DOCSLIB.ORG

Exploiting Tree Decomposition and Soft Local Consistency in Weighted CSP

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Exploiting Tree Decomposition and Soft Local Consistency in Weighted CSP Exploiting Tree Decomposition and Soft Local Consistency in Weighted CSP Simon de Givry and Thomas Schiex Gerard Verfaillie INRA, Toulouse, France ONERA - DCSD, Toulouse, France degivry,tschiex @toulouse.inra.fr [email protected] { } Abstract proved practical time and space complexity inside the origi- nal bounds. Note however that in the context of tree-search, Several recent approaches for processing graphical these bounds are obtained at the cost of restricted freedom models (constraint and Bayesian networks) simultane- in variable assignment ordering (see (Bacchus, Dalmao, & ously exploit graph decomposition and local consis- tency enforcing. Graph decomposition exploits the Pitassi 2003) on this topic). problem structure and offers space and time complex- In this paper, we are specifically interested in optimiza- ity bounds while hard information propagation provides tion problems in the framework of valued and weighted practical improvements of space and time behavior in- constraint networks (Schiex, Fargier, & Verfaillie 1995). side these theoretical bounds. Weighted constraint networks (WCN) provide a very general Concurrently, the extension of local consistency to model with several applications in domains such as resource weighted constraint networks has led to important im- allocation, combinatorial auctions and bioinformatics. provements in branch and bound based solvers. In- In a first part, we show that, with a limited weakening of deed, soft local consistencies give incrementally com- existing theoretical complexity bounds, an alternative com- puted strong lower bounds providing inexpensive yet bination of tree decomposition and branch and bound can be powerful pruning and better informed heuristics. defined. In a second part, we exploit the extension of local In this paper, we consider combinations of tree decom- consistency to WCN.This extension has lead to increasingly position based approaches and soft local consistency en- forcing for solving weighted constraint problems. The efficient branch and bound algorithms using increasingly intricacy of weighted information processing leads to strong local consistency properties (Cooper & Schiex 2004; different approaches, with different theoretical proper- Larrosa & Schiex 2004; 2003).They offer incrementally ties. It appears that the most promising combination maintained strong bounds and also contribute to better in- sacrifices a bit of theory for improved practical effi- formed variable and value ordering. Introduced in tree de- ciency. composition based approaches, they should enhance their practical time and space complexities and also provide better guidance. Introduction Taken together, these two parts show the increased intri- Graphical model processing is a central problem in AI. In cacy of weighted information processing, leading to differ- the last years, in order to solve satisfaction, optimization or ent possible combinations, each having different theoretical counting problems, several algorithms have been proposed properties and practical efficiencies. that simultaneously exploit a decomposition of the graph of the problem and the propagation of hard information using Preliminaries local consistency enforcing. This includes algorithms such as Recursive Conditioning (RC) (Darwiche 2001), Back- A weighted binary CSP (WCSP) is a triplet (X, D, W). track bounded by Tree Decomposition (BTD) (Terrioux X = 1,...,n is a set of n variables. Each variable i X {has a finite} domain D D of values than can be & Jegou 2003), AND-OR tree and graph search (Mari- ∈ i ∈ nescu & Dechter 2005b; 2005a), all related to Pseudo-tree assigned to it. The maximum domain size is d. W is a set of search (Freuder & Quinn 1985). soft constraints (or cost functions). A binary soft constraint W W is a function W : D D [0,k] where k is a Implicitly or not, all these algorithms exploit the proper- ij ∈ ij i × j → ties of Tree Decompositions (Bodlaender 2005) which cap- given maximum integer cost corresponding to a completely ture structural independence information, in order to ob- forbidden assignment (expressing hard constraints). If they tain theoretical bounds on the time and space complexity do not exist, we add to W one unary cost function for every of the algorithms. The combination with hard informa- variable such that Wi : Di [0,k] and a zero arity con- straint W (a constant cost payed→ by any assignment). All tion propagation provides additional pruning leading to im- ∅ these additional cost functions have initial value 0, leaving Copyright c 2006, American Association for Artificial Intelli- the semantics of the problem unchanged. gence (www.aaai.org). All rights reserved. The problem is then to find a complete assignment 22 with a minimum cost: min W + (a1,a2,...,an) i Di ∅ n ∈ { Wi(ai)+ Wij(ai,aj) , an optimization i=1 Wij W } problem with an associated∈ NP-complete decision problem. The constraint graph of a WCSP is a graph G =(X, E) with one vertex for each variable and one edge (i, j) for every binary constraint Wij W . A tree decomposi- tion of this graph is defined by∈ a tree (C, T). The set of nodes of the tree is C = C ,...,C where C is a { 1 m} e set of variables (Ce X) called a cluster. T is a set of edges connecting clusters⊂ and forming a tree (a con- nected acyclic graph). The set of clusters C must cover all the variables ( C = X) and all the constraints Ce C e ( (i, j) E, C ∈C s.t. i, j C ). Furthermore, if a ∀ ∈ ∃ e∈ ∈ e variable i appears in two clusters Ce and Cg, i must also ap- pear in all the clusters Cf on the unique path from Ce to Cg in (C, T). The rationale behind this definition is that acyclic prob- lems can be solved with a good theoretical complexity. Thus, a tree decomposition decomposes a problem in sub- problems (clusters) organized in an acyclic graph and in such a way that the variables that relate two adjacent clusters (variables whose removal disconnects the subproblems) are obtained by intersecting the two clusters. This is illustrated on Figure 1 where the graph of a fre- quency assignment problem is covered by clusters defining Figure 1: The constraint graph of preprocessed RLFAP a tree decomposition. Because of the usual emphasis on SCEN-06 problem covered by clusters and the associated purely random problems, one may think that such nice de- tree decomposition using C1 as root. Problem P5 is outlined compositions are exceptional but the emphasis on modular- with separator S5 = 57, 72 . The constraint W57,72 inside S is not part of P . P{ has a} separator S = 69 . ity in design problems and the (spatial or temporal) locality 5 5 6 6 { } of many problems easily induce such structures. The tree-width of a tree decomposition (C, T) is equal to the variables of a cluster C are assigned before any of the max C 1, denoted w in the sequel. The tree- e Ce C e remaining variables in its son clusters and consider a cur- width w∈ {|of G|}is − the minimum tree-width over all tree de- ∗ rent assignment A. Then, for any cluster C Sons(C ), compositions of G (also called the induced-width of G). If f e and for the current assignment A of the separator∈ S , the a root C C is chosen, the maximum number of variables f f r subproblem P under assignment A (denoted P /A ) can appearing∈ in a path starting from C is called the tree-height f f f f r be solved independently from the rest of the problem. If of the decomposition. Finding a minimum tree-width or a memory allows, the optimal cost of P /A may be recorded minimum tree-height decomposition is NP-hard. f f which means it will never be solved again for the same as- signment of S . Exploiting tree decompositions f If d is the maximum domain size, h the decomposition For a given WCSP, we consider a rooted tree decomposition tree-height and w the tree-width, this idea applied recur- (C, T) with an arbitrary root C . We denote by Father(C ) 1 e sively with no recording guarantees that a cluster C will (resp. Sons(C )) the parent (resp. set of sons) of C in T . e e e never be visited more than d to the power of the number The separator of C is the set S = C Father(C ). The set e e e e of variables in the path from the root to C (proper vari- of proper variables of C is V = C ∩S . Note that the V e e e e e e ables excluded). This gives a guaranteed bound on the family defines a partition of X. For a\ given variable i X, number of visited nodes which is dominated by O(dh). we denote by [i] the index of the cluster such that i V∈ . [i] This covers Pseudo-Tree or AND-OR tree search, also ap- The essential property of tree decomposition is∈ that as- plied to WCSP in (Larrosa, Meseguer, & Sanchez 2002; signing separates the initial problem in two subproblems Se Marinescu & Dechter 2005b). which can then be solved independently. The first subprob- lem, denoted Pe, is defined by the variables of Ce and all With recording, a cluster Ce will never be visited more its descendant clusters in T and by all the cost functions in- than the number of assignments of Se. Given the number of volving at least one proper variable of these clusters. The variables in Ve, this means that the number of nodes is dom- remaining constraints, together with the variables they in- inated by O(dw+1), w +1being the size of the largest clus- volve, define the remaining subproblem. ter. This gives algorithms such as RC with full recording, This property has been exploited by many related algo- BTD and AND-OR graph search, which can be considered rithms.
Load more

Recommended publications
  • Structural Parameterizations of Clique Coloring
    Structural Parameterizations of Clique Coloring Lars Jaffke University of Bergen, Norway lars.jaff[email protected] Paloma T. Lima University of Bergen, Norway [email protected] Geevarghese Philip Chennai Mathematical Institute, India UMI ReLaX, Chennai, India [email protected] Abstract A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed q ≥ 2, we give an O?(qtw)-time algorithm when the input graph is given together with one of its tree decompositions of width tw. We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is XP parameterized by clique-width. 2012 ACM Subject Classification Mathematics of computing → Graph coloring Keywords and phrases clique coloring, treewidth, clique-width, structural parameterization, Strong Exponential Time Hypothesis Digital Object Identifier 10.4230/LIPIcs.MFCS.2020.49 Related Version A full version of this paper is available at https://arxiv.org/abs/2005.04733. Funding Lars Jaffke: Supported by the Trond Mohn Foundation (TMS). Acknowledgements The work was partially done while L. J. and P. T. L. were visiting Chennai Mathematical Institute. 1 Introduction Vertex coloring problems are central in algorithmic graph theory, and appear in many variants. One of these is Clique Coloring, which given a graph G and an integer k asks whether G has a clique coloring with k colors, i.e.
    [Show full text]
  • Clique-Width and Tree-Width of Sparse Graphs
    Clique-width and tree-width of sparse graphs Bruno Courcelle Labri, CNRS and Bordeaux University∗ 33405 Talence, France email: [email protected] June 10, 2015 Abstract Tree-width and clique-width are two important graph complexity mea- sures that serve as parameters in many fixed-parameter tractable (FPT) algorithms. The same classes of sparse graphs, in particular of planar graphs and of graphs of bounded degree have bounded tree-width and bounded clique-width. We prove that, for sparse graphs, clique-width is polynomially bounded in terms of tree-width. For planar and incidence graphs, we establish linear bounds. Our motivation is the construction of FPT algorithms based on automata that take as input the algebraic terms denoting graphs of bounded clique-width. These algorithms can check properties of graphs of bounded tree-width expressed by monadic second- order formulas written with edge set quantifications. We give an algorithm that transforms tree-decompositions into clique-width terms that match the proved upper-bounds. keywords: tree-width; clique-width; graph de- composition; sparse graph; incidence graph Introduction Tree-width and clique-width are important graph complexity measures that oc- cur as parameters in many fixed-parameter tractable (FPT) algorithms [7, 9, 11, 12, 14]. They are also important for the theory of graph structuring and for the description of graph classes by forbidden subgraphs, minors and vertex-minors. Both notions are based on certain hierarchical graph decompositions, and the associated FPT algorithms use dynamic programming on these decompositions. To be usable, they need input graphs of "small" tree-width or clique-width, that furthermore are given with the relevant decompositions.
    [Show full text]
  • Parameterized Algorithms for Recognizing Monopolar and 2
    Parameterized Algorithms for Recognizing Monopolar and 2-Subcolorable Graphs∗ Iyad Kanj1, Christian Komusiewicz2, Manuel Sorge3, and Erik Jan van Leeuwen4 1School of Computing, DePaul University, Chicago, USA, [email protected] 2Institut für Informatik, Friedrich-Schiller-Universität Jena, Germany, [email protected] 3Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Germany, [email protected] 4Max-Planck-Institut für Informatik, Saarland Informatics Campus, Saarbrücken, Germany, [email protected] October 16, 2018 Abstract A graph G is a (ΠA, ΠB)-graph if V (G) can be bipartitioned into A and B such that G[A] satisfies property ΠA and G[B] satisfies property ΠB. The (ΠA, ΠB)-Recognition problem is to recognize whether a given graph is a (ΠA, ΠB )-graph. There are many (ΠA, ΠB)-Recognition problems, in- cluding the recognition problems for bipartite, split, and unipolar graphs. We present efficient algorithms for many cases of (ΠA, ΠB)-Recognition based on a technique which we dub inductive recognition. In particular, we give fixed-parameter algorithms for two NP-hard (ΠA, ΠB)-Recognition problems, Monopolar Recognition and 2-Subcoloring, parameter- ized by the number of maximal cliques in G[A]. We complement our algo- rithmic results with several hardness results for (ΠA, ΠB )-Recognition. arXiv:1702.04322v2 [cs.CC] 4 Jan 2018 1 Introduction A (ΠA, ΠB)-graph, for graph properties ΠA, ΠB , is a graph G = (V, E) for which V admits a partition into two sets A, B such that G[A] satisfies ΠA and G[B] satisfies ΠB. There is an abundance of (ΠA, ΠB )-graph classes, and important ones include bipartite graphs (which admit a partition into two independent ∗A preliminary version of this paper appeared in SWAT 2016, volume 53 of LIPIcs, pages 14:1–14:14.
    [Show full text]
  • Sub-Coloring and Hypo-Coloring Interval Graphs⋆
    Sub-coloring and Hypo-coloring Interval Graphs? Rajiv Gandhi1, Bradford Greening, Jr.1, Sriram Pemmaraju2, and Rajiv Raman3 1 Department of Computer Science, Rutgers University-Camden, Camden, NJ 08102. E-mail: [email protected]. 2 Department of Computer Science, University of Iowa, Iowa City, Iowa 52242. E-mail: [email protected]. 3 Max-Planck Institute for Informatik, Saarbr¨ucken, Germany. E-mail: [email protected]. Abstract. In this paper, we study the sub-coloring and hypo-coloring problems on interval graphs. These problems have applications in job scheduling and distributed computing and can be used as “subroutines” for other combinatorial optimization problems. In the sub-coloring problem, given a graph G, we want to partition the vertices of G into minimum number of sub-color classes, where each sub-color class induces a union of disjoint cliques in G. In the hypo-coloring problem, given a graph G, and integral weights on vertices, we want to find a partition of the vertices of G into sub-color classes such that the sum of the weights of the heaviest cliques in each sub-color class is minimized. We present a “forbidden subgraph” characterization of graphs with sub-chromatic number k and use this to derive a a 3-approximation algorithm for sub-coloring interval graphs. For the hypo-coloring problem on interval graphs, we first show that it is NP-complete and then via reduction to the max-coloring problem, show how to obtain an O(log n)-approximation algorithm for it. 1 Introduction Given a graph G = (V, E), a k-sub-coloring of G is a partition of V into sub-color classes V1,V2,...,Vk; a subset Vi ⊆ V is called a sub-color class if it induces a union of disjoint cliques in G.
    [Show full text]
  • Shrub-Depth a Successful Depth Measure for Dense Graphs Graphs Petr Hlinˇen´Y Faculty of Informatics, Masaryk University Brno, Czech Republic
    page.19 Shrub-Depth a successful depth measure for dense graphs graphs Petr Hlinˇen´y Faculty of Informatics, Masaryk University Brno, Czech Republic Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 1 / 19 Shrub-depth measure for dense graphs page.19 Shrub-Depth a successful depth measure for dense graphs graphs Petr Hlinˇen´y Faculty of Informatics, Masaryk University Brno, Czech Republic Ingredients: joint results with J. Gajarsk´y,R. Ganian, O. Kwon, J. Neˇsetˇril,J. Obdrˇz´alek, S. Ordyniak, P. Ossona de Mendez Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 1 / 19 Shrub-depth measure for dense graphs page.19 Measuring Width or Depth? • Being close to a TREE { \•-width" sparse dense tree-width / branch-width { showing a structure clique-width / rank-width { showing a construction Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 2 / 19 Shrub-depth measure for dense graphs page.19 Measuring Width or Depth? • Being close to a TREE { \•-width" sparse dense tree-width / branch-width { showing a structure clique-width / rank-width { showing a construction • Being close to a STAR { \•-depth" sparse dense tree-depth { containment in a structure ??? (will show) Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 2 / 19 Shrub-depth measure for dense graphs page.19 1 Recall: Width Measures Tree-width tw(G) ≤ k if whole G can be covered by bags of size ≤ k + 1, arranged in a \tree-like fashion". Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 3 / 19 Shrub-depth measure for dense graphs page.19 1 Recall: Width Measures Tree-width tw(G) ≤ k if whole G can be covered by bags of size ≤ k + 1, arranged in a \tree-like fashion".
    [Show full text]
  • A Fast Algorithm for the Maximum Clique Problem Patric R
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Applied Mathematics 120 (2002) 197–207 A fast algorithm for the maximum clique problem Patric R. J. Osterg%# ard ∗ Department of Computer Science and Engineering, Helsinki University of Technology, P.O. Box 5400, 02015 HUT, Finland Received 12 October 1999; received in revised form 29 May 2000; accepted 19 June 2001 Abstract Given a graph, in the maximum clique problem, one desires to ÿnd the largest number of vertices, any two of which are adjacent. A branch-and-bound algorithm for the maximum clique problem—which is computationally equivalent to the maximum independent (stable) set problem—is presented with the vertex order taken from a coloring of the vertices and with a new pruning strategy. The algorithm performs successfully for many instances when applied to random graphs and DIMACS benchmark graphs. ? 2002 Elsevier Science B.V. All rights reserved. 1. Introduction We denote an undirected graph by G =(V; E), where V is the set of vertices and E is the set of edges. Two vertices are said to be adjacent if they are connected by an edge. A clique of a graph is a set of vertices, any two of which are adjacent. Cliques with the following two properties have been studied over the last three decades: maximal cliques, whose vertices are not a subset of the vertices of a larger clique, and maximum cliques, which are the largest among all cliques in a graph (maximum cliques are clearly maximal).
    [Show full text]
  • Cliques and Clubs⋆
    Cliques and Clubs? Petr A. Golovach1, Pinar Heggernes1, Dieter Kratsch2, and Arash Rafiey1 1 Department of Informatics, University of Bergen, Norway, fpetr.golovach,pinar.heggernes,[email protected] 2 LITA, Universit´ede Lorraine - Metz, France, [email protected] Abstract. Clubs are generalizations of cliques. For a positive integer s, an s-club in a graph G is a set of vertices that induces a subgraph of G of diameter at most s. The importance and fame of cliques are evident, whereas clubs provide more realistic models for practical applications. Computing an s-club of maximum cardinality is an NP-hard problem for every fixed s ≥ 1, and this problem has attracted significant attention recently. We present new positive results for the problem on large and important graph classes. In particular we show that for input G and s, a maximum s-club in G can be computed in polynomial time when G is a chordal bipartite or a strongly chordal or a distance hereditary graph. On a superclass of these graphs, weakly chordal graphs, we obtain a polynomial-time algorithm when s is an odd integer, which is best possible as the problem is NP-hard on this clas for even values of s. We complement these results by proving the NP-hardness of the problem for every fixed s on 4-chordal graphs, a superclass of weakly chordal graphs. Finally, if G is an AT-free graph, we prove that the problem can be solved in polynomial time when s ≥ 2, which gives an interesting contrast to the fact that the problem is NP-hard for s = 1 on this graph class.
    [Show full text]
  • Clique Colourings of Geometric Graphs
    CLIQUE COLOURINGS OF GEOMETRIC GRAPHS COLIN MCDIARMID, DIETER MITSCHE, AND PAWELPRA LAT Abstract. A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number. Given n points x1;:::; xn in the plane, and a threshold r > 0, the corre- sponding geometric graph has vertex set fv1; : : : ; vng, and distinct vi and vj are adjacent when the Euclidean distance between xi and xj is at most r. We investigate the clique chromatic number of such graphs. We first show that the clique chromatic number is at most 9 for any geometric graph in the plane, and briefly consider geometric graphs in higher dimensions. Then we study the asymptotic behaviour of the clique chromatic number for the random geometric graph G(n; r) in the plane, where n random points are independently and uniformly distributed in a suitable square. We see that as r increases from 0, with high probability the clique chromatic number is 1 for very small r, then 2 for small r, then at least 3 for larger r, and finally drops back to 2. 1. Introduction and main results In this section we introduce clique colourings and geometric graphs; and we present our main results, on clique colourings of deterministic and random geometric graphs. Recall that a proper colouring of a graph is a labeling of its vertices with colours such that no two vertices sharing the same edge have the same colour; and the smallest number of colours in a proper colouring of a graph G = (V; E) is its chromatic number, denoted by χ(G).
    [Show full text]
  • Algorithmic Graph Theory Part III Perfect Graphs and Their Subclasses
    Algorithmic Graph Theory Part III Perfect Graphs and Their Subclasses Martin Milanicˇ [email protected] University of Primorska, Koper, Slovenia Dipartimento di Informatica Universita` degli Studi di Verona, March 2013 1/55 What we’ll do 1 THE BASICS. 2 PERFECT GRAPHS. 3 COGRAPHS. 4 CHORDAL GRAPHS. 5 SPLIT GRAPHS. 6 THRESHOLD GRAPHS. 7 INTERVAL GRAPHS. 2/55 THE BASICS. 2/55 Induced Subgraphs Recall: Definition Given two graphs G = (V , E) and G′ = (V ′, E ′), we say that G is an induced subgraph of G′ if V ⊆ V ′ and E = {uv ∈ E ′ : u, v ∈ V }. Equivalently: G can be obtained from G′ by deleting vertices. Notation: G < G′ 3/55 Hereditary Graph Properties Hereditary graph property (hereditary graph class) = a class of graphs closed under deletion of vertices = a class of graphs closed under taking induced subgraphs Formally: a set of graphs X such that G ∈ X and H < G ⇒ H ∈ X . 4/55 Hereditary Graph Properties Hereditary graph property (Hereditary graph class) = a class of graphs closed under deletion of vertices = a class of graphs closed under taking induced subgraphs Examples: forests complete graphs line graphs bipartite graphs planar graphs graphs of degree at most ∆ triangle-free graphs perfect graphs 5/55 Hereditary Graph Properties Why hereditary graph classes? Vertex deletions are very useful for developing algorithms for various graph optimization problems. Every hereditary graph property can be described in terms of forbidden induced subgraphs. 6/55 Hereditary Graph Properties H-free graph = a graph that does not contain H as an induced subgraph Free(H) = the class of H-free graphs Free(M) := H∈M Free(H) M-free graphT = a graph in Free(M) Proposition X hereditary ⇐⇒ X = Free(M) for some M M = {all (minimal) graphs not in X} The set M is the set of forbidden induced subgraphs for X.
    [Show full text]
  • Graph Theory
    Stephen P. Borgatti Graph Theory This is FIRST draft. Very likely to contain errors. lthough graph theory is one of the younger branches of mathematics, it is fundamental to a number of applied fields, including operations research, A computer science, and social network analysis. In this chapter we discuss the basic concepts of graph theory from the point of view of social network analysis. Graphs The fundamental concept of graph theory is the graph, which (despite the name) is best thought of as a mathematical object rather than a diagram, even though graphs have a very natural graphical representation. A graph – usually denoted G(V,E) or G = (V,E) – consists of set of vertices V together with a set of edges E. Vertices are also known as nodes, points and (in social networks) as actors, agents or players. Edges are also known as lines and (in social networks) as ties or links. An edge e = (u,v) is defined by the unordered pair of vertices that serve as its end points. Two vertices u and v are adjacent if there exists an edge (u,v) that connects them. An edge e = (u,u) that links a vertex to itself is known as a self-loop or reflexive tie. The number of vertices in a graph is usually denoted n while the number of edges is usually denoted m. As an example, the graph depicted in Figure 1 has vertex set V={a,b,c,d,e.f} and edge set E = {(a,b),(b,c),(c,d),(c,e),(d,e),(e,f)}.
    [Show full text]
  • Linear Clique-Width for Hereditary Classes of Cographs 2
    LINEAR CLIQUE-WIDTH FOR HEREDITARY CLASSES OF COGRAPHS Robert Brignall∗ Nicholas Korpelainen∗ Department of Mathematics and Statistics Mathematics Department The Open University University of Derby Milton Keynes, UK Derby, UK Vincent Vatter∗† Department of Mathematics University of Florida Gainesville, Florida USA The class of cographs is known to have unbounded linear clique-width. We prove that a hereditary class of cographs has bounded linear clique- width if and only if it does not contain all quasi-threshold graphs or their complements. The proof borrows ideas from the enumeration of permu- tation classes. 1. INTRODUCTION A variety of measures of the complexity of graph classes have been introduced and have proved useful for algorithmic problems [3–5, 7, 8, 11, 17, 18, 22, 23]. We are concerned with clique-width, introduced by Courcelle, Engelfriet, and Rozenberg [6], and more pertinently, the linear version of this parameter, due to Lozin and Rautenbach [20], and studied further by Gurski and Wanke [14]. The linear clique-width of a graph G, lcw(G), is the size of the smallest alphabet Σ such that G can arXiv:1305.0636v5 [math.CO] 25 Jan 2016 be constructed by a sequence of the following three operations: • add a new vertex labeled by a letter in Σ, • add edges between all vertices labeled i and all vertices labeled j (for i 6= j), and • give all vertices labeled i the label j. ∗All three authors were partially supported by EPSRC via the grant EP/J006130/1. †Vatter’s research was also partially supported by the National Security Agency under Grant Number H98230-12-1-0207 and the National Science Foundation under Grant Number DMS-1301692.
    [Show full text]
  • Topics in Graph Theory – 2 January 14 and 16, 2014 for Interval Graphs
    Topics in Graph Theory { 2 January 14 and 16, 2014 For interval graphs, we obtained a perfect colouring using a very special type of ordering, with the property that all uncoloured neighbours form a clique. This idea carries over to other classes of graphs. For a graph G = (V; E), a perfect elimination ordering of G is an ordering v1; v2; : : : ; vn so that, for each vertex vi, N(vi) \ fv1; : : : ; vi−1g is a clique. Theorem 1. If a graph G has a perfect elimination ordering then G is perfect. Proof. If G = (V; E) has a perfect elimination ordering, v1; v2; : : : ; vn, then this ordering has the property that, for each vertex vi, N(vi) \ fv1; : : : ; vi−1g has size at most !(G) − 1. Thus, the greedy colouring uses at !(G) colours, and χ(G) = !(G). To complete the proof, note that if G has a perfect elimination ordering, then so does each induced subgraph of G. A subgraph of a graph G = (V; E) is a graph H = (VH ;EH ) so that VH ⊆ VG and EH ⊆ EG. A subgraph H is an induced subgraph if every edge in EG with endpoints in VH is included in EH . A subgraph H is a spanning subgraph if VH = VG. An induced cycle is an induced subgraph which is a cycle. This means that the cycle has no chords, i.e. no other edges than the cycle edges connecting the vertices of the cycle. A graph G = (V; E) is chordal if it has no induced cycles of size larger than 3.
    [Show full text]