# Computer Science - Discrete Mathematics Publications (50)

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## Computer Science - Discrete Mathematics Publications

The paper develops a new technique to extract a characteristic subset from a random source that repeatedly samples from a set of elements. Here a characteristic subset is a set that when containing an element contains all elements that have the same probability. With this technique at hand the paper looks at the special case of the tournament isomorphism problem that stands in the way towards a polynomial-time algorithm for the graph isomorphism problem. Read More

We present a general approximation framework for weighted integer covering problems. In a weighted integer covering problem, the goal is to determine a non-negative integer solution $x$ to system $\{ Ax \geq r \}$ minimizing a non-negative cost function $c^T x$ (of appropriate dimensions). All coefficients in matrix $A$ are assumed to be non-negative. Read More

A set $D$ of vertices in a graph $G$ is a dominating set if every vertex of $G$, which is not in $D$, has a neighbor in $D$. A set of vertices $D$ in $G$ is convex (respectively, isometric), if all vertices in all shortest paths (respectively, all vertices in one of the shortest paths) between any two vertices in $D$ lie in $D$. The problem of finding a minimum convex dominating (respectively, isometric dominating) set is considered in this paper from algorithmic point of view. Read More

A unique sink orientation (USO) is an orientation of the $n$-dimensional cube graph ($n$-cube) such that every face (subcube) has a unique sink. The number of unique sink orientations is $n^{\Theta(2^n)}$. If a cube orientation is not a USO, it contains a pseudo unique sink orientation (PUSO): an orientation of some subcube such that every proper face of it has a unique sink, but the subcube itself hasn't. Read More

Through the development of efficient algorithms, data structures and preprocessing techniques, real-world shortest path problems in street networks are now very fast to solve. But in reality, the exact travel times along each arc in the network may not be known. This lead to the development of robust shortest path problems, where all possible arc travel times are contained in a so-called uncertainty set of possible outcomes. Read More

In data-parallel computing frameworks, intermediate parallel data is often produced at various stages which needs to be transferred among servers in the datacenter network (e.g. the shuffle phase in MapReduce). Read More

Robust network flows are a concept for dealing with uncertainty and unforeseen failures in the network infrastructure. They and their dual counterpart, network flow interdiction, have received steady attention within the operations research community over the past years. One of the most basic models is the Maximum Robust Flow Problem: Given a network and an integer k, the task is to find a path flow of maximum robust value, i. Read More

Let $\mathcal{B}$ denote a set of bicolorings of $[n]$, where each bicoloring is a mapping of the points in $[n]$ to $\{-1,+1\}$. For each $B \in \mathcal{B}$, let $Y_B=(B(1),\ldots,B(n))$. For each $A \subseteq [n]$, let $X_A \in \{0,1\}^n$ denote the incidence vector of $A$. Read More

We introduce and investigate reroutable flows, a robust version of network flows in which link failures can be mitigated by rerouting the affected flow. Given a capacitated network, a path flow is reroutable if after failure of an arbitrary arc, we can reroute the interrupted flow from the tail of that arc to the sink, without modifying the flow that is not affected by the failure. Similar types of restoration, which are often termed "local", were previously investigated in the context of network design, such as min-cost capacity planning. Read More

It has long been known that Feedback Vertex Set can be solved in time $2^{\mathcal{O}(w\log w)}n^{\mathcal{O}(1)}$ on graphs of treewidth $w$, but it was only recently that this running time was improved to $2^{\mathcal{O}(w)}n^{\mathcal{O}(1)}$, that is, to single-exponential parameterized by treewidth. We investigate which generalizations of Feedback Vertex Set can be solved in a similar running time. Formally, for a class of graphs $\mathcal{P}$, the Bounded $\mathcal{P}$-Block Vertex Deletion problem asks, given a graph $G$ on $n$ vertices and positive integers $k$ and $d$, whether $G$ contains a set $S$ of at most $k$ vertices such that each block of $G-S$ has at most $d$ vertices and is in $\mathcal{P}$. Read More

We propose a pure 0-1 formulation for the wireless network design problem, i.e. the problem of configuring a set of transmitters to provide service coverage to a set of receivers. Read More

**Authors:**Fabio D'Andreagiovanni

We revisit the mathematical models for wireless network jamming introduced by Commander et al.: we first point out the strong connections with classical wireless network design and then we propose a new model based on the explicit use of signal-to-interference quantities. Moreover, to address the intrinsic uncertain nature of the jamming problem and tackle the peculiar right-hand-side (RHS) uncertainty of the problem, we propose an original robust cutting-plane algorithm drawing inspiration from Multiband Robust Optimization. Read More

We consider an issue of much current concern: could fairness, an issue that is already difficult to guarantee, worsen when algorithms run much of our lives? We consider this in the context of resource-allocation problems; we show that algorithms can guarantee certain types of fairness in a verifiable way. Our conceptual contribution is a simple approach to fairness in this context, which only requires that all users trust some public lottery. Our technical contributions are in ways to address the $k$-center and knapsack-center problems that arise in this context: we develop a novel dependent-rounding technique that, via the new ingredients of "slowing down" and additional randomization, guarantees stronger correlation properties than known before. Read More

We study the problem of approximating the partition function of the ferromagnetic Ising model in graphs and hypergraphs. Our first result is a deterministic approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters $\beta$ (the interaction) and $\lambda$ (the external field), except for the case $\vert{\lambda}\vert=1$ (the "zero-field" case). A randomized algorithm (FPRAS) for all graphs, and all $\beta,\lambda$, has long been known. Read More

We study the shared processor scheduling problem with a single shared processor where a unit time saving (weight) obtained by processing a job on the shared processor depends on the job. A polynomial-time optimization algorithm has been given for the problem with equal weights in the literature. This paper extends that result by showing an $O(n \log n)$ optimization algorithm for a class of instances in which non-decreasing order of jobs with respect to processing times provides a non-increasing order with respect to weights --- this instance generalizes the unweighted case of the problem. Read More

Singleton arc consistency is an important type of local consistency which has been recently shown to solve all constraint satisfaction problems (CSPs) over constraint languages of bounded width. We aim to characterise all classes of CSPs defined by a forbidden pattern that are solved by singleton arc consistency and closed under removing constraints. We identify five new patterns whose absence ensures solvability by singleton arc consistency, four of which are provably maximal and three of which generalise 2-SAT. Read More

A graph $G$ is equitably $k$-colorable if its vertices can be partitioned into $k$ independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest integer $k$ for which such a coloring exists is known as the \emph{equitable chromatic number} of $G$ and it is denoted by $\chi_{=}(G)$. In this paper the problem of determinig the value of equitable chromatic number for multicoronas of cubic graphs $G \circ^l H$ is studied. Read More

In 1947 Nathan Fine gave a beautiful product for the number of binomial coefficients $\binom{n}{m}$, for $m$ in the range $0 \leq m \leq n$, that are not divisible by $p$. We give a matrix product that generalizes Fine's formula, simultaneously counting binomial coefficients with $p$-adic valuation $\alpha$ for each $\alpha \geq 0$. For each $n$ this information is naturally encoded in a polynomial generating function, and the sequence of these polynomials is $p$-regular in the sense of Allouche and Shallit. Read More

Partly in service of exploring the formal basis for Georgetown University's AvesTerra database structure, we formalize a recursive hypergraph data structure, which we call an ubergraph. Read More

The generalized hierarchical product of graphs was introduced by L. Barri\'ere et al in 2009. In this paper, reformulated first Zagreb index of generalized hierarchical product of two connected graphs and hence as a special case cluster product of graphs are obtained. Read More

Not all approximations arise from information systems. The problem of fitting approximations, subjected to some rules (and related data), to information systems in a rough scheme of things is known as the \emph{inverse problem}. The inverse problem is more general than the duality (or abstract representation) problems and was introduced by the present author in her earlier papers. Read More

A zero-one matrix $A$ contains another zero-one matrix $P$ if some submatrix of $A$ can be transformed to $P$ by changing some ones to zeros. $A$ avoids $P$ if $A$ does not contain $P$. The Pattern Avoidance Game is played by two players. Read More

We say a zero-one matrix $A$ avoids another zero-one matrix $P$ if no submatrix of $A$ can be transformed to $P$ by changing some ones to zeros. A fundamental problem is to study the extremal function $ex(n,P)$, the maximum number of nonzero entries in an $n \times n$ zero-one matrix $A$ which avoids $P$. To calculate exact values of $ex(n,P)$ for specific values of $n$, we need containment algorithms which tell us whether a given $n \times n$ matrix $A$ contains a given pattern matrix $P$. Read More

In this paper, we study various parallelization schemes for the Variable Neighborhood Search (VNS) metaheuristic on a CPU-GPU system via OpenMP and OpenACC. A hybrid parallel VNS method is applied to recent benchmark problem instances for the multi-product dynamic lot sizing problem with product returns and recovery, which appears in reverse logistics and is known to be NP-hard. We report our findings regarding these parallelization approaches and present promising computational results. Read More

A vector composition of a vector $\mathbf{\ell}$ is a matrix $\mathbf{A}$ whose rows sum to $\mathbf{\ell}$. We define a weighted vector composition as a vector composition in which the column values of $\mathbf{A}$ may appear in different colors. We study vector compositions from different viewpoints: (1) We show how they are related to sums of random vectors and (2) how they allow to derive formulas for partial derivatives of composite functions. Read More

We use techniques from the theory of electrical networks to give tight bounds for the transience class of the Abelian sandpile model on the two-dimensional grid up to polylogarithmic factors. The Abelian sandpile model is a discrete process on graphs that is intimately related to the phenomenon of self-organized criticality. In this process vertices receive grains of sand, and once the number of grains exceeds their degree, they topple by sending grains to their neighbors. Read More

The Wireless Network Design Problem (WND) consists in choosing values of radio-electrical parameters of transmitters of a wireless network, to maximize network coverage. We present a pure 0-1 Linear Programming formulation for the WND that may contain an exponential number of constraints. Violated inequalities of this formulation are hard to separate both theoretically and in practice. Read More

A parallel genetic algorithm (GA) implemented on GPU clusters is proposed to solve the Uncapacitated Single Allocation p-Hub Median problem. The GA uses binary and integer encoding and genetic operators adapted to this problem. Our GA is improved by generated initial solution with hubs located at middle nodes. Read More

The complexity of testing whether a graph contains an induced odd cycle of length at least five is currently unknown. In this paper we show that this can be done in polynomial time if the input graph has no induced subgraph isomorphic to the bull (a triangle with two disjoint pendant edges). Read More

A directed odd cycle transversal of a directed graph (digraph) $D$ is a vertex set $S$ that intersects every odd directed cycle of $D$. In the Directed Odd Cycle Transversal (DOCT) problem, the input consists of a digraph $D$ and an integer $k$. The objective is to determine whether there exists a directed odd cycle transversal of $D$ of size at most $k$. Read More

N distinguishable players are randomly fitted with a white or black hat, where the probabilities of getting a white or black hat may be different for each player, but known to all the players. All players guess simultaneously the color of their own hat observing only the hat colors of the other N-1 players. It is also allowed for each player to pass: no color is guessed. Read More

We consider the recent works of \cite{AIJACM,HV,Harmonic} that provide tools for analyzing focused stochastic local search algorithms that arise from algorithmizations of the Lovasz Local Lemma \cite{LLL} (LLL) in general probability spaces. These are algorithms which search a state space probabilistically by repeatedly selecting a "flaw" that is currently present and moving to a random nearby state in an effort to address it and, eventually, reach a flawless state. While the original Moser-Tardos \cite{MT} (MT) algorithm is amenable to the analysis of these abstract frameworks, many follow-up results \cite{Haeupler_jacm,EnuHarris,szege_meet,determ,distributed,ParallelHarris} that further enhance, or exploit, our understanding of the MT process are not transferable to these general settings. Read More

This paper contains an axiomatic study of consistent approval-based multi-winner rules, i.e., voting rules that select a fixed-size group of candidates based on approval ballots. Read More

In a model of network communication based on a random walk in an undirected graph, what subset of nodes (subject to constraints on the set size), enables the fastest spread of information? In this paper, we assume the dynamics of spread is described by a network consensus process, but to find the most effective seeds we consider the target set of a random walk--the process dual to network consensus spread. Thus an optimal set $A$ minimizes the sum of the expected first hitting times $F(A)$ of random walks that start at nodes outside the set. We introduce a submodular, non-decreasing rank function $\rho$, that permits some comparison between the solution obtained by the classical greedy algorithm and one obtained by our methods. Read More

**Affiliations:**

^{1}CNRS, GIPSA-CICS,

^{2}CNRS, GIPSA-VIBS,

^{3}CNRS, GIPSA-CICS

We consider the problem of sampling k-bandlimited graph signals, \ie, linear combinations of the first k graph Fourier modes. We know that a set of k nodes embedding all k-bandlimited signals always exists, thereby enabling their perfect reconstruction after sampling. Unfortunately, to exhibit such a set, one needs to partially diagonalize the graph Laplacian, which becomes prohibitive at large scale. Read More

We characterise finite axiomatisability and intractability of deciding membership for universal Horn classes generated by finite loop-free hypergraphs. Read More

This paper is devoted to understanding the shortest augmenting path approach for computing a maximum matching. Despite its apparent potential for designing efficient matching and flow algorithms, it has been poorly understood. Chaudhuri et. Read More

In this paper, we propose computational approaches for the zero forcing problem, the connected zero forcing problem, and the problem of forcing a graph within a specified number of timesteps. Our approaches are based on a combination of integer programming models and combinatorial algorithms, and include formulations for zero forcing as a dynamic process, and as a set-covering problem. We explore several solution strategies for these models, and numerically compare them to the well-known Wavefront algorithm for zero forcing developed by Grout et al. Read More

In a seminal paper, Chen, Roughgarden and Valiant studied cost sharing protocols for network design with the objective to implement a low-cost Steiner forest as a Nash equilibrium of an induced cost-sharing game. One of the most intriguing open problems up to date is to understand the power of budget-balanced and separable cost sharing protocols in order to induce low-cost Steiner forests. In this work, we focus on undirected networks and analyze topological properties of the underlying graph so that an optimal Steiner forest can be implemented as a Nash equilibrium (by some separable cost sharing protocol) independent of the edge costs. Read More

We prove that the regular $n\times n$ square grid of points in the integer lattice $\mathbb{Z}^{2}$ cannot be recovered from an arbitrary $n^{2}$-element subset of $\mathbb{Z}^{2}$ via a mapping with prescribed Lipschitz constant (independent of $n$). This answers negatively a question of Feige from 2002. Our resolution of Feige's question takes place largely in a continuous setting and is based on new results for Lipschitz mappings falling into two broad areas of interest, which we study independently. Read More

A classic result due to Schaefer (1978) classifies all constraint satisfaction problems (CSPs) over the Boolean domain as being either in $\mathsf{P}$ or $\mathsf{NP}$-hard. This paper considers a promise-problem variant of CSPs called PCSPs. A PCSP over a finite set of pairs of constraints $\Gamma$ consists of a pair $(\Psi_P, \Psi_Q)$ of CSPs with the same set of variables such that for every $(P, Q) \in \Gamma$, $P(x_{i_1}, . Read More

For a set ${\cal F}$ of graphs, an instance of the ${\cal F}$-{\sc free Sandwich Problem} is a pair $(G_1,G_2)$ consisting of two graphs $G_1$ and $G_2$ with the same vertex set such that $G_1$ is a subgraph of $G_2$, and the task is to determine an ${\cal F}$-free graph $G$ containing $G_1$ and contained in $G_2$, or to decide that such a graph does not exist. Initially motivated by the graph sandwich problem for trivially perfect graphs, which are the $\{ P_4,C_4\}$-free graphs, we study the complexity of the ${\cal F}$-{\sc free Sandwich Problem} for sets ${\cal F}$ containing two non-isomorphic graphs of order four. We show that if ${\cal F}$ is one of the sets $\left\{ {\rm diamond},K_4\right\}$, $\left\{ {\rm diamond},C_4\right\}$, $\left\{ {\rm diamond},{\rm paw}\right\}$, $\left\{ K_4,\overline{K_4}\right\}$, $\left\{ P_4,C_4\right\}$, $\left\{ P_4,\overline{\rm claw}\right\}$, $\left\{ P_4,\overline{\rm paw}\right\}$, $\left\{ P_4,\overline{\rm diamond}\right\}$, $\left\{ {\rm paw},C_4\right\}$, $\left\{ {\rm paw},{\rm claw}\right\}$, $\left\{ {\rm paw},\overline{{\rm claw}}\right\}$, $\left\{ {\rm paw},\overline{\rm paw}\right\}$, $\left\{ C_4,\overline{C_4}\right\}$, $\left\{ {\rm claw},\overline{{\rm claw}}\right\}$, and $\left\{ {\rm claw},\overline{C_4}\right\}$, then the ${\cal F}$-{\sc free Sandwich Problem} can be solved in polynomial time, and, if ${\cal F}$ is one of the sets $\left\{ C_4,K_4\right\}$, $\left\{ {\rm paw},K_4\right\}$, $\left\{ {\rm paw},\overline{K_4}\right\}$, $\left\{ {\rm paw},\overline{C_4}\right\}$, $\left\{ {\rm diamond},\overline{C_4}\right\}$, $\left\{ {\rm paw},\overline{\rm diamond}\right\}$, and $\left\{ {\rm diamond},\overline{\rm diamond}\right\}$, then the decision version of the ${\cal F}$-{\sc free Sandwich Problem} is NP-complete. Read More

We introduce a generalization of the celebrated Lov\'asz theta number of a graph to simplicial complexes of arbitrary dimension. Our generalization takes advantage of real simplicial cohomology theory, in particular combinatorial Laplacians, and provides a semidefinite programming upper bound of the independence number of a simplicial complex. We consider properties of the graph theta number such as the relationship to Hoffman's ratio bound and to the chromatic number and study how they extend to higher dimensions. Read More

Seymour's second neighbourhood conjecture asserts that every oriented graph has a vertex whose second out-neighbourhood is at least as large as its out-neighbourhood. In this paper, we prove that the conjecture holds for quasi-transitive oriented graphs, which is a superclass of tournaments and transitive acyclic digraphs. A digraph $D$ is called quasi-transitive is for every pair $xy,yz$ of arcs between distinct vertices $x,y,z$, $xz$ or $zx$ ("or" is inclusive here) is in $D$. Read More

Diluted mean-field models are spin systems whose geometry of interactions is induced by a sparse random graph or hypergraph. Such models play an eminent role in the statistical mechanics of disordered systems as well as in combinatorics and computer science. In a path-breaking paper based on the non-rigorous `cavity method', physicists predicted not only the existence of a replica symmetry breaking phase transition in such models but also sketched a detailed picture of the evolution of the Gibbs measure within the replica symmetric phase and its impact on important problems in combinatorics, computer science and physics [Krzakala et al. Read More

Given graphs $G$ and $H$, we consider the problem of decomposing a properly edge-colored graph $G$ into few parts consisting of rainbow copies of $H$ and single edges. We establish a close relation to the previously studied problem of minimum $H$-decompositions, where an edge coloring does not matter and one is merely interested in decomposing graphs into copies of $H$ and single edges. Read More

A Cayley graph over a group G is said to be central if its connection set is a normal subset of G. It is proved that for any two central Cayley graphs over explicitly given almost simple groups of order n, the set of all isomorphisms from the first graph onto the second can be found in time poly(n). Read More

This paper addresses automatic summarization and search in visual data comprising of videos, live streams and image collections in a unified manner. In particular, we propose a framework for multi-faceted summarization which extracts key-frames (image summaries), skims (video summaries) and entity summaries (summarization at the level of entities like objects, scenes, humans and faces in the video). The user can either view these as extractive summarization, or query focused summarization. Read More

We consider the problem of matching applicants to posts where applicants have preferences over posts. Thus the input to our problem is a bipartite graph G = (A U P,E), where A denotes a set of applicants, P is a set of posts, and there are ranks on edges which denote the preferences of applicants over posts. A matching M in G is called rank-maximal if it matches the maximum number of applicants to their rank 1 posts, subject to this the maximum number of applicants to their rank 2 posts, and so on. Read More