IA Scholar Query: Between clique-width and linear clique-width of bipartite graphs.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSun, 27 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Maximum independent set (stable set) problem: A mathematical programming model with valid inequalities; Computational testing with binary search and alternate optimal basic solutions (extreme points)
https://scholar.archive.org/work/ppk4axwzt5czrajdurejs4evci
This paper deals with the maximum independent set (M.I.S.) problem, also known as the stable set problem. The basic mathematical programming model that captures this problem is an Integer Program (I.P.) with zero-one variables and only the edge inequalities. We present an enhanced model by adding a polynomial number of linear constraints, known as valid inequalities; this new model is still polynomial in the number of vertices in the graph. We carried out computational testing of the Linear Relaxation of the new Integer Program. We tested about 7000 instances of randomly generated (and connected) graphs with up to 64 vertices (as well as all 64, 128, and 256-vertex instances at the "challenge" website OEIS.org). In each of these instances, the Linear Relaxation returned an optimal solution with (i) every variable having an integer value, and (ii) the optimal solution value of the Linear Relaxation was the same as that of the original (basic) Integer Program. Our computational experience has been that a binary search on the objective function value is a powerful tool which yields a (weakly) polynomial algorithm.Prabhu Manyemwork_ppk4axwzt5czrajdurejs4evciSun, 27 Nov 2022 00:00:00 GMTFair allocation of indivisible items with conflict graphs
https://scholar.archive.org/work/7ckzprkjq5cfpc2pbkec7sgaae
We consider the fair allocation of indivisible items to several agents and add a graph theoretical perspective to this classical problem. Namely, we introduce an incompatibility relation between pairs of items described in terms of a conflict graph. Every subset of items assigned to one agent has to form an independent set in this graph. Thus, the allocation of items to the agents corresponds to a partial coloring of the conflict graph. Every agent has its own profit valuation for every item. Aiming at a fair allocation, our goal is the maximization of the lowest total profit of items allocated to any one of the agents. The resulting optimization problem contains, as special cases, both Partition and Independent Set. In our contribution we derive complexity and algorithmic results depending on the properties of the given graph. We show that the problem is strongly NP-hard for bipartite graphs and their line graphs, and solvable in pseudo-polynomial time for the classes of chordal graphs, cocomparability graphs, biconvex bipartite graphs, and graphs of bounded treewidth. Each of the pseudo-polynomial algorithms can also be turned into a fully polynomial approximation scheme (FPTAS).Nina Chiarelli, Matjaž Krnc, Martin Milanič, Ulrich Pferschy, Nevena Pivač, Joachim Schauerwork_7ckzprkjq5cfpc2pbkec7sgaaeSat, 26 Nov 2022 00:00:00 GMTComplexity Framework For Forbidden Subgraphs
https://scholar.archive.org/work/kis4wfc2pbhz3fsir7gy3vhmea
For any finite set ℋ = {H_1,...,H_p} of graphs, a graph is ℋ-subgraph-free if it does not contain any of H_1,...,H_p as a subgraph. We propose a meta-theorem to classify if problems are "efficiently solvable" or "computationally hard" on ℋ-subgraph-free graphs. The conditions are that the problem should be efficiently solvable on graphs of bounded treewidth, computationally hard on subcubic graphs, and computational hardness is preserved under edge subdivision. We show that all problems satisfying these conditions are efficiently solvable if ℋ contains a disjoint union of one or more paths and subdivided claws, and are computationally hard otherwise. To illustrate the broad applicability of our framework, we study covering or packing problems, network design problems and width parameter problems. We apply the framework to obtain a dichotomy between polynomial-time solvability and NP-completeness. For other problems we obtain a dichotomy between almost-linear-time solvability and having no subquadratic-time algorithm (conditioned on some hardness hypotheses). In this way we strengthen results in the literature.Matthew Johnson and Barnaby Martin and Jelle J. Oostveen and Sukanya Pandey and Daniël Paulusma and Siani Smith and Erik Jan van Leeuwenwork_kis4wfc2pbhz3fsir7gy3vhmeaWed, 23 Nov 2022 00:00:00 GMTEdge Multiway Cut and Node Multiway Cut are NP-complete on subcubic graphs
https://scholar.archive.org/work/jnipfmnok5d7zokrcpq2imtlny
We show that Edge Multiway Cut (also called Multiterminal Cut) and Node Multiway Cut are NP-complete on graphs of maximum degree 3 (also known as subcubic graphs). This improves on a previous degree bound of 11. Our NP-completeness result holds even for subcubic graphs that are planar.Matthew Johnson, Barnaby Martin, Siani Smith, Sukanya Pandey, Daniel Paulusma, Erik Jan van Leeuwenwork_jnipfmnok5d7zokrcpq2imtlnyTue, 22 Nov 2022 00:00:00 GMTOn Structural Parameterizations of Star Coloring
https://scholar.archive.org/work/gkww76kyobaohmuajnezuxjm5q
A Star Coloring of a graph G is a proper vertex coloring such that every path on four vertices uses at least three distinct colors. The minimum number of colors required for such a star coloring of G is called star chromatic number, denoted by χ_s(G). Given a graph G and a positive integer k, the STAR COLORING PROBLEM asks whether G has a star coloring using at most k colors. This problem is NP-complete even on restricted graph classes such as bipartite graphs. In this paper, we initiate a study of STAR COLORING from the parameterized complexity perspective. We show that STAR COLORING is fixed-parameter tractable when parameterized by (a) neighborhood diversity, (b) twin-cover, and (c) the combined parameters clique-width and the number of colors.Sriram Bhyravarapu, I. Vinod Reddywork_gkww76kyobaohmuajnezuxjm5qTue, 22 Nov 2022 00:00:00 GMTOn the Complexity of Problems on Tree-structured Graphs
https://scholar.archive.org/work/223ftdgzmvggvbf46jmkfncs3y
In this paper, we introduce a new class of parameterized problems, which we call XALP: the class of all parameterized problems that can be solved in f(k)n^O(1) time and f(k)log n space on a non-deterministic Turing Machine with access to an auxiliary stack (with only top element lookup allowed). Various natural problems on 'tree-structured graphs' are complete for this class: we show that List Coloring and All-or-Nothing Flow parameterized by treewidth are XALP-complete. Moreover, Independent Set and Dominating Set parameterized by treewidth divided by log n, and Max Cut parameterized by cliquewidth are also XALP-complete. Besides finding a 'natural home' for these problems, we also pave the road for future reductions. We give a number of equivalent characterisations of the class XALP, e.g., XALP is the class of problems solvable by an Alternating Turing Machine whose runs have tree size at most f(k)n^O(1) and use f(k)log n space. Moreover, we introduce 'tree-shaped' variants of Weighted CNF-Satisfiability and Multicolor Clique that are XALP-complete.Hans L. Bodlaender and Carla Groenland and Hugo Jacob and Marcin Pilipczuk and Michal Pilipczukwork_223ftdgzmvggvbf46jmkfncs3ySun, 20 Nov 2022 00:00:00 GMTCounting Subgraphs in Somewhere Dense Graphs
https://scholar.archive.org/work/rvbsk2qgrbdejbks47ldc45kca
We study the problems of counting copies and induced copies of a small pattern graph H in a large host graph G. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns H. In this work, we address the more challenging task of analysing the complexity for restricted patterns and restricted hosts. Specifically we ask which families of allowed patterns and hosts imply fixed-parameter tractability, i.e., the existence of an algorithm running in time f(H)· |G|^O(1) for some computable function f. Our main results present exhaustive and explicit complexity classifications for families that satisfy natural closure properties. Among others, we identify the problems of counting small matchings and independent sets in subgraph-closed graph classes 𝒢 as our central objects of study and establish the following crisp dichotomies as consequences of the Exponential Time Hypothesis: (1) Counting k-matchings in a graph G∈𝒢 is fixed-parameter tractable if and only if 𝒢 is nowhere dense. (2) Counting k-independent sets in a graph G∈𝒢 is fixed-parameter tractable if and only if 𝒢 is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if 𝒢 is somewhere dense, i.e., not nowhere dense. These base cases of our classifications subsume a wide variety of previous results on the matching and independent set problem, such as counting k-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in F-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs (Bressan, Roth; FOCS 21), as well as counting k-independent sets in bipartite graphs (Curticapean et al.; Algorithmica 19).Marco Bressan, Leslie Ann Goldberg, Kitty Meeks, Marc Rothwork_rvbsk2qgrbdejbks47ldc45kcaWed, 16 Nov 2022 00:00:00 GMTBayesian Reconstruction and Differential Testing of Excised mRNA
https://scholar.archive.org/work/n4rvlx5bofbw7hnjfrox3fo764
Characterizing the differential excision of mRNA is critical for understanding the functional complexity of a cell or tissue, from normal developmental processes to disease pathogenesis. Most transcript reconstruction methods infer full-length transcripts from high-throughput sequencing data. However, this is a challenging task due to incomplete annotations and the differential expression of transcripts across cell-types, tissues, and experimental conditions. Several recent methods circumvent these difficulties by considering local splicing events, but these methods lose transcript-level splicing information and may conflate transcripts. We develop the first probabilistic model that reconciles the transcript and local splicing perspectives. First, we formalize the sequence of mRNA excisions (SME) reconstruction problem, which aims to assemble variable-length sequences of mRNA excisions from RNA-sequencing data. We then present a novel hierarchical Bayesian admixture model for the Reconstruction of Excised mRNA (BREM). BREM interpolates between local splicing events and full-length transcripts and thus focuses only on SMEs that have high posterior probability. We develop posterior inference algorithms based on Gibbs sampling and local search of independent sets and characterize differential SME usage using generalized linear models based on converged BREM model parameters. We show that BREM achieves higher F1 score for reconstruction tasks and improved accuracy and sensitivity in differential splicing when compared with four state-of-the-art transcript and local splicing methods on simulated data. Lastly, we evaluate BREM on both bulk and scRNA sequencing data based on transcript reconstruction, novelty of transcripts produced, model sensitivity to hyperparameters, and a functional analysis of differentially expressed SMEs, demonstrating that BREM captures relevant biological signal.Marjan Hosseini, Devin McConnell, Derek Aguiarwork_n4rvlx5bofbw7hnjfrox3fo764Mon, 14 Nov 2022 00:00:00 GMTAn Improved Parameterized Algorithm for Treewidth
https://scholar.archive.org/work/5htjoqcjabb2xehfkon7gjsxau
We give an algorithm that takes as input an n-vertex graph G and an integer k, runs in time 2^O(k^2) n^O(1), and outputs a tree decomposition of G of width at most k, if such a decomposition exists. This resolves the long-standing open problem of whether there is a 2^o(k^3) n^O(1) time algorithm for treewidth. In particular, our algorithm is the first improvement on the dependency on k in algorithms for treewidth since the 2^O(k^3) n^O(1) time algorithm given by Bodlaender and Kloks [ICALP 1991] and Lagergren and Arnborg [ICALP 1991]. We also give an algorithm that given an n-vertex graph G, an integer k, and a rational ε∈ (0,1), in time k^O(k/ε) n^O(1) either outputs a tree decomposition of G of width at most (1+ε)k or determines that the treewidth of G is larger than k. Prior to our work, no approximation algorithms for treewidth with approximation ratio less than 2, other than the exact algorithms, were known. Both of our algorithms work in polynomial space.Tuukka Korhonen, Daniel Lokshtanovwork_5htjoqcjabb2xehfkon7gjsxauMon, 14 Nov 2022 00:00:00 GMTTowards near-term quantum simulation of materials
https://scholar.archive.org/work/pzwomhk5nvcylcmk7cimxgt2om
Simulation of materials is one of the most promising applications of quantum computers. On near-term hardware the crucial constraint on these simulations is circuit depth. Many quantum simulation algorithms rely on a layer of unitary evolutions generated by each term in a Hamiltonian. This appears in time-dynamics as a single Trotter step, and in variational quantum eigensolvers under the Hamiltonian variational ansatz as a single ansatz layer. We present a new quantum algorithm design for materials modelling where the depth of a layer is independent of the system size. This design takes advantage of the locality of materials in the Wannier basis and employs a tailored fermionic encoding that preserves locality. We analyse the circuit costs of this approach and present a compiler that transforms density functional theory data into quantum circuit instructions – connecting the physics of the material to the simulation circuit. The compiler automatically optimises circuits at multiple levels, from the base gate level to optimisations derived from the physics of the specific target material. We present numerical results for materials spanning a wide structural and technological range. Our results demonstrate a reduction of many orders of magnitude in circuit depth over standard prior methods that do not consider the structure of the Hamiltonian. For example our results improve resource requirements for Strontium Vanadate (SrVO_3) from 864 to 180 qubits for a 3×3×3 lattice, and the circuit depth of a single Trotter or variational layer from 7.5× 10^8 to depth 884. Although this is still beyond current hardware, our results show that materials simulation may be feasible on quantum computers without necessarily requiring scalable, fault-tolerant quantum computers, provided quantum algorithm design incorporates understanding of the materials and applications.Laura Clinton, Toby Cubitt, Brian Flynn, Filippo Maria Gambetta, Joel Klassen, Ashley Montanaro, Stephen Piddock, Raul A. Santos, Evan Sheridanwork_pzwomhk5nvcylcmk7cimxgt2omThu, 10 Nov 2022 00:00:00 GMTDirected Acyclic Outerplanar Graphs Have Constant Stack Number
https://scholar.archive.org/work/4mvdsdxv3fhbzpyliyjlr5enci
The stack number of a directed acyclic graph G is the minimum k for which there is a topological ordering of G and a k-coloring of the edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological ordering. We prove that the stack number of directed acyclic outerplanar graphs is bounded by a constant, which gives a positive answer to a conjecture by Heath, Pemmaraju and Trenk [SIAM J. Computing, 1999]. As an immediate consequence, this shows that all upward outerplanar graphs have constant stack number, answering a question by Bhore et al. [GD 2021] and thereby making significant progress towards the problem for general upward planar graphs originating from Nowakowski and Parker [Order, 1989]. As our main tool we develop the novel technique of directed H-partitions, which might be of independent interest. We complement the bounded stack number for directed acyclic outerplanar graphs by constructing a family of directed acyclic 2-trees that have unbounded stack number, thereby refuting a conjecture by Nöllenburg and Pupyrev [arXiv:2107.13658, 2021].Paul Jungeblut, Laura Merker, Torsten Ueckerdtwork_4mvdsdxv3fhbzpyliyjlr5enciWed, 09 Nov 2022 00:00:00 GMTTight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs
https://scholar.archive.org/work/3npf6q457nbwja2f7l5kkusn3a
We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. For sets σ,ρ of non-negative integers, a (σ,ρ)-set of a graph G is a set S of vertices such that |N(u)∩ S|∈σ for every u∈ S, and |N(v)∩ S|∈ρ for every v∉S. The problem of finding a (σ,ρ)-set (of a certain size) unifies standard problems such as Independent Set, Dominating Set, Independent Dominating Set, and many others. For all pairs of finite or cofinite sets (σ,ρ), we determine (under standard complexity assumptions) the best possible value c_σ,ρ such that there is an algorithm that counts (σ,ρ)-sets in time c_σ,ρ^ tw· n^O(1) (if a tree decomposition of width tw is given in the input). For example, for the Exact Independent Dominating Set problem (also known as Perfect Code) corresponding to σ={0} and ρ={1}, we improve the 3^ tw· n^O(1) algorithm of [van Rooij, 2020] to 2^ tw· n^O(1). Despite the unusually delicate definition of c_σ,ρ, we show that our algorithms are most likely optimal, i.e., for any pair (σ, ρ) of finite or cofinite sets where the problem is non-trivial, and any ε>0, a (c_σ,ρ-ε)^ tw· n^O(1)-algorithm counting the number of (σ,ρ)-sets would violate the Counting Strong Exponential-Time Hypothesis (#SETH). For finite sets σ and ρ, our lower bounds also extend to the decision version, showing that our algorithms are optimal in this setting as well. In contrast, for many cofinite sets, we show that further significant improvements for the decision and optimization versions are possible using the technique of representative sets.Jacob Focke, Dániel Marx, Fionn Mc Inerney, Daniel Neuen, Govind S. Sankar, Philipp Schepper, Philip Wellnitzwork_3npf6q457nbwja2f7l5kkusn3aTue, 08 Nov 2022 00:00:00 GMTHardness of Graph-Structured Algebraic and Symbolic Problems
https://scholar.archive.org/work/n5vmph5rjzg23hwa4xh37pduye
In this paper, we study the hardness of solving graph-structured linear systems with coefficients over a finite field ℤ_p and over a polynomial ring 𝔽[x_1,...,x_t]. We reduce solving general linear systems in ℤ_p to solving unit-weight low-degree graph Laplacians over ℤ_p with a polylogarithmic overhead on the number of non-zeros. Given the hardness of solving general linear systems in ℤ_p [Casacuberta-Kyng 2022], this result shows that it is unlikely that we can generalize Laplacian solvers over ℝ, or finite-element based methods over ℝ in general, to a finite-field setting. We also reduce solving general linear systems over ℤ_p to solving linear systems whose coefficient matrices are walk matrices (matrices with all ones on the diagonal) and normalized Laplacians (Laplacians that are also walk matrices) over ℤ_p. We often need to apply linear system solvers to random linear systems, in which case the worst case analysis above might be less relevant. For example, we often need to substitute variables in a symbolic matrix with random values. Here, a symbolic matrix is simply a matrix whose entries are in a polynomial ring 𝔽[x_1, ..., x_t]. We formally define the reducibility between symbolic matrix classes, which are classified in terms of the degrees of the entries and the number of occurrences of the variables. We show that the determinant identity testing problem for symbolic matrices with polynomial degree 1 and variable multiplicity at most 3 is at least as hard as the same problem for general matrices over ℝ.Jingbang Chen, Yu Gao, Yufan Huang, Richard Peng, Runze Wangwork_n5vmph5rjzg23hwa4xh37pduyeTue, 08 Nov 2022 00:00:00 GMTThe Parameterized Complexity of the Survivable Network Design Problem
https://scholar.archive.org/work/ycreqzb64rcrfjojzathek4c5y
For the well-known Survivable Network Design Problem (SNDP) we are given an undirected graph G with edge costs, a set R of terminal vertices, and an integer demand d_s,t for every terminal pair s,t∈ R. The task is to compute a subgraph H of G of minimum cost, such that there are at least d_s,t disjoint paths between s and t in H. If the paths are required to be edge-disjoint we obtain the edge-connectivity variant (EC-SNDP), while internally vertex-disjoint paths result in the vertex-connectivity variant (VC-SNDP). Another important case is the element-connectivity variant (LC-SNDP), where the paths are disjoint on edges and non-terminals. In this work we shed light on the parameterized complexity of the above problems. We consider several natural parameters, which include the solution size ℓ, the sum of demands D, the number of terminals k, and the maximum demand d_max. Using simple, elegant arguments, we prove the following results. - We give a complete picture of the parameterized tractability of the three variants w.r.t. parameter ℓ: both EC-SNDP and LC-SNDP are FPT, while VC-SNDP is W[1]-hard. - We identify some special cases of VC-SNDP that are FPT: * when d_max≤ 3 for parameter ℓ, * on locally bounded treewidth graphs (e.g., planar graphs) for parameter ℓ, and * on graphs of treewidth tw for parameter tw+D. - The well-known Directed Steiner Tree (DST) problem can be seen as single-source EC-SNDP with d_max=1 on directed graphs, and is FPT parameterized by k [Dreyfus Wagner 1971]. We show that in contrast, the 2-DST problem, where d_max=2, is W[1]-hard, even when parameterized by ℓ.Andreas Emil Feldmann, Anish Mukherjee, Erik Jan van Leeuwenwork_ycreqzb64rcrfjojzathek4c5yTue, 08 Nov 2022 00:00:00 GMTVariable Markov dynamics as a multi-focal lens to map multi-scale complex networks
https://scholar.archive.org/work/gkaz6zug7fbkbblc72q56xsm4q
From traffic flows on road networks to electrical signals in brain networks, many real-world networks contain modular structures of different sizes and densities. In the networks where modular structures emerge due to coupling between nodes with similar dynamical functions, we can identify them using flow-based community detection methods. However, these methods implicitly assume that communities are dense or clique-like which can shatter sparse communities due to a field-of-view limit inherent in one-step dynamics. Taking multiple steps with shorter or longer Markov time enables us to effectively zoom in or out to capture small or long-range communities. However, zooming out to avoid the field-of-view limit comes at the expense of introducing or increasing a lower resolution limit. Here we relax the constant Markov time constraint and introduce variable Markov dynamics as a multi-focal lens to capture functional communities in networks with a higher range of scales. With variable Markov time, a random walker can keep one-step dynamics in dense areas to avoid the resolution limit and move faster in sparse areas to detect long-range modular structures and prevent the field-of-view limit. We analyze the performance of variable Markov time using the flow-based community detection method called the map equation. We have implemented the map equation with variable Markov time in the search algorithm Infomap without any complexity overhead and tested its performance on synthetic and real-world networks from different domains. Results show that it outperforms the standard map equation in networks with constrained structures and locally sparse regions. In addition, the method estimates the optimal Markov time and avoids parameter tuning.Daniel Edler, Jelena Smiljanić, Anton Holmgren, Alexandre Antonelli, Martin Rosvallwork_gkaz6zug7fbkbblc72q56xsm4qTue, 08 Nov 2022 00:00:00 GMTSketches, metrics and fast algorithms
https://scholar.archive.org/work/q2ke5ehfxbhdzeqzosxoqh756q
As it has become easier and cheaper to collect big datasets in the last few decades, designing efficient and low-cost algorithms for these datasets has attracted unprecedented attention. However, in most applications, even storing datasets as acquired has become extremely costly and inefficient, which motivates the study of sublinear algorithms. This thesis focuses on studying two fundamental graph problems in the sublinear regime. Furthermore, it presents a fast kernel density estimation algorithm and data structure. The first part of this thesis focuses on graph spectral sparsification in dynamic streams. Our algorithm achieves almost optimal runtime and space simultaneously in a single pass. Our method is based on a novel bucketing scheme that enables us to recover high effective resistance edges faster. This contribution presents a novel approach to the effective resistance embedding of the graph, using locality-sensitive hash functions, with possible further future applications. The second part of this thesis presents spanner construction results in the dynamic streams and the simultaneous communication models. First, we show how one can construct a O(n 2/3 )-spanner using the above-mentioned almost-optimal single-pass spectral sparsifier, resulting in the first single-pass algorithm for non-trivial spanner construction in the literature. Then, we generalize this result to constructing O(n 2/3(1−α) )-spanners using O(n 1+α ) space for any α ∈ [0, 1], providing a smooth trade-off between distortion and memory complexity. Moreover, we study the simultaneous communication model and propose a novel protocol with low per player information. Also, we show how one can leverage more rounds of communication in this setting to achieve better distortion guarantees. Finally, in the third part of this thesis, we study the kernel density estimation problem. In this problem, given a kernel function, an input dataset imposes a kernel density on the space. The goal is to design fast and memory-efficient data structures that can output approximations to the kernel density at queried points. This thesis presents a data structure based on the classical near neighbor search and localitysensitive hashing techniques that improves or matches the query time and space complexity for radial kernels considered in the literature. The approach is based on an implementation of (approximate) importance sampling for each distance range and then using near neighbor search algorithms to recover points from these distance ranges. Later, we show how to improve the runtime, using recent advances in the data-dependent near neighbor search data structures, for a class of radial kernels that includes the Gaussian kernel.Navid Nouriwork_q2ke5ehfxbhdzeqzosxoqh756qMon, 07 Nov 2022 00:00:00 GMTPerfectly Matched Sets in Graphs: Parameterized and Exact Computation
https://scholar.archive.org/work/ytayw2ljdjhvbk2zzxhxxfjhvu
In an undirected graph G=(V,E), we say (A,B) is a pair of perfectly matched sets if A and B are disjoint subsets of V and every vertex in A (resp. B) has exactly one neighbor in B (resp. A). The size of a pair of perfectly matched sets (A,B) is |A|=|B|. The PERFECTLY MATCHED SETS problem is to decide whether a given graph G has a pair of perfectly matched sets of size k. We show that PMS is W[1]-hard when parameterized by solution size k even when restricted to split graphs and bipartite graphs. We observe that PMS is FPT when parameterized by clique-width, and give FPT algorithms with respect to the parameters distance to cluster, distance to co-cluster and treewidth. Complementing FPT results, we show that PMS does not admit a polynomial kernel when parameterized by vertex cover number unless NP⊆ coNP/poly. We also provide an exact exponential algorithm running in time O^*(1.966^n). Finally, considering graphs with structural assumptions, we show that PMS remains NP-hard on planar graphs.N.R. Aravind, Roopam Saxenawork_ytayw2ljdjhvbk2zzxhxxfjhvuMon, 07 Nov 2022 00:00:00 GMTBalancing graph Voronoi diagrams with one more vertex
https://scholar.archive.org/work/tecmdbv4grelznxzndlc5mnzki
Let G=(V,E) be a graph with unit-length edges and nonnegative costs assigned to its vertices. Being given a list of pairwise different vertices S=(s_1,s_2,...,s_p), the prioritized Voronoi diagram of G with respect to S is the partition of G in p subsets V_1,V_2,...,V_p so that, for every i with 1 ≤ i ≤ p, a vertex v is in V_i if and only if s_i is a closest vertex to v in S and there is no closest vertex to v in S within the subset {s_1,s_2,...,s_i-1}. For every i with 1 ≤ i ≤ p, the load of vertex s_i equals the sum of the costs of all vertices in V_i. The load of S equals the maximum load of a vertex in S. We study the problem of adding one more vertex v at the end of S in order to minimize the load. This problem occurs in the context of optimally locating a new service facility (e.g., a school or a hospital) while taking into account already existing facilities, and with the goal of minimizing the maximum congestion at a site. There is a brute-force algorithm for solving this problem in O(nm) time on n-vertex m-edge graphs. We prove a matching time lower bound for the special case where m=n^1+o(1) and p=1, assuming the so called Hitting Set Conjecture of Abboud et al. On the positive side, we present simple linear-time algorithms for this problem on cliques, paths and cycles, and almost linear-time algorithms for trees, proper interval graphs and (assuming p to be a constant) bounded-treewidth graphs.Guillaume Ducoffework_tecmdbv4grelznxzndlc5mnzkiSun, 06 Nov 2022 00:00:00 GMTCompound Logics for Modification Problems
https://scholar.archive.org/work/lpjop6xt6zafjiqvz7ly5s4jvu
We introduce a novel model-theoretic framework inspired from graph modification and based on the interplay between model theory and algorithmic graph minors. The core of our framework is a new compound logic operating with two types of sentences, expressing graph modification: the modulator sentence, defining some property of the modified part of the graph, and the target sentence, defining some property of the resulting graph. In our framework, modulator sentences are in counting monadic second-order logic (CMSOL) and have models of bounded treewidth, while target sentences express first-order logic (FOL) properties along with minor-exclusion. Our logic captures problems that are not definable in first-order logic and, moreover, may have instances of unbounded treewidth. Also, it permits the modeling of wide families of problems involving vertex/edge removals, alternative modulator measures (such as elimination distance or 𝒢-treewidth), multistage modifications, and various cut problems. Our main result is that, for this compound logic, model-checking can be done in quadratic time. All derived algorithms are constructive and this, as a byproduct, extends the constructibility horizon of the algorithmic applications of the Graph Minors theorem of Robertson and Seymour. The proposed logic can be seen as a general framework to capitalize on the potential of the irrelevant vertex technique. It gives a way to deal with problem instances of unbounded treewidth, for which Courcelle's theorem does not apply. The proof of our meta-theorem combines novel combinatorial results related to the Flat Wall theorem along with elements of the proof of Courcelle's theorem and Gaifman's theorem. We finally prove extensions where the target property is expressible in FOL+DP, i.e., the enhancement of FOL with disjoint-paths predicates.Fedor V. Fomin and Petr A. Golovach and Ignasi Sau and Giannos Stamoulis and Dimitrios M. Thilikoswork_lpjop6xt6zafjiqvz7ly5s4jvuFri, 04 Nov 2022 00:00:00 GMTA Framework for Approximation Schemes on Disk Graphs
https://scholar.archive.org/work/otomykizzfcrvepuoxq6yroyoa
We initiate a systematic study of approximation schemes for fundamental optimization problems on disk graphs, a common generalization of both planar graphs and unit-disk graphs. Our main contribution is a general framework for designing efficient polynomial-time approximation schemes (EPTASes) for vertex-deletion problems on disk graphs, which results in EPTASes for many problems including Vertex Cover, Feedback Vertex Set, Small Cycle Hitting (in particular, Triangle Hitting), P_k-Hitting for k∈{3,4,5}, Path Deletion, Pathwidth 1-Deletion, Component Order Connectivity, Bounded Degree Deletion, Pseudoforest Deletion, Finite-Type Component Deletion, etc. All EPTASes obtained using our framework are robust in the sense that they do not require a realization of the input graph. To the best of our knowledge, prior to this work, the only problems known to admit (E)PTASes on disk graphs are Maximum Clique, Independent Set, Dominating set, and Vertex Cover, among which the existing PTAS [Erlebach et al., SICOMP'05] and EPTAS [Leeuwen, SWAT'06] for Vertex Cover require a realization of the input disk graph (while ours does not). The core of our framework is a reduction for a broad class of (approximation) vertex-deletion problems from (general) disk graphs to disk graphs of bounded local radius, which is a new invariant of disk graphs introduced in this work. Disk graphs of bounded local radius can be viewed as a mild generalization of planar graphs, which preserves certain nice properties of planar graphs. Specifically, we prove that disk graphs of bounded local radius admit the Excluded Grid Minor property and have locally bounded treewidth. This allows existing techniques for designing approximation schemes on planar graphs (e.g., bidimensionality and Baker's technique) to be directly applied to disk graphs of bounded local radius.Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, Meirav Zehaviwork_otomykizzfcrvepuoxq6yroyoaFri, 04 Nov 2022 00:00:00 GMT