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Parameterized Algorithms for Coordinated Motion Planning: Minimizing Energy arXiv.cs.DM Pub Date : 2024-04-24 Argyrios Deligkas, Eduard Eiben, Robert Ganian, Iyad Kanj, M. S. Ramanujan
We study the parameterized complexity of a generalization of the coordinated motion planning problem on graphs, where the goal is to route a specified subset of a given set of $k$ robots to their destinations with the aim of minimizing the total energy (i.e., the total length traveled). We develop novel techniques to push beyond previously-established results that were restricted to solid grids. We
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Detecting Disjoint Shortest Paths in Linear Time and More arXiv.cs.DM Pub Date : 2024-04-24 Shyan Akmal, Virginia Vassilevska Williams, Nicole Wein
In the $k$-Disjoint Shortest Paths ($k$-DSP) problem, we are given a weighted graph $G$ on $n$ nodes and $m$ edges with specified source vertices $s_1, \dots, s_k$, and target vertices $t_1, \dots, t_k$, and are tasked with determining if $G$ contains vertex-disjoint $(s_i,t_i)$-shortest paths. For any constant $k$, it is known that $k$-DSP can be solved in polynomial time over undirected graphs and
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Linear Search for an Escaping Target with Unknown Speed arXiv.cs.DM Pub Date : 2024-04-22 Jared Coleman, Dmitry Ivanov, Evangelos Kranakis, Danny Krizanc, Oscar Morales-Ponce
We consider linear search for an escaping target whose speed and initial position are unknown to the searcher. A searcher (an autonomous mobile agent) is initially placed at the origin of the real line and can move with maximum speed $1$ in either direction along the line. An oblivious mobile target that is moving away from the origin with an unknown constant speed $v<1$ is initially placed by an adversary
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An algorithm with a delay of $\mathcal{O}(kΔ)$ for enumerating connected induced subgraphs of size $k$ arXiv.cs.DM Pub Date : 2024-04-19 Chenglong Xiao, Chengyong Mao, Shanshan Wang
The problem of enumerating connected subgraphs of a given size in a graph has been extensively studied in recent years. In this short communication, we propose an algorithm with a delay of $\mathcal{O}(k\Delta)$ for enumerating all connected induced subgraphs of size $k$ in an undirected graph $G=(V, E)$, where $k$ and $\Delta$ are respectively the size of subgraphs and the maximum degree of $G$. The
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Bit catastrophes for the Burrows-Wheeler Transform arXiv.cs.DM Pub Date : 2024-04-16 Sara Giuliani, Shunsuke Inenaga, Zsuzsanna Lipták, Giuseppe Romana, Marinella Sciortino, Cristian Urbina
A bit catastrophe, loosely defined, is when a change in just one character of a string causes a significant change in the size of the compressed string. We study this phenomenon for the Burrows-Wheeler Transform (BWT), a string transform at the heart of several of the most popular compressors and aligners today. The parameter determining the size of the compressed data is the number of equal-letter
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Robot Positioning Using Torus Packing for Multisets arXiv.cs.DM Pub Date : 2024-04-15 Chung Shue Chen, Peter Keevash, Sean Kennedy, Élie de Panafieu, Adrian Vetta
We consider the design of a positioning system where a robot determines its position from local observations. This is a well-studied problem of considerable practical importance and mathematical interest. The dominant paradigm derives from the classical theory of de Bruijn sequences, where the robot has access to a window within a larger code and can determine its position if these windows are distinct
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Integrating On-demand Ride-sharing with Mass Transit at-Scale arXiv.cs.DM Pub Date : 2024-04-11 Danushka Edirimanna, Hins Hu, Samitha Samaranayake
We are in the midst of a technology-driven transformation of the urban mobility landscape. However, unfortunately these new innovations are still dominated by car-centric personal mobility, which leads to concerns such as environmental sustainability, congestion, and equity. On the other hand, mass transit provides a means to move large amounts of travelers very efficiently, but is not very versatile
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Parameterized Complexity of Submodular Minimization under Uncertainty arXiv.cs.DM Pub Date : 2024-04-11 Naonori Kakimura, Ildikó Schlotter
This paper studies the computational complexity of a robust variant of a two-stage submodular minimization problem that we call Robust Submodular Minimizer. In this problem, we are given $k$ submodular functions $f_1,\dots,f_k$ over a set family $2^V$, which represent $k$ possible scenarios in the future when we will need to find an optimal solution for one of these scenarios, i.e., a minimizer for
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Exploring Repetitiveness Measures for Two-Dimensional Strings arXiv.cs.DM Pub Date : 2024-04-10 Giuseppe Romana, Marinella Sciortino, Cristian Urbina
Detecting and measuring repetitiveness of strings is a problem that has been extensively studied in data compression and text indexing. However, when the data are structured in a non-linear way, like in the context of two-dimensional strings, inherent redundancy offers a rich source for compression, yet systematic studies on repetitiveness measures are still lacking. In the paper we introduce extensions
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An asymptotically optimal algorithm for generating bin cardinalities arXiv.cs.DM Pub Date : 2024-04-10 Luc Devroye, Dimitrios Los
In the balls-into-bins setting, $n$ balls are thrown uniformly at random into $n$ bins. The na\"{i}ve way to generate the final load vector takes $\Theta(n)$ time. However, it is well-known that this load vector has with high probability bin cardinalities of size $\Theta(\frac{\log n}{\log \log n})$. Here, we present an algorithm in the RAM model that generates the bin cardinalities of the final load
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The steady-states of splitter networks arXiv.cs.DM Pub Date : 2024-04-08 Basile Couëtoux, Bastien Gastaldi, Guyslain Naves
We introduce splitter networks, which abstract the behavior of conveyor belts found in the video game Factorio. Based on this definition, we show how to compute the steady-state of a splitter network. Then, leveraging insights from the players community, we provide multiple designs of splitter networks capable of load-balancing among several conveyor belts, and prove that any load-balancing network
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Stability in Graphs with Matroid Constraints arXiv.cs.DM Pub Date : 2024-04-05 Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, Saket Saurabh
We study the following Independent Stable Set problem. Let G be an undirected graph and M = (V(G),I) be a matroid whose elements are the vertices of G. For an integer k\geq 1, the task is to decide whether G contains a set S\subseteq V(G) of size at least k which is independent (stable) in G and independent in M. This problem generalizes several well-studied algorithmic problems, including Rainbow
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A Temporal Graph Model to Study the Dynamics of Collective Behavior and Performance in Team Sports: An Application to Basketball arXiv.cs.DM Pub Date : 2024-04-02 Quentin BourgeaisCETAPS, Eric SanlavilleLITIS, Rodolphe CharrierLITIS, Ludovic SeifertCETAPS
In this study, a temporal graph model is designed to model the behavior of collective sports teams based on the networks of player interactions. The main motivation for the model is to integrate the temporal dimension into the analysis of players' passing networks in order to gain deeper insights into the dynamics of system behavior, particularly how a system exploits the degeneracy property to self-regulate
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Sumsets in the Hypercube arXiv.cs.DM Pub Date : 2024-03-25 Noga Alon, Or Zamir
A subset $S$ of the Boolean hypercube $\mathbb{F}_2^n$ is a sumset if $S = A+A = \{a + b \ | \ a, b\in A\}$ for some $A \subseteq \mathbb{F}_2^n$. We prove that the number of sumsets in $\mathbb{F}_2^n$ is asymptotically $(2^n-1)2^{2^{n-1}}$. Furthermore, we show that the family of sumsets in $\mathbb{F}_2^n$ is almost identical to the family of all subsets of $\mathbb{F}_2^n$ that contain a complete
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Robust Filter Design for Graph Signals arXiv.cs.DM Pub Date : 2024-03-25 Lucia Testa, Stefania Sardellitti, Sergio Barbarossa
Our goal in this paper is the robust design of filters acting on signals observed over graphs subject to small perturbations of their edges. The focus is on developing a method to identify spectral and polynomial graph filters that can adapt to the perturbations in the underlying graph structure while ensuring the filters adhere to the desired spectral mask. To address this, we propose a novel approach
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Establishing a leader in a pairwise comparisons method arXiv.cs.DM Pub Date : 2024-03-21 Jacek Szybowski, Konrad Kułakowski, Jiri Mazurek, Sebastian Ernst
Abstract Like electoral systems, decision-making methods are also vulnerable to manipulation by decision-makers. The ability to effectively defend against such threats can only come from thoroughly understanding the manipulation mechanisms. In the presented article, we show two algorithms that can be used to launch a manipulation attack. They allow for equating the weights of two selected alternatives
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Rescue Craft Allocation in Tidal Waters of the North and Baltic Sea arXiv.cs.DM Pub Date : 2024-03-21 Tom Mucke, Alexander Renneke, Finn Seesemann, Felix Engelhardt
This paper aims to improve the average response time for naval accidents in the North and Baltic Sea. To do this we optimize the strategic distribution of the vessel fleet used by the Deutsche Gesellschaft zur Rettung Schiffbr\"uchiger (German Maritime Search and Rescue Service) (DGzRS) across several home stations. Based on these locations, in case of an incoming distress call the vessel with the
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Proper Rainbow Saturation Numbers for Cycles arXiv.cs.DM Pub Date : 2024-03-22 Anastasia Halfpap, Bernard Lidický, Tomáš Masařík
We say that an edge-coloring of a graph $G$ is proper if every pair of incident edges receive distinct colors, and is rainbow if no two edges of $G$ receive the same color. Furthermore, given a fixed graph $F$, we say that $G$ is rainbow $F$-saturated if $G$ admits a proper edge-coloring which does not contain any rainbow subgraph isomorphic to $F$, but the addition of any edge to $G$ makes such an
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The ultrametric backbone is the union of all minimum spanning forests arXiv.cs.DM Pub Date : 2024-03-19 Jordan C Rozum, Luis M Rocha
Minimum spanning trees and forests are powerful sparsification techniques that remove cycles from weighted graphs to minimize total edge weight while preserving node connectivity. They have applications in computer science, network science, and graph theory. Despite their utility and ubiquity, they have several limitations, including that they are only defined for undirected networks, they significantly
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Home Spaces and Invariants to Analyze Parameterized Petri Nets arXiv.cs.DM Pub Date : 2024-03-18 Gerard Memmi
This article focuses on comparing the notions of home spaces and invariants, in Transition Systems and more particularly, in Petri Nets as well as a variety of derived Petri Nets. After recalling basic notions of Petri Nets and semiflows, we then discuss important characteristics of finite generating sets for F, the set of all semiflows with integer coordinates of a given Petri Net. Then, we particularly
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Solving the Multiobjective Quasi-Clique Problem arXiv.cs.DM Pub Date : 2024-03-16 Daniela Scherer dos Santos, Kathrin Klamroth, Pedro Martins, Luís Paquete
Given a simple undirected graph $G$, a quasi-clique is a subgraph of $G$ whose density is at least $\gamma$ $(0 < \gamma \leq 1)$. Finding a maximum quasi-clique has been addressed from two different perspectives: $i)$ maximizing vertex cardinality for a given edge density; and $ii)$ maximizing edge density for a given vertex cardinality. However, when no a priori preference information about cardinality
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On maximum-sum matchings of bichromatic points arXiv.cs.DM Pub Date : 2024-03-13 Oscar Chacón-Rivera, Pablo Pérez-Lantero
Huemer et al. (Discrete Math, 2019) proved that for any two finite point sets $R$ and $B$ in the plane with $|R| = |B|$, the perfect matching that matches points of $R$ with points of $B$, and maximizes the total squared Euclidean distance of the matched pairs, has the property that all the disks induced by the matching have a nonempty common intersection. A pair of matched points induces the disk
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Improved Dynamics for the Maximum Common Subgraph Problem arXiv.cs.DM Pub Date : 2024-03-13 Davide Guidobene, Guido Cera
The Maximum Common Subgraph (MCS) problem plays a crucial role across various domains, bridging theoretical exploration and practical applications in fields like bioinformatics and social network analysis. Despite its wide applicability, MCS is notoriously challenging and is classified as an NP-Complete (NPC) problem. This study introduces new heuristics aimed at mitigating these challenges through
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Ensuring connectedness for the Maximum Quasi-clique and Densest $k$-subgraph problems arXiv.cs.DM Pub Date : 2024-03-13 Daniela Scherer dos Santos, Kathrin Klamroth, Pedro Martins, Luís Paquete
Given an undirected graph $G$, a quasi-clique is a subgraph of $G$ whose density is at least $\gamma$ $(0 < \gamma \leq 1)$. Two optimization problems can be defined for quasi-cliques: the Maximum Quasi-Clique (MQC) Problem, which finds a quasi-clique with maximum vertex cardinality, and the Densest $k$-Subgraph (DKS) Problem, which finds the densest subgraph given a fixed cardinality constraint. Most
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Approximating Maximum Edge 2-Coloring by Normalizing Graphs arXiv.cs.DM Pub Date : 2024-03-11 Tobias Mömke, Alexandru Popa, Aida Roshany-Tabrizi, Michael Ruderer, Roland Vincze
In a simple, undirected graph G, an edge 2-coloring is a coloring of the edges such that no vertex is incident to edges with more than 2 distinct colors. The problem maximum edge 2-coloring (ME2C) is to find an edge 2-coloring in a graph G with the goal to maximize the number of colors. For a relevant graph class, ME2C models anti-Ramsey numbers and it was considered in network applications. For the
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Arborescences and Shortest Path Trees when Colors Matter arXiv.cs.DM Pub Date : 2024-03-11 P. S. Ardra, Jasine Babu, Kritika Kashyap, R. Krithika, Sreejith K. Pallathumadam, Deepak Rajendraprasad
Color-constrained subgraph problems are those where we are given an edge-colored (directed or undirected) graph and the task is to find a specific type of subgraph, like a spanning tree, an arborescence, a single-source shortest path tree, a perfect matching etc., with constraints on the number of edges of each color. Some of these problems, like color-constrained spanning tree, have elegant solutions
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Hamiltonicity, Path Cover, and Independence Number: An FPT Perspective arXiv.cs.DM Pub Date : 2024-03-09 Fedor V. Fomin, Petr A. Golovach, Danil Sagunov, Kirill Simonov
The connection between Hamiltonicity and the independence numbers of graphs has been a fundamental aspect of Graph Theory since the seminal works of the 1960s. This paper presents a novel algorithmic perspective on these classical problems. Our contributions are twofold. First, we establish that a wide array of problems in undirected graphs, encompassing problems such as Hamiltonian Path and Cycle
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Parameterized Algorithms for Balanced Cluster Edge Modification Problems arXiv.cs.DM Pub Date : 2024-03-06 Jayakrishnan Madathil, Kitty Meeks
We introduce Cluster Edge Modification problems with constraints on the size of the clusters and study their complexity. A graph $G$ is a cluster graph if every connected component of $G$ is a clique. In a typical Cluster Edge Modification problem such as the widely studied Cluster Editing, we are given a graph $G$ and a non-negative integer $k$ as input, and we have to decide if we can turn $G$ into
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Graph Visualization for Blockchain Data arXiv.cs.DM Pub Date : 2024-03-06 Marcell Dietl, Andre Gemünd, Daniel Oeltz, Felix M. Thiele, Christian Werner
In this report, we introduce a novel approach to visualize extremely large graphs efficiently. Our method combines two force-directed algorithms, Kamada-Kawai and ForceAtlas2, to handle different graph components based on their node count. Additionally, we suggest utilizing the Fast Multipole method to enhance the speed of ForceAtlas2. Although initially designed for analyzing bitcoin transaction graphs
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Payment Scheduling in the Interval Debt Model arXiv.cs.DM Pub Date : 2024-03-04 Tom Friedetzky, David C. Kutner, George B. Mertzios, Iain A. Stewart, Amitabh Trehan
The network-based study of financial systems has received considerable attention in recent years but has seldom explicitly incorporated the dynamic aspects of such systems. We consider this problem setting from the temporal point of view and introduce the Interval Debt Model (IDM) and some scheduling problems based on it, namely: Bankruptcy Minimization/Maximization, in which the aim is to produce
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Graph drawing applications in combinatorial theory of maturity models arXiv.cs.DM Pub Date : 2024-03-04 Špela Kajzer, Alexander Dobler, Janja Jerebic, Martin Nöllenburg, Joachim Orthaber, Drago Bokal
In this paper, we introduce tiled graphs as models of learning and maturing processes. We show how tiled graphs can combine graphs of learning spaces or antimatroids (partial hypercubes) and maturity models (total orders) to yield models of learning processes. For the visualization of these processes it is a natural approach to aim for certain optimal drawings. We show for most of the more detailed
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Matching Algorithms in the Sparse Stochastic Block Model arXiv.cs.DM Pub Date : 2024-03-04 Anna Brandenberger, Byron Chin, Nathan S. Sheffield, Divya Shyamal
The stochastic block model (SBM) is a generalization of the Erd\H{o}s--R\'enyi model of random graphs that describes the interaction of a finite number of distinct communities. In sparse Erd\H{o}s--R\'enyi graphs, it is known that a linear-time algorithm of Karp and Sipser achieves near-optimal matching sizes asymptotically almost surely, giving a law-of-large numbers for the matching sizes of such
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Popularity and Perfectness in One-sided Matching Markets with Capacities arXiv.cs.DM Pub Date : 2024-03-01 Gergely Csáji
We consider many-to-one matching problems, where one side corresponds to applicants who have preferences and the other side to houses who do not have preferences. We consider two different types of this market: one, where the applicants have capacities, and one where the houses do. First, we answer an open question by Manlove and Sng (2006) (partly solved Paluch (2014) for preferences with ties), that
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PosSLP and Sum of Squares arXiv.cs.DM Pub Date : 2024-02-29 Markus Bläser, Julian Dörfler, Gorav Jindal
The problem PosSLP is the problem of determining whether a given straight-line program (SLP) computes a positive integer. PosSLP was introduced by Allender et al. to study the complexity of numerical analysis (Allender et al., 2009). PosSLP can also be reformulated as the problem of deciding whether the integer computed by a given SLP can be expressed as the sum of squares of four integers, based on
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Fractional Linear Matroid Matching is in quasi-NC arXiv.cs.DM Pub Date : 2024-02-28 Rohit Gurjar, Taihei Oki, Roshan Raj
The matching and linear matroid intersection problems are solvable in quasi-NC, meaning that there exist deterministic algorithms that run in polylogarithmic time and use quasi-polynomially many parallel processors. However, such a parallel algorithm is unknown for linear matroid matching, which generalizes both of these problems. In this work, we propose a quasi-NC algorithm for fractional linear
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Output-Sensitive Enumeration of Potential Maximal Cliques in Polynomial Space arXiv.cs.DM Pub Date : 2024-02-28 Caroline Brosse, Alessio Conte, Vincent Limouzy, Giulia Punzi, Davide Rucci
A set of vertices in a graph forms a potential maximal clique if there exists a minimal chordal completion in which it is a maximal clique. Potential maximal cliques were first introduced as a key tool to obtain an efficient, though exponential-time algorithm to compute the treewidth of a graph. As a byproduct, this allowed to compute the treewidth of various graph classes in polynomial time. In recent
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Symbolic Listings as Computation arXiv.cs.DM Pub Date : 2024-02-24 Hamilton Sawczuk, Edinah Gnang
We propose an algebraic model of computation which formally relates symbolic listings, complexity of Boolean functions, and low depth arithmetic circuit complexity. In this model algorithms are arithmetic formula expressing symbolic listings of YES instances of Boolean functions, and computation is executed via partial differential operators. We consider the Chow rank of an arithmetic formula as a
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The Complexity of Diameter on H-free graphs arXiv.cs.DM Pub Date : 2024-02-26 Jelle J. Oostveen, Daniël Paulusma, Erik Jan van Leeuwen
The intensively studied Diameter problem is to find the diameter of a given connected graph. We investigate, for the first time in a structured manner, the complexity of Diameter for H-free graphs, that is, graphs that do not contain a fixed graph H as an induced subgraph. We first show that if H is not a linear forest with small components, then Diameter cannot be solved in subquadratic time for H-free
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A mathematical model for simultaneous personnel shift planning and unrelated parallel machine scheduling arXiv.cs.DM Pub Date : 2024-02-24 Maziyar Khadivi, Mostafa Abbasi, Todd Charter, Homayoun Najjaran
This paper addresses a production scheduling problem derived from an industrial use case, focusing on unrelated parallel machine scheduling with the personnel availability constraint. The proposed model optimizes the production plan over a multi-period scheduling horizon, accommodating variations in personnel shift hours within each time period. It assumes shared personnel among machines, with one
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Tight Inapproximability of Target Set Reconfiguration arXiv.cs.DM Pub Date : 2024-02-23 Naoto Ohsaka
Given a graph $G$ with a vertex threshold function $\tau$, consider a dynamic process in which any inactive vertex $v$ becomes activated whenever at least $\tau(v)$ of its neighbors are activated. A vertex set $S$ is called a target set if all vertices of $G$ would be activated when initially activating vertices of $S$. In the Minmax Target Set Reconfiguration problem, for a graph $G$ and its two target
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Extending the definition of set tolerances arXiv.cs.DM Pub Date : 2024-02-22 Gerold Jäger, Marcel Turkensteen
Optimal solutions of combinatorial optimization problems can be sensitive to changes in the cost of one or more elements. Single and set tolerances measure the largest / smallest possible change such that the current solution remains optimal and other solutions become non-optimal for cost changes in one or more elements, respectively. The current definition only applies to subsets of elements. In this
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Towards Linear Spanners in All Temporal Cliques arXiv.cs.DM Pub Date : 2024-02-21 Sebastian Angrick, Ben Bals, Tobias Friedrich, Hans Gawendowicz, Niko Hastrich, Nicolas Klodt, Pascal Lenzner, Jonas Schmidt, George Skretas, Armin Wells
Many real-world networks, like transportation networks and social networks, are dynamic in the sense that the edge set may change over time, but these changes are known in advance. This behavior is captured by the temporal graphs model, which has recently become a trending topic in theoretical computer science. A core open problem in the field is to prove the existence of linear-size temporal spanners
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Edge-Disjoint Paths in Eulerian Digraphs arXiv.cs.DM Pub Date : 2024-02-21 Dario Cavallaro, Ken-ichi Kawarabayashi, Stephan Kreutzer
Disjoint paths problems are among the most prominent problems in combinatorial optimization. The edge- as well as vertex-disjoint paths problem, are NP-complete on directed and undirected graphs. But on undirected graphs, Robertson and Seymour (Graph Minors XIII) developed an algorithm for the vertex- and the edge-disjoint paths problem that runs in cubic time for every fixed number $p$ of terminal
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On the Permutation Algorithm for Online Facility Assignment on a Line arXiv.cs.DM Pub Date : 2024-02-20 Tsubasa Harada
In the online facility assignment on a line (OFAL) with a set $S$ of $k$ servers and a capacity $c:S\to\mathbb{N}$, each server $s\in S$ with a capacity $c(s)$ is placed on a line and a request arrives on a line one-by-one. The task of an online algorithm is to irrevocably assign a current request to one of the servers with vacancies before the next request arrives. An algorithm can assign up to $c(s)$
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Distance Recoloring arXiv.cs.DM Pub Date : 2024-02-20 Niranka Banerjee, Christian Engels, Duc A. Hoang
Coloring a graph is a well known problem and used in many different contexts. Here we want to assign $k \geq 1$ colors to each vertex of a graph $G$ such that each edge has two different colors at each endpoint. Such a vertex-coloring, if exists, is called a feasible coloring of $G$. \textsc{Distance Coloring} is an extension to the standard \textsc{Coloring} problem. Here we want to enforce that every
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Optimal PSPACE-hardness of Approximating Set Cover Reconfiguration arXiv.cs.DM Pub Date : 2024-02-20 Shuichi Hirahara, Naoto Ohsaka
In the Minmax Set Cover Reconfiguration problem, given a set system $\mathcal{F}$ over a universe and its two covers $\mathcal{C}^\mathsf{start}$ and $\mathcal{C}^\mathsf{goal}$ of size $k$, we wish to transform $\mathcal{C}^\mathsf{start}$ into $\mathcal{C}^\mathsf{goal}$ by repeatedly adding or removing a single set of $\mathcal{F}$ while covering the universe in any intermediate state. Then, the
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Spectral Independence Beyond Uniqueness with. the topological method -- An extended view arXiv.cs.DM Pub Date : 2024-02-18 Charilaos Efthymiou
We present novel results for fast mixing of Glauber dynamics using the newly introduced and powerful Spectral Independence method from [Anari, Liu, Oveis-Gharan: FOCS 2020]. We mainly focus on the Hard-core model and the Ising model. We obtain bounds for fast mixing with the parameters expressed in terms of the spectral radius of the adjacency matrix, improving on the seminal work in [Hayes: FOCS 2006]
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Faster algorithms on linear delta-matroids arXiv.cs.DM Pub Date : 2024-02-18 Tomohiro Koana, Magnus Wahlström
We show new algorithms and constructions over linear delta-matroids. We observe an alternative representation for linear delta-matroids, as a contraction representation over a skew-symmetric matrix. This is equivalent to the more standard "twist representation" up to $O(n^\omega)$-time transformations, but is much more convenient for algorithmic tasks. For instance, the problem of finding a max-weight
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Computational complexity of the Weisfeiler-Leman dimension arXiv.cs.DM Pub Date : 2024-02-18 Moritz Lichter, Simon Raßmann, Pascal Schweitzer
The Weisfeiler-Leman dimension of a graph $G$ is the least number $k$ such that the $k$-dimensional Weisfeiler-Leman algorithm distinguishes $G$ from every other non-isomorphic graph. The dimension is a standard measure of the descriptive complexity of a graph and recently finds various applications in particular in the context of machine learning. In this paper, we study the computational complexity
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Unbalanced Random Matching Markets with Partial Preferences arXiv.cs.DM Pub Date : 2024-02-15 Aditya Potukuchi, Shikha Singh
Properties of stable matchings in the popular random-matching-market model have been studied for over 50 years. In a random matching market, each agent has complete preferences drawn uniformly and independently at random. Wilson (1972), Knuth (1976) and Pittel (1989) proved that in balanced random matching markets, the proposers are matched to their $\ln n$th choice on average. In this paper, we consider
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Tight (Double) Exponential Bounds for Identification Problems: Locating-Dominating Set and Test Cover arXiv.cs.DM Pub Date : 2024-02-13 Dipayan Chakraborty, Florent Foucaud, Diptapriyo Majumdar, Prafullkumar Tale
We investigate fine-grained algorithmic aspects of identification problems in graphs and set systems, with a focus on Locating-Dominating Set and Test Cover. We prove, among other things, the following three (tight) conditional lower bounds. \begin{enumerate} \item \textsc{Locating-Dominating Set} does not admit an algorithm running in time $2^{o(k^2)} \cdot poly(n)$, nor a polynomial time kernelization
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Growth Rate of the Number of Empty Triangles in the Plane arXiv.cs.DM Pub Date : 2024-02-12 Bhaswar B. Bhattacharya, Sandip Das, Sk Samim Islam, Saumya Sen
Given a set $P$ of $n$ points in the plane, in general position, denote by $N_\Delta(P)$ the number of empty triangles with vertices in $P$. In this paper we investigate by how much $N_\Delta(P)$ changes if a point $x$ is removed from $P$. By constructing a graph $G_P(x)$ based on the arrangement of the empty triangles incident on $x$, we transform this geometric problem to the problem of counting
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Greedy Matchings in Bipartite Graphs with Ordered Vertex Sets arXiv.cs.DM Pub Date : 2024-02-09 Hans U. Simon
We define and study greedy matchings in vertex-ordered bipartite graphs. It is shown that each vertex-ordered bipartite graph has a unique greedy matching. The proof uses (a weak form of) Newman's lemma. The vertex ordering is called a preference relation. Given a vertex-ordered bipartite graph, the goal is to match every vertex of one vertex class but to leave unmatched as many as possible vertices
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An approximation algorithm for Maximum DiCut vs. Cut arXiv.cs.DM Pub Date : 2024-02-12 Tamio-Vesa Nakajima, Stanislav Živný
Goemans and Williamson designed a 0.878-approximation algorithm for Max-Cut in undirected graphs [JACM'95]. Khot, Kindler, Mosel, and O'Donnel showed that the approximation ratio of the Goemans-Williamson algorithm is optimal assuming Khot's Unique Games Conjecture [SICOMP'07]. In the problem of maximum cuts in directed graphs (Max-DiCut), in which we seek as many edges going from one particular side
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Complexity of Boolean automata networks under block-parallel update modes arXiv.cs.DM Pub Date : 2024-02-09 Kévin Perrot, Sylvain Sené, Léah Tapin
Boolean automata networks (aka Boolean networks) are space-time discrete dynamical systems, studied as a model of computation and as a representative model of natural phenomena. A collection of simple entities (the automata) update their 0-1 states according to local rules. The dynamics of the network is highly sensitive to update modes, i.e., to the schedule according to which the automata apply their
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Expressivity of Geometric Inhomogeneous Random Graphs -- Metric and Non-Metric arXiv.cs.DM Pub Date : 2024-02-06 Benjamin Dayan, Marc Kaufmann, Ulysse Schaller
Recently there has been increased interest in fitting generative graph models to real-world networks. In particular, Bl\"asius et al. have proposed a framework for systematic evaluation of the expressivity of random graph models. We extend this framework to Geometric Inhomogeneous Random Graphs (GIRGs). This includes a family of graphs induced by non-metric distance functions which allow capturing
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Further Constructions of AMUBs for Non-prime power Composite Dimensions arXiv.cs.DM Pub Date : 2024-02-06 Ajeet Kumar, Subhamoy Maitra
Construction of a large class of Mutually Unbiased Bases (MUBs) for non-prime power composite dimensions ($d = k\times s$) is a long standing open problem, which leads to different construction methods for the class Approximate MUBs (AMUBs) by relaxing the criterion that the absolute value of the dot product between two vectors chosen from different bases should be $\leq \frac{\beta}{\sqrt{d}}$. In
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Fast solutions to k-parity and k-synchronisation using parallel automata networks arXiv.cs.DM Pub Date : 2024-02-05 Pacôme Perrotin, Eurico Ruivo, Pedro Paulo Balbi
We present a family of automata networks that solve the k-parity problem when run in parallel. These solutions are constructed by connecting cliques in a non-cyclical fashion. The size of the local neighbourhood is linear in the size of the alphabet, and the convergence time is proven to always be the diameter of the interaction graph. We show that this family of solutions can be slightly altered to
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Independent set reconfiguration in H-free graphs arXiv.cs.DM Pub Date : 2024-02-05 Valentin Bartier, Nicolas Bousquet, Moritz Mühlenthaler
Given a graph $G$ and two independent sets of $G$, the independent set reconfiguration problem asks whether one independent set can be transformed into the other by moving a single vertex at a time, such that at each intermediate step we have an independent set of $G$. We study the complexity of this problem for $H$-free graphs under the token sliding and token jumping rule. Our contribution is twofold
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Morse frames arXiv.cs.DM Pub Date : 2024-02-05 Gilles BertrandLIGM, Laurent NajmanLIGM
In the context of discrete Morse theory, we introduce Morse frames, which are maps that associate a set of critical simplexes to all simplexes. The main example of Morse frames are the Morse references. In particular, these Morse references allow computing Morse complexes, an important tool for homology. We highlight the link between Morse references and gradient flows. We also propose a novel presentation