IA Scholar Query: Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization.
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Internet Archive Scholar query results feedeninfo@archive.orgMon, 05 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Resolving Infeasibility of Linear Systems: A Parameterized Approach
https://scholar.archive.org/work/bjiqf3snbnab5jar2rvgxgtfby
Deciding feasibility of large systems of linear equations and inequalities is one of the most fundamental algorithmic tasks. However, due to data inaccuracies or modeling errors, in practical applications one often faces linear systems that are infeasible. Extensive theoretical and practical methods have been proposed for post-infeasibility analysis of linear systems. This generally amounts to detecting a feasibility blocker of small size k, which is a set of equations and inequalities whose removal or perturbation from the large system of size m yields a feasible system. This motivates a parameterized approach towards post-infeasibility analysis, where we aim to find a feasibility blocker of size at most k in fixed-parameter time f(k) · m^O(1). We establish parameterized intractability (W[1]- and NP-hardness) results already in very restricted settings for different choices of the parameters maximum size of a deletion set, number of positive/negative right-hand sides, treewidth, pathwidth and treedepth. Additionally, we rule out a polynomial compression for MinFB parameterized by the size of a deletion set and the number of negative right-hand sides. Furthermore, we develop fixed-parameter algorithms parameterized by various combinations of these parameters when every row of the system corresponds to a difference constraint. Our algorithms capture the case of Directed Feedback Arc Set, a fundamental parameterized problem whose fixed-parameter tractability was shown by Chen et al. (STOC 2008).Kristóf Bérczi, Alexander Göke, Lydia Mirabel Mendoza-Cadena, Matthias Mnichwork_bjiqf3snbnab5jar2rvgxgtfbyMon, 05 Sep 2022 00:00:00 GMTAlmost Consistent Systems of Linear Equations
https://scholar.archive.org/work/4ktke6denbdqbmcnglhojd3t7e
Checking whether a system of linear equations is consistent is a basic computational problem with ubiquitous applications. When dealing with inconsistent systems, one may seek an assignment that minimizes the number of unsatisfied equations. This problem is NP-hard and UGC-hard to approximate within any constant even for two-variable equations over the two-element field. We study this problem from the point of view of parameterized complexity, with the parameter being the number of unsatisfied equations. We consider equations defined over Euclidean domains - a family of commutative rings that generalize finite and infinite fields including the rationals, the ring of integers, and many other structures. We show that if every equation contains at most two variables, the problem is fixed-parameter tractable. This generalizes many eminent graph separation problems such as Bipartization, Multiway Cut and Multicut parameterized by the size of the cutset. To complement this, we show that the problem is W[1]-hard when three or more variables are allowed in an equation, as well as for many commutative rings that are not Euclidean domains. On the technical side, we introduce the notion of important balanced subgraphs, generalizing important separators of Marx [Theor. Comput. Sci. 2006] to the setting of biased graphs. Furthermore, we use recent results on parameterized MinCSP [Kim et al., SODA 2021] to efficiently solve a generalization of Multicut with disjunctive cut requests.Konrad K. Dabrowski, Peter Jonsson, Sebastian Ordyniak, George Osipov, Magnus Wahlströmwork_4ktke6denbdqbmcnglhojd3t7eThu, 04 Aug 2022 00:00:00 GMTSubexponential Parameterized Algorithms for Cut and Cycle Hitting Problems on H-Minor-Free Graphs
https://scholar.archive.org/work/5ealjkbcerh2titoxqrtbehdqi
We design the first subexponential-time (parameterized) algorithms for several cut and cycle-hitting problems on H-minor free graphs. In particular, we obtain the following results (where k is the solution-size parameter). 1. 2^O(√(k)log k)· n^O(1) time algorithms for Edge Bipartization and Odd Cycle Transversal; 2. a 2^O(√(k)log^4 k)· n^O(1) time algorithm for Edge Multiway Cut and a 2^O(r √(k)log k)· n^O(1) time algorithm for Vertex Multiway Cut, where r is the number of terminals to be separated; 3. a 2^O((r+√(k))log^4 (rk))· n^O(1) time algorithm for Edge Multicut and a 2^O((√(rk)+r) log (rk))· n^O(1) time algorithm for Vertex Multicut, where r is the number of terminal pairs to be separated; 4. a 2^O(√(k)log g log^4 k)· n^O(1) time algorithm for Group Feedback Edge Set and a 2^O(g √(k)log(gk))· n^O(1) time algorithm for Group Feedback Vertex Set, where g is the size of the group. 5. In addition, our approach also gives n^O(√(k)) time algorithms for all above problems with the exception of n^O(r+√(k)) time for Edge/Vertex Multicut and (ng)^O(√(k)) time for Group Feedback Edge/Vertex Set. We obtain our results by giving a new decomposition theorem on graphs of bounded genus, or more generally, an h-almost-embeddable graph for any fixed constant h. In particular we show the following. Let G be an h-almost-embeddable graph for a constant h. Then for every p∈ℕ, there exist disjoint sets Z_1,...,Z_p ⊆ V(G) such that for every i ∈{1,...,p} and every Z'⊆ Z_i, the treewidth of G/(Z_i\ Z') is O(p+|Z'|). Here G/(Z_i\ Z') is the graph obtained from G by contracting edges with both endpoints in Z_i \ Z'.Sayan Bandyapadhyay, William Lochet, Daniel Lokshtanov, Saket Saurabh, Jie Xuework_5ealjkbcerh2titoxqrtbehdqiMon, 04 Jul 2022 00:00:00 GMTA Survey on the k-Path Vertex Cover Problem
https://scholar.archive.org/work/jirft327pzcwzd2yssdkw3a7eu
Given an integer k ≥ 2, a k-path is a path on k vertices. A set of vertices in a graph G is called a k-path vertex cover if it includes at least one vertex of every k-path of G. A minimum k-path vertex cover in G is a k-path vertex cover having the smallest possible number of vertices and its cardinality is called the k-path vertex cover number of G. In the k-path vertex cover problem, the goal is to find a minimum k-path vertex cover in a given graph. In this paper, we present a brief survey of the current state of the art in the study of the k-path vertex cover problem and the k-path vertex cover number.Jianhua Tuwork_jirft327pzcwzd2yssdkw3a7euWed, 20 Apr 2022 00:00:00 GMTDevelopments in the Tensor Network – from Statistical Mechanics to Quantum Entanglement
https://scholar.archive.org/work/zdwdgts6obd5bd5ojofjvupgr4
Tensor networks (TNs) have become one of the most essential building blocks for various fields of theoretical physics such as condensed matter theory, statistical mechanics, quantum information, and quantum gravity. This review provides a unified description of a series of developments in the TN from the statistical mechanics side. In particular, we begin with the variational principle for the transfer matrix of the 2D Ising model, which naturally leads us to the matrix product state (MPS) and the corner transfer matrix (CTM). We then explain how the CTM can be evolved to such MPS-based approaches as density matrix renormalization group (DMRG) and infinite time-evolved block decimation. We also elucidate that the finite-size DMRG played an intrinsic role for incorporating various quantum information concepts in subsequent developments in the TN. After surveying higher-dimensional generalizations like tensor product states or projected entangled pair states, we describe tensor renormalization groups (TRGs), which are a fusion of TNs and Kadanoff-Wilson type real-space renormalization groups, focusing on their fixed point structures. We then discuss how the difficulty in TRGs for critical systems can be overcome in the tensor network renormalization and the multi-scale entanglement renormalization ansatz.Kouichi Okunishi, Tomotoshi Nishino, Hiroshi Uedawork_zdwdgts6obd5bd5ojofjvupgr4Wed, 16 Mar 2022 00:00:00 GMTA Survey on the k-Path Vertex Cover Problem
https://scholar.archive.org/work/spmuj2dhofffjagy7gbdoqhnmm
Given a graph G=(V,E) and a positive integer k≥2, a k-path vertex cover is a subset of vertices F such that every path on k vertices in G contains at least one vertex from F. A minimum k-path vertex cover in G is a k-path vertex cover with minimum cardinality and its cardinality is called the k-path vertex cover number of G. In the k-path vertex cover problem, it is required to find a minimum k-path vertex cover in a given graph. In this paper, we present a brief survey of the current state of the art in the study of the k-path vertex cover problem and the k-path vertex cover number.Jianhua Tuwork_spmuj2dhofffjagy7gbdoqhnmmWed, 12 Jan 2022 00:00:00 GMTA Framework for Parameterized Subexponential Algorithms for Generalized Cycle Hitting Problems on Planar Graphs
https://scholar.archive.org/work/h6dhxhpk7bf4lbfgzdz2bcm66e
Subexponential parameterized algorithms are known for a wide range of natural problems on planar graphs, but the techniques are usually highly problem specific. The goal of this paper is to introduce a framework for obtaining n^O(√(k)) time algorithms for a family of graph modification problems that includes problems that can be seen as generalized cycle hitting problems. Our starting point is the Node Unique Label Cover problem (that is, given a CSP instance where each constraint is a permutation of values on two variables, the task is to delete k variables to make the instance satisfiable). We introduce a variant of the problem where k vertices have to be deleted such that every 2-connected component of the remaining instance is satisfiable. Then we extend the problem with cardinality constraints that restrict the number of times a certain value can be used (globally or within a 2-connected component of the solution). We show that there is an n^O(√(k)) time algorithm on planar graphs for any problem that can be formulated this way, which includes a large number of well-studied problems, for example, Odd Cycle Transversal, Subset Feedback Vertex Set, Group Feedback Vertex Set, Subset Group Feedback Vertex Set, Vertex Multiway Cut, and Component Order Connectivity. For those problems that admit appropriate (quasi)polynomial kernels (that increase the parameter only linearly and preserve planarity), our results immediately imply 2^O(√(k)·polylog(k))n^O(1) time parameterized algorithms on planar graphs. In particular, we use or adapt known kernelization results to obtain 2^O(√(k)·polylog(k)) n^O(1) time (randomized) algorithms for Vertex Multiway Cut, Group Feedback Vertex Set, and Subset Feedback Vertex Set.Dániel Marx, Pranabendu Misra, Daniel Neuen, Prafullkumar Talework_h6dhxhpk7bf4lbfgzdz2bcm66eThu, 28 Oct 2021 00:00:00 GMT