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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
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Freeze-Tag in $L_1$ has Wake-up Time Five arXiv.cs.DM Pub Date : 2024-02-05 Nicolas Bonichon, Arnaud Casteigts, Cyril Gavoille, Nicolas Hanusse
The Freeze-Tag Problem, introduced in Arkin et al. (SODA'02) consists of waking up a swarm of $n$ robots, starting from a single active robot. In the basic geometric version, every robot is given coordinates in the plane. As soon as a robot is awakened, it can move towards inactive robots to wake them up. The goal is to minimize the wake-up time of the last robot, the makespan. Despite significant
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RDNF Oriented Analytics to Random Boolean Functions arXiv.cs.DM Pub Date : 2024-02-01 Levon Aslanyan, Irina Arsenyan, Vilik Karakhanyan, Hasmik Sahakyan
Dominant areas of computer science and computation systems are intensively linked to the hypercube-related studies and interpretations. This article presents some transformations and analytics for some example algorithms and Boolean domain problems. Our focus is on the methodology of complexity evaluation and integration of several types of postulations concerning special hypercube structures. Our
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Allocation of Indivisible Items with a Common Preference Graph: Minimizing Total Dissatisfaction arXiv.cs.DM Pub Date : 2024-02-01 Nina Chiarelli, Clément Dallard, Andreas Darmann, Stefan Lendl, Martin Milanič, Peter Muršič, Ulrich Pferschy
Allocating indivisible items among a set of agents is a frequently studied discrete optimization problem. In the setting considered in this work, the agents' preferences over the items are assumed to be identical. We consider a very recent measure for the overall quality of an allocation which does not rely on numerical valuations of the items. Instead, it captures the agents' opinion by a directed
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A two-phase Volume of Fluid approach to model rigid-perfectly plastic granular materials arXiv.cs.DM Pub Date : 2024-02-01 Wibke Düsterhöft-Wriggers, Svenja Schubert, Thomas Rung
Granular flow problems characterized by large deformations are widespread in various applications, including coastal and geotechnical engineering. The paper deals with the application of a rigid-perfectly plastic two-phase model extended by the Drucker-Prager yield criterion to simulate granular media with a finite volume flow solver (FV). The model refers to the combination of a Bingham fluid and
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Heuristics for the Run-length Encoded Burrows-Wheeler Transform Alphabet Ordering Problem arXiv.cs.DM Pub Date : 2024-01-26 Lily Major, Amanda Clare, Jacqueline W. Daykin, Benjamin Mora, Christine Zarges
The Burrows-Wheeler Transform (BWT) is a string transformation technique widely used in areas such as bioinformatics and file compression. Many applications combine a run-length encoding (RLE) with the BWT in a way which preserves the ability to query the compressed data efficiently. However, these methods may not take full advantage of the compressibility of the BWT as they do not modify the alphabet
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From Tripods to Bipods: Reducing the Queue Number of Planar Graphs Costs Just One Leg arXiv.cs.DM Pub Date : 2024-01-29 Henry Förster
As an alternative to previously existing planar graph product structure theorems, we prove that every planar graph $G$ is a subgraph of the strong product of $K_2$, a path and a planar subgraph of a $4$-tree. As an application, we show that the queue number of planar graphs is at most $38$ whereas the queue number of planar bipartite graphs is at most $25$.
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Transit Functions and Clustering Systems arXiv.cs.DM Pub Date : 2024-01-28 Manoj Changat, Ameera Vaheeda Shanavas, Peter F. Stadler
Transit functions serve not only as abstractions of betweenness and convexity but are also closely connected with clustering systems. Here, we investigate the canonical transit functions of binary clustering systems inspired by pyramids, i.e., interval hypergraphs. We provide alternative characterizations of weak hierarchies, and describe union-closed binary clustering systems as a subclass of pyramids
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On the Hardness of Gray Code Problems for Combinatorial Objects arXiv.cs.DM Pub Date : 2024-01-26 Arturo Merino, Namrata, Aaron Williams
Can a list of binary strings be ordered so that consecutive strings differ in a single bit? Can a list of permutations be ordered so that consecutive permutations differ by a swap? Can a list of non-crossing set partitions be ordered so that consecutive partitions differ by refinement? These are examples of Gray coding problems: Can a list of combinatorial objects (of a particular type and size) be
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Searching in trees with monotonic query times arXiv.cs.DM Pub Date : 2024-01-24 Dariusz Dereniowski, Izajasz Wrosz
We consider the following generalization of binary search in sorted arrays to tree domains. In each step of the search, an algorithm is querying a vertex $q$, and as a reply, it receives an answer, which either states that $q$ is the desired target, or it gives the neighbor of $q$ that is closer to the target than $q$. A further generalization assumes that a vertex-weight function $\omega$ gives the
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Generation of weighted trees, block trees and block graphs arXiv.cs.DM Pub Date : 2024-01-18 Tınaz Ekim, Mordechai Shalom, Mehmet Aziz Yirik
We present a general framework to generate trees every vertex of which has a non-negative weight and a color. The colors are used to impose certain restrictions on the weight and colors of other vertices. We first extend the enumeration algorithms of unweighted trees given in [19, 20] to generate weighted trees that allow zero weight. We avoid isomorphisms by generalizing the concept of centroids to
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Token Jumping in Planar Graphs has Linear Sized Kernels arXiv.cs.DM Pub Date : 2024-01-17 Daniel W. Cranston
Let $G$ be a planar graph and $I_s$ and $I_t$ be two independent sets in $G$, each of size $k$. We begin with a ``token'' on each vertex of $I_s$ and seek to move all tokens to $I_t$, by repeated ``token jumping'', removing a single token from one vertex and placing it on another vertex. We require that each intermediate arrangement of tokens again specifies an independent set of size $k$. Given $G$
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Graph Structure of an Inversive Pseudorandom Number Generator over Ring $\mathbb{Z}_{p^{e}}$ arXiv.cs.DM Pub Date : 2024-01-16 Xiaoxiong Lu, Chengqing Li, Bo Zhou
Generating random and pseudorandom numbers with a deterministic system is a long-standing challenge in theoretical research and engineering applications. Several pseudorandom number generators based on the inversive congruential method have been designed as attractive alternatives to those based on the classical linear congruential method. This paper discloses the least period of sequences generated
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Deterministic Minimum Steiner Cut in Maximum Flow Time arXiv.cs.DM Pub Date : 2023-12-27 Matthew Ding, Jason Li
We devise a deterministic algorithm for minimum Steiner cut which uses polylogarithmic maximum flow calls and near-linear time outside of these maximum flow calls. This improves on Li and Panigrahi's (FOCS 2020) algorithm which takes $O(m^{1+\epsilon})$ time outside of maximum flow calls. Our algorithm thus shows that deterministic minimum Steiner cut can be solved in maximum flow time up to polylogarithmic
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Unique Triangulated 1-Planar Graphs arXiv.cs.DM Pub Date : 2023-12-25 Franz J. Brandenburg
It is well-known that every 3-connected planar graph has a unique planar embedding on the sphere. We study the extension to triangulated 1-planar graphs, T1P graphs for short, which admit an embedding in which each edge is crossed at most once and each face is a triangle, and obtain an algorithmic solution by a cubic time recognition algorithm that also counts the number of T1P embeddings. In particular
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Embedding 1-Planar Graphs in Ten Pages arXiv.cs.DM Pub Date : 2023-12-25 Franz J. Brandenburg
Every planar graph has a 4-page book embedding and this bound is tight. We show that every 1-planar graph, which is a graph that admits a drawing with at most one crossing per edge, has a 10-page book embedding. In addition, four pages are sometimes necessary and always sufficient if the planar skeleton, obtained from a 1-planar drawing by removing all crossed edges, has a Hamiltonian cycle.
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A Spigot-Algorithm for Square-Roots: Explained and Extended arXiv.cs.DM Pub Date : 2023-12-23 Mayer Goldberg
This work presents and extends a known spigot-algorithm for computing square-roots, digit-by-digit, that is suitable for calculation by hand or an abacus, using only addition and subtraction. We offer an elementary proof of correctness for the original algorithm, then present a corresponding spigot-algorithm for computing cube-roots. Finally, we generalize the algorithm, so as to find $r$-th roots
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Scheduling with conflicts: formulations, valid inequalities, and computational experiments arXiv.cs.DM Pub Date : 2023-12-22 Phablo F. S. Moura, Roel Leus, Hande Yaman
Given an undirected graph $G=(V,E)$ (i.e. the conflict graph) where $V$ is a set of $n$ vertices (representing the jobs), processing times $p \colon V \to \mathbb{Z}_>$, and $m\geq 2$ identical machines the Parallel Machine Scheduling with Conflicts (PMC) consists in finding an assignment $c \colon V \to [m]:=\{1,\ldots, m\}$ with $c(u)\neq c(v)$ for all $\{u,v\} \in E$ that minimizes the makespan
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Deciding the Feasibility and Minimizing the Height of Tangles arXiv.cs.DM Pub Date : 2023-12-23 Oksana Firman, Philipp Kindermann, Boris Klemz, Alexander Ravsky, Alexander Wolff, Johannes Zink
We study the following combinatorial problem. Given a set of $n$ y-monotone \emph{wires}, a \emph{tangle} determines the order of the wires on a number of horizontal \emph{layers} such that the orders of the wires on any two consecutive layers differ only in swaps of neighboring wires. Given a multiset~$L$ of \emph{swaps} (that is, unordered pairs of wires) and an initial order of the wires, a tangle
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Theoretical analysis of git bisect arXiv.cs.DM Pub Date : 2023-12-21 Paul DorbecGREYC, Julien CourtielGREYC, Romain LecoqGREYC
In this paper, we consider the problem of finding a regression in a version control system (VCS), such as \texttt{git}. The set of versions is modelled by a Directed Acyclic Graph (DAG) where vertices represent versions of the software, and arcs are the changes between different versions. We assume that somewhere in the DAG, a bug was introduced, which persists in all of its subsequent versions. It
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Moving a Derivation Along a Derivation Preserves the Spine in Adhesive High-level Replacement Systems arXiv.cs.DM Pub Date : 2023-12-21 Hans-Jörg Kreowski, Aaron Lye, Aljoscha Windhorst
In this paper, we investigate the relationship between two elementary operations on derivations in adhesive high-level replacement systems that are well-known in the context of graph transformation: moving a derivation along a derivation based on parallel and sequential independence on one hand and restriction of a derivation with respect to a monomorphism into the start object on the other hand. Intuitively
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How to apply tree decomposition ideas in large networks? arXiv.cs.DM Pub Date : 2023-12-19 Johannes Carmesin, Sarah Frenkel
Graph decompositions are the natural generalisation of tree decompositions where the decomposition tree is replaced by a genuine graph. Recently they found theoretical applications in the theory of sparsity, topological graph theory, structural graph theory and geometric group theory. We demonstrate applicability of graph decompositions on large networks by implementing an efficient algorithm and testing