Exploring Algorithms for Branch Decompositions of Planar Graphs (132 Pp.)
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Fast Dynamic Programming on Graph Decompositions
Fast Dynamic Programming on Graph Decompositions∗ Johan M. M. van Rooij† Hans L. Bodlaender† Erik Jan van Leeuwen‡ Peter Rossmanith§ Martin Vatshelle‡ June 6, 2018 Abstract In this paper, we consider three types of graph decompositions, namely tree decompo- sitions, branch decompositions, and clique decompositions. We improve the running time of dynamic programming algorithms on these graph decompositions for a large number of problems as a function of the treewidth, branchwidth, or cliquewidth, respectively. Such algorithms have many practical and theoretical applications. On tree decompositions of width k, we improve the running time for Dominating Set to O∗(3k). Hereafter, we give a generalisation of this result to [ρ,σ]-domination problems with finite or cofinite ρ and σ. For these problems, we give O∗(sk)-time algorithms. Here, s is the number of ‘states’ a vertex can have in a standard dynamic programming algorithm for such a problems. Furthermore, we give an O∗(2k)-time algorithm for counting the number of perfect matchings in a graph, and generalise this result to O∗(2k)-time algorithms for many clique covering, packing, and partitioning problems. On branch decompositions of width k, we give ∗ ω k ∗ ω k an O (3 2 )-time algorithm for Dominating Set, an O (2 2 )-time algorithm for counting ∗ ω k the number of perfect matchings, and O (s 2 )-time algorithms for [ρ,σ]-domination problems involving s states with finite or cofinite ρ and σ. Finally, on clique decompositions of width k, we give O∗(4k)-time algorithms for Dominating Set, Independent Dominating Set, and Total Dominating Set. -
Total Domination on Some Graph Operators
mathematics Article Total Domination on Some Graph Operators José M. Sigarreta Faculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame 5, Col. La Garita, C. P. 39350 Acapulco, Mexico; [email protected]; Tel.: +52-744-159-2272 Abstract: Let G = (V, E) be a graph; a set D ⊆ V is a total dominating set if every vertex v 2 V has, at least, one neighbor in D. The total domination number gt(G) is the minimum cardinality among all total dominating sets. Given an arbitrary graph G, we consider some operators on this graph; S(G), R(G), and Q(G), and we give bounds or the exact value of the total domination number of these new graphs using some parameters in the original graph G. Keywords: domination theory; total domination; graph operators 1. Introduction The study of the mathematical and structural properties of operators in graphs was initiated in Reference [1]. This study is a current research topic, mainly because of its multiple theoretical and practical applications (see References [2–6]). Motivated by the previous investigations, we work here on the total domination number of some graph operators. Other important parameters have also been studied in this type of graphs (see References [7,8]). We denote by G = (V, E) a finite simple graph of order n = jVj and size m = jEj. For any two adjacent vertices u, v 2 V, uv denotes the corresponding edge, and we denote N(v) = fu 2 V : uv 2 Eg and N[v] = N(v) [ fvg. We denote by D, d the maximum and Citation: Sigarreta, J.M. -
Locating Domination in Bipartite Graphs and Their Complements
Locating domination in bipartite graphs and their complements C. Hernando∗ M. Moray I. M. Pelayoz Abstract A set S of vertices of a graph G is distinguishing if the sets of neighbors in S for every pair of vertices not in S are distinct. A locating-dominating set of G is a dominating distinguishing set. The location-domination number of G, λ(G), is the minimum cardinality of a locating-dominating set. In this work we study relationships between λ(G) and λ(G) for bipartite graphs. The main result is the characterization of all connected bipartite graphs G satisfying λ(G) = λ(G) + 1. To this aim, we define an edge-labeled graph GS associated with a distinguishing set S that turns out to be very helpful. Keywords: domination; location; distinguishing set; locating domination; complement graph; bipartite graph. AMS subject classification: 05C12, 05C35, 05C69. ∗Departament de Matem`atiques. Universitat Polit`ecnica de Catalunya, Barcelona, Spain, car- [email protected]. Partially supported by projects Gen. Cat. DGR 2017SGR1336 and MTM2015- arXiv:1711.01951v2 [math.CO] 18 Jul 2018 63791-R (MINECO/FEDER). yDepartament de Matem`atiques. Universitat Polit`ecnica de Catalunya, Barcelona, Spain, [email protected]. Partially supported by projects Gen. Cat. DGR 2017SGR1336, MTM2015-63791-R (MINECO/FEDER) and H2020-MSCA-RISE project 734922 - CONNECT. zDepartament de Matem`atiques. Universitat Polit`ecnica de Catalunya, Barcelona, Spain, ig- [email protected]. Partially supported by projects MINECO MTM2014-60127-P, igna- [email protected]. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 734922. -
Treewidth-Erickson.Pdf
Computational Topology (Jeff Erickson) Treewidth Or il y avait des graines terribles sur la planète du petit prince . c’étaient les graines de baobabs. Le sol de la planète en était infesté. Or un baobab, si l’on s’y prend trop tard, on ne peut jamais plus s’en débarrasser. Il encombre toute la planète. Il la perfore de ses racines. Et si la planète est trop petite, et si les baobabs sont trop nombreux, ils la font éclater. [Now there were some terrible seeds on the planet that was the home of the little prince; and these were the seeds of the baobab. The soil of that planet was infested with them. A baobab is something you will never, never be able to get rid of if you attend to it too late. It spreads over the entire planet. It bores clear through it with its roots. And if the planet is too small, and the baobabs are too many, they split it in pieces.] — Antoine de Saint-Exupéry (translated by Katherine Woods) Le Petit Prince [The Little Prince] (1943) 11 Treewidth In this lecture, I will introduce a graph parameter called treewidth, that generalizes the property of having small separators. Intuitively, a graph has small treewidth if it can be recursively decomposed into small subgraphs that have small overlap, or even more intuitively, if the graph resembles a ‘fat tree’. Many problems that are NP-hard for general graphs can be solved in polynomial time for graphs with small treewidth. Graphs embedded on surfaces of small genus do not necessarily have small treewidth, but they can be covered by a small number of subgraphs, each with small treewidth. -
Dominating Cliques in Graphs
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 86 (1990) 101-116 101 North-Holland DOMINATING CLIQUES IN GRAPHS Margaret B. COZZENS Mathematics Department, Northeastern University, Boston, MA 02115, USA Laura L. KELLEHER Massachusetts Maritime Academy, USA Received 2 December 1988 A set of vertices is a dominating set in a graph if every vertex not in the dominating set is adjacent to one or more vertices in the dominating set. A dominating clique is a dominating set that induces a complete subgraph. Forbidden subgraph conditions sufficient to imply the existence of a dominating clique are given. For certain classes of graphs, a polynomial algorithm is given for finding a dominating clique. A forbidden subgraph characterization is given for a class of graphs that have a connected dominating set of size three. Introduction A set of vertices in a simple, undirected graph is a dominating set if every vertex in the graph which is not in the dominating set is adjacent to one or more vertices in the dominating set. The domination number of a graph G is the minimum number of vertices in a dominating set. Several types of dominating sets have been investigated, including independent dominating sets, total dominating sets, and connected dominating sets, each of which has a corresponding domination number. Dominating sets have applications in a variety of fields, including communication theory and political science. For more background on dominating sets see [3, 5, 151 and other articles in this issue. For arbitrary graphs, the problem of finding the size of a minimum dominating set in the graph is an NP-complete problem [9]. -
A Complete Anytime Algorithm for Treewidth
UAI 2004 GOGATE & DECHTER 201 A Complete Anytime Algorithm for Treewidth Vibhav Gogate and Rina Dechter School of Information and Computer Science, University of California, Irvine, CA 92967 fvgogate,[email protected] Abstract many algorithms that solve NP-hard problems in Arti¯cial Intelligence, Operations Research, In this paper, we present a Branch and Bound Circuit Design etc. are exponential only in the algorithm called QuickBB for computing the treewidth. Examples of such algorithm are Bucket treewidth of an undirected graph. This al- elimination [Dechter, 1999] and junction-tree elimina- gorithm performs a search in the space of tion [Lauritzen and Spiegelhalter, 1988] for Bayesian perfect elimination ordering of vertices of Networks and Constraint Networks. These algo- the graph. The algorithm uses novel prun- rithms operate in two steps: (1) constructing a ing and propagation techniques which are good tree-decomposition and (2) solving the prob- derived from the theory of graph minors lem on this tree-decomposition, with the second and graph isomorphism. We present a new step generally exponential in the treewidth of the algorithm called minor-min-width for com- tree-decomposition computed in step 1. In this puting a lower bound on treewidth that is paper, we present a complete anytime algorithm used within the branch and bound algo- called QuickBB for computing a tree-decomposition rithm and which improves over earlier avail- of a graph that can feed into any algorithm like able lower bounds. Empirical evaluation of Bucket elimination [Dechter, 1999] that requires a QuickBB on randomly generated graphs and tree-decomposition. benchmarks in Graph Coloring and Bayesian The majority of recent literature on Networks shows that it is consistently bet- the treewidth problem is devoted to ¯nd- ter than complete algorithms like Quick- ing constant factor approximation algo- Tree [Shoikhet and Geiger, 1997] in terms of rithms [Amir, 2001, Becker and Geiger, 2001]. -
The Bidimensionality Theory and Its Algorithmic
The Bidimensionality Theory and Its Algorithmic Applications by MohammadTaghi Hajiaghayi B.S., Sharif University of Technology, 2000 M.S., University of Waterloo, 2001 Submitted to the Department of Mathematics in partial ful¯llment of the requirements for the degree of DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2005 °c MohammadTaghi Hajiaghayi, 2005. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author.............................................................. Department of Mathematics April 29, 2005 Certi¯ed by. Erik D. Demaine Associate Professor of Electrical Engineering and Computer Science Thesis Supervisor Accepted by . Rodolfo Ruben Rosales Chairman, Applied Mathematics Committee Accepted by . Pavel I. Etingof Chairman, Department Committee on Graduate Students 2 The Bidimensionality Theory and Its Algorithmic Applications by MohammadTaghi Hajiaghayi Submitted to the Department of Mathematics on April 29, 2005, in partial ful¯llment of the requirements for the degree of DOCTOR OF PHILOSOPHY Abstract Our newly developing theory of bidimensional graph problems provides general techniques for designing e±cient ¯xed-parameter algorithms and approximation algorithms for NP- hard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k £ k grid graph (and similar graphs) grows with k, typically as (k2), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex- removal parameters, dominating set, edge dominating set, r-dominating set, connected dominating set, connected edge dominating set, connected r-dominating set, and unweighted TSP tour (a walk in the graph visiting all vertices). -
Combinatorial Optimization
Prof. F. Eisenbrand EPFL - DISOPT Combinatorial Optimization Adrian Bock Fall 2011 Sheet 6 December 1, 2011 General remark: In order to obtain a bonus for the final grading, you may hand in written solutions to the exercises marked with a star at the beginning of the exercise session on December 13. Exercise 1 Trace the steps of algorithm from the lecture to compute a minimum weight arborescence rooted at r in the following example. r 6 a 5 2 2 2 2 2 b c d 4 2 1 3 1 2 e f g 1 3 Prove the optimality of your solution! Solution Follow algorithm for min cost arborescence. We find an r-arborescence of weight 16. The following 16 r-cuts prove the opti- mality of this solution: {a}, {b}, {d}, {g}, {a, b, d} two times each and {c}, {e}, {f}, {c, e, f}, {a,b,c,e,f,g}, {a,b,c,d,e,f,g} once. Exercise 2 Let G = (A ∪ B, E) be a bipartite graph. We define two partition matroids M1 =(E, I1) and M2 =(E, I2) with I1 = {I ⊆ E : |I ∩ δ(a)| ≤ 1 for all a ∈ A} and I2 = {I ⊆ E : |I ∩ δ(b)| ≤ 1 for all b ∈ B}. What is a set I ∈ I1 ∩ I2 in terms of graph theory? Can you maximize a weight function w : E → R over the intersection? Remark: This is a special case of the optimization over the intersection of two matroids. It can be shown that all such matroid intersection problems can be solved efficiently. -
Polynomial-Time Data Reduction for Dominating Set∗
Journal of the ACM, Vol. 51(3), pp. 363–384, 2004 Polynomial-Time Data Reduction for Dominating Set∗ Jochen Alber† Michael R. Fellows‡ Rolf Niedermeier§ Abstract Dealing with the NP-complete Dominating Set problem on graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set restricted to planar graphs has a so-called problem kernel of linear size, achieved by two simple and easy to implement reduction rules. Moreover, having implemented our reduction rules, first experiments indicate the impressive practical potential of these rules. Thus, this work seems to open up a new and prospective way how to cope with one of the most important problems in graph theory and combinatorial optimization. 1 Introduction Motivation. A core tool for practically solving NP-hard problems is data reduction through preprocessing. Weihe [40, 41] gave a striking example when dealing with the NP-complete Red/Blue Dominating Set problem appearing in the context of the European railroad network. In a preprocess- ing phase, he applied two simple data reduction rules again and again until ∗An extended abstract of this work entitled “Efficient Data Reduction for Dominating Set: A Linear Problem Kernel for the Planar Case” appeared in the Proceedings of the 8th Scandinavian Workshop on Algorithm Theory (SWAT 2002), Lecture Notes in Computer Science (LNCS) 2368, pages 150–159, Springer-Verlag 2002. †Wilhelm-Schickard-Institut f¨ur Informatik, Universit¨at T¨ubingen, Sand 13, D- 72076 T¨ubingen, Fed. Rep. of Germany. Email: [email protected]. -
Upper Dominating Set: Tight Algorithms for Pathwidth and Sub-Exponential Approximation
Upper Dominating Set: Tight Algorithms for Pathwidth and Sub-Exponential Approximation Louis Dublois1, Michael Lampis1, and Vangelis Th. Paschos1 Universit´eParis-Dauphine, PSL University, CNRS, LAMSADE, Paris, France flouis.dublois,michail.lampis,[email protected] Abstract. An upper dominating set is a minimal dominating set in a graph. In the Upper Dominating Set problem, the goal is to find an upper dominating set of maximum size. We study the complexity of parameterized algorithms for Upper Dominating Set, as well as its sub-exponential approximation. First, we prove that, under ETH, k- cannot be solved in time O(no(k)) (improving Upper Dominatingp Set on O(no( k))), and in the same time we show under the same complex- ity assumption that for any constant ratio r and any " > 0, there is no 1−" r-approximation algorithm running in time O(nk ). Then, we settle the problem's complexity parameterized by pathwidth by giving an al- gorithm running in time O∗(6pw) (improving the current best O∗(7pw)), and a lower bound showing that our algorithm is the best we can get under the SETH. Furthermore, we obtain a simple sub-exponential ap- proximation algorithm for this problem: an algorithm that produces an r-approximation in time nO(n=r), for any desired approximation ratio r < n. We finally show that this time-approximation trade-off is tight, up to an arbitrarily small constant in the second exponent: under the randomized ETH, and for any ratio r > 1 and " > 0, no algorithm 1−" can output an r-approximation in time n(n=r) . -
Tree-Decomposition Graph Minor Theory and Algorithmic Implications
DIPLOMARBEIT Tree-Decomposition Graph Minor Theory and Algorithmic Implications Ausgeführt am Institut für Diskrete Mathematik und Geometrie der Technischen Universität Wien unter Anleitung von Univ.Prof. Dipl.-Ing. Dr.techn. Michael Drmota durch Miriam Heinz, B.Sc. Matrikelnummer: 0625661 Baumgartenstraße 53 1140 Wien Datum Unterschrift Preface The focus of this thesis is the concept of tree-decomposition. A tree-decomposition of a graph G is a representation of G in a tree-like structure. From this structure it is possible to deduce certain connectivity properties of G. Such information can be used to construct efficient algorithms to solve problems on G. Sometimes problems which are NP-hard in general are solvable in polynomial or even linear time when restricted to trees. Employing the tree-like structure of tree-decompositions these algorithms for trees can be adapted to graphs of bounded tree-width. This results in many important algorithmic applications of tree-decomposition. The concept of tree-decomposition also proves to be useful in the study of fundamental questions in graph theory. It was used extensively by Robertson and Seymour in their seminal work on Wagner’s conjecture. Their Graph Minors series of papers spans more than 500 pages and results in a proof of the graph minor theorem, settling Wagner’s conjecture in 2004. However, it is not only the proof of this deep and powerful theorem which merits mention. Also the concepts and tools developed for the proof have had a major impact on the field of graph theory. Tree-decomposition is one of these spin-offs. Therefore, we will study both its use in the context of graph minor theory and its several algorithmic implications. -
A Polynomial Excluded-Minor Approximation of Treedepth
A Polynomial Excluded-Minor Approximation of Treedepth Ken-ichi Kawarabayashi∗ Benjamin Rossmany National Institute of Informatics University of Toronto k [email protected] [email protected] November 1, 2017 Abstract Treedepth is a well-studied graph invariant in the family of \width measures" that includes treewidth and pathwidth. Understanding these invariants in terms of excluded minors has been an active area of research. The recent Grid Minor Theorem of Chekuri and Chuzhoy [12] establishes that treewidth is polynomially approximated by the largest k × k grid minor. In this paper, we give a similar polynomial excluded-minor approximation for treedepth in terms of three basic obstructions: grids, tree, and paths. Specifically, we show that there is a constant c such that every graph of treedepth ≥ kc contains one of the following minors (each of treedepth ≥ k): • the k × k grid, • the complete binary tree of height k, • the path of order 2k. Let us point out that we cannot drop any of the above graphs for our purpose. Moreover, given a graph G we can, in randomized polynomial time, find either an embedding of one of these minors or conclude that treedepth of G is at most kc. This result has potential applications in a variety of settings where bounded treedepth plays a role. In addition to some graph structural applications, we describe a surprising application in circuit complexity and finite model theory from recent work of the second author [28]. 1 Introduction Treedepth is a well-studied graph invariant with several equivalent definitions. It appears in the literature under various names including vertex ranking number [30], ordered chromatic number [17], and minimum elimination tree height [23], before being systematically studied under the name treedepth by Ossona de Mendes and Neˇsetˇril[20].