and Negations arXiv.cs.DM Pub Date : 2021-03-23 Jacob Focke, Leslie Ann Goldberg, Marc Roth, Stanislav Živný
We study the complexity of approximating the number of answers to a small query $\varphi$ in a large database $\mathcal{D}$. We establish an exhaustive classification into tractable and intractable cases if $\varphi$ is a conjunctive query with disequalities and negations: $\bullet$ If there is a constant bound on the arity of $\varphi$, and if the randomised Exponential Time Hypothesis (rETH) holds
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From Coupling to Spectral Independence and Blackbox Comparison with the Down-Up Walk arXiv.cs.DM Pub Date : 2021-03-22 Kuikui Liu
We show that the existence of a ``good'' coupling w.r.t. Hamming distance for any local Markov chain on a discrete product space implies rapid mixing of the Glauber dynamics in a blackbox fashion. More specifically, we only require the expected distance between successive iterates under the coupling to be summable, as opposed to being one-step contractive in the worst case. Combined with recent local-to-global
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A Markov chain on the solution space of edge-colorings of bipartite graphs arXiv.cs.DM Pub Date : 2021-03-22 Letong Hong, Istvan Miklos
In this paper, we exhibit an irreducible Markov chain $M$ on the edge $k$-colorings of bipartite graphs based on certain properties of the solution space. We show that diameter of this Markov chain grows linearly with the number of edges in the graph. We also prove a polynomial upper bound on the inverse of acceptance ratio of the Metropolis-Hastings algorithm when the algorithm is applied on $M$ with
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Integer and Constraint Programming Revisited for Mutually Orthogonal Latin Squares arXiv.cs.DM Pub Date : 2021-03-19 Noah Rubin, Curtis Bright, Kevin K. H. Cheung, Brett Stevens
In this paper we provide results on using integer programming (IP) and constraint programming (CP) to search for sets of mutually orthogonal latin squares (MOLS). Both programming paradigms have previously successfully been used to search for MOLS, but solvers for IP and CP solvers have significantly improved in recent years and data on how modern IP and CP solvers perform on the MOLS problem is lacking
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Higher-order Homophily is Combinatorially Impossible arXiv.cs.DM Pub Date : 2021-03-22 Nate Veldt, Austin R. Benson, Jon Kleinberg
Homophily is the seemingly ubiquitous tendency for people to connect with similar others, which is fundamental to how society organizes. Even though many social interactions occur in groups, homophily has traditionally been measured from collections of pairwise interactions involving just two individuals. Here, we develop a framework using hypergraphs to quantify homophily from multiway, group interactions
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$2$-distance $(Δ+1)$-coloring of sparse graphs using the potential method arXiv.cs.DM Pub Date : 2021-03-22 Hoang La, Mickael Montassier
A $2$-distance $k$-coloring of a graph is a proper $k$-coloring of the vertices where vertices at distance at most 2 cannot share the same color. We prove the existence of a $2$-distance ($\Delta+1$)-coloring for graphs with maximum average degree less than $\frac{18}{7}$ and maximum degree $\Delta\geq 7$. As a corollary, every planar graph with girth at least $9$ and $\Delta\geq 7$ admits a $2$-distance
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Continuous mean distance of a weighted graph arXiv.cs.DM Pub Date : 2021-03-22 Delia Garijo, Alberto Márquez, Rodrigo I. Silveira
We study the concept of the continuous mean distance of a weighted graph. For connected unweighted graphs, the mean distance can be defined as the arithmetic mean of the distances between all pairs of vertices. This parameter provides a natural measure of the compactness of the graph, and has been intensively studied, together with several variants, including its version for weighted graphs. The continuous
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Fourier Growth of Parity Decision Trees arXiv.cs.DM Pub Date : 2021-03-22 Uma Girish, Avishay Tal, Kewen Wu
We prove that for every parity decision tree of depth $d$ on $n$ variables, the sum of absolute values of Fourier coefficients at level $\ell$ is at most $d^{\ell/2} \cdot O(\ell \cdot \log(n))^\ell$. Our result is nearly tight for small values of $\ell$ and extends a previous Fourier bound for standard decision trees by Sherstov, Storozhenko, and Wu (STOC, 2021). As an application of our Fourier bounds
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Fairmandering: A column generation heuristic for fairness-optimized political districting arXiv.cs.DM Pub Date : 2021-03-21 Wes Gurnee, David B. Shmoys
The American winner-take-all congressional district system empowers politicians to engineer electoral outcomes by manipulating district boundaries. Existing computational solutions mostly focus on drawing unbiased maps by ignoring political and demographic input, and instead simply optimize for compactness. We claim that this is a flawed approach because compactness and fairness are orthogonal qualities
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A systematic association of subgraph counts over a network arXiv.cs.DM Pub Date : 2021-03-19 Dimitris Floros, Nikos Pitsianis, Xiaobai Sun
We associate all small subgraph counting problems with a systematic graph encoding/representation system which makes a coherent use of graphlet structures. The system can serve as a unified foundation for studying and connecting many important graph problems in theory and practice. We describe topological relations among graphlets (graph elements) in rigorous mathematics language and from the perspective
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Frobenius Numbers and Automatic Sequences arXiv.cs.DM Pub Date : 2021-03-19 Jeffrey Shallit
The Frobenius number $g(S)$ of a set $S$ of non-negative integers with $\gcd 1$ is the largest integer not expressible as a linear combination of elements of $S$. Given a sequence ${\bf s} = (s_i)_{i \geq 0}$, we can define the associated sequence $G_{\bf s} (i) = g(\{ s_i,s_{i+1},\ldots \})$. In this paper we compute $G_{\bf s} (i)$ for some classical automatic sequences: the evil numbers, the odious
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Bhattacharyya parameter of monomials codes for the Binary Erasure Channel: from pointwise to average reliability arXiv.cs.DM Pub Date : 2021-03-19 Vlad-Florin Dragoi, Gabriela Cristescu
Monomial codes were recently equipped with partial order relations, fact that allowed researchers to discover structural properties and efficient algorithm for constructing polar codes. Here, we refine the existing order relations in the particular case of Binary Erasure Channel. The new order relation takes us closer to the ultimate order relation induced by the pointwise evaluation of the Bhattacharyya
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On a recolouring version of Hadwiger's conjecture arXiv.cs.DM Pub Date : 2021-03-19 Marthe Bonamy, Marc Heinrich, Clément Legrand-Duchesne, Jonathan Narboni
We prove that for any $\varepsilon>0$, for any large enough $t$, there is a graph $G$ that admits no $K_t$-minor but admits a $(\frac32-\varepsilon)t$-colouring that is "frozen" with respect to Kempe changes, i.e. any two colour classes induce a connected component. This disproves three conjectures of Las Vergnas and Meyniel from 1981.
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The equidistribution of some Mahonian statistics over permutations avoiding a pattern of length three arXiv.cs.DM Pub Date : 2021-03-18 Phan Thuan Do, Thi Thu Huong Tran, Vincent Vajnovszki
We prove the equidistribution of several multistatistics over some classes of permutations avoiding a $3$-length pattern. We deduce the equidistribution, on the one hand of inv and foze" statistics, and on the other hand that of maj and makl statistics, over these classes of pattern avoiding permutations. Here inv and maj are the celebrated Mahonian statistics, foze" is one of the statistics defined
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Learning to Schedule Heuristics in Branch-and-Bound arXiv.cs.DM Pub Date : 2021-03-18 Antonia Chmiela, Elias B. Khalil, Ambros Gleixner, Andrea Lodi, Sebastian Pokutta
Primal heuristics play a crucial role in exact solvers for Mixed Integer Programming (MIP). While solvers are guaranteed to find optimal solutions given sufficient time, real-world applications typically require finding good solutions early on in the search to enable fast decision-making. While much of MIP research focuses on designing effective heuristics, the question of how to manage multiple MIP
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Comparative Design-Choice Analysis of Color Refinement Algorithms Beyond the Worst Case arXiv.cs.DM Pub Date : 2021-03-18 Markus Anders, Pascal Schweitzer, Florian Wetzels
Color refinement is a crucial subroutine in symmetry detection in theory as well as practice. It has further applications in machine learning and in computational problems from linear algebra. While tight lower bounds for the worst case complexity are known [Berkholz, Bonsma, Grohe, ESA2013] no comparative analysis of design choices for color refinement algorithms is available. We devise two models
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Graphs with unique zero forcing sets and Grundy dominating sets arXiv.cs.DM Pub Date : 2021-03-18 Boštjan Brešar, Tanja Dravec
The concept of zero forcing was introduced in the context of linear algebra, and was further studied by both graph theorists and linear algebraists. It is based on the process of activating vertices of a graph $G$ starting from a set of vertices that are already active, and applying the rule that an active vertex with exactly one non-active neighbor forces that neighbor to become active. A set $S\subset
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Faster Quantum Concentration via Grover's Search arXiv.cs.DM Pub Date : 2021-03-16 Cem M. Unsal, A. Yavuz Oruc
We present quantum algorithms for routing concentration assignments on full capacity fat-and-slim concentrators, bounded fat-and-slim concentrators, and regular fat-and-slim concentrators. Classically, the concentration assignment takes $O(n)$ time on all these concentrators, where $n$ is the number of inputs. Powered by Grover's quantum search algorithm, our algorithms take $O(\sqrt{nc}\ln{c})$ time
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Weighted Sparse and Lightweight Spanners with Local Additive Error arXiv.cs.DM Pub Date : 2021-03-17 Reyan Ahmed, Greg Bodwin, Keaton Hamm, Stephen Kobourov, Richard Spence
An \emph{additive $+\beta$ spanner} of a graph $G$ is a subgraph which preserves shortest paths up to an additive $+\beta$ error. Additive spanners are well-studied in unweighted graphs but have only recently received attention in weighted graphs [Elkin et al.\ 2019 and 2020, Ahmed et al.\ 2020]. This paper makes two new contributions to the theory of weighted additive spanners. For weighted graphs
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