Clique-width Research Articles

Computing the Clique-Width on Series-Parallel Graphs

The clique-width (cwd) is an invariant of graphs which, similar to other invariants like the tree-width (twd) establishes a parameter for the complexity of a problem. For example, several problems with bounded clique-width can be solved in polynomial time. There is a well known relation between tree-width and clique-width denoted as cwd(G) ≤ 3 · 2 twd(G)−1 . Serial-parallel graphs have tree-width of at most 2, so its clique–width is at most 6 according to the previous relation. In this paper, we improve the bound for this particular case, showing that the clique-width of series-parallel graphs is smaller or equal to 5.

Uncountably Many Minimal Hereditary Classes of Graphs of Unbounded Clique-Width

Given an infinite word over the alphabet $\{0,1,2,3\}$, we define a class of bipartite hereditary graphs $\mathcal{G}^\alpha$, and show that $\mathcal{G}^\alpha$ has unbounded clique-width unless $\alpha$ contains at most finitely many non-zero letters.
 We also show that $\mathcal{G}^\alpha$ is minimal of unbounded clique-width if and only if $\alpha$ belongs to a precisely defined collection of words $\Gamma$. The set $\Gamma$ includes all almost periodic words containing at least one non-zero letter, which both enables us to exhibit uncountably many pairwise distinct minimal classes of unbounded clique width, and also proves one direction of a conjecture due to Collins, Foniok, Korpelainen, Lozin and Zamaraev. Finally, we show that the other direction of the conjecture is false, since $\Gamma$ also contains words that are not almost periodic.

Vertex coloring (4K1, hole-twin, 5-wheel)-free graphs

There has been recently keen interest in finding polynomial-time algorithms for the VERTEX COLORING problem for graphs G that are F-free for a given set F of graphs. If L is a set of four-vertex graphs, then the complexity of VERTEX COLORING for L-free graphs is known with three exceptions: L1 = {claw, 4K1}, L2 = {claw, 4K1, co-diamond}, and L3 = {4K1, C4}. In this paper, we study a problem arising from the class L3. A hole is an induced cycle with at least four vertices. A hole-twin is the graph obtained from a hole by adding a vertex that forms true twins with some vertex of the hole. A 5-wheel is the graph obtained from a C5 by adding a vertex that is adjacent to all vertices of the C5. We prove that a (4K1, hole-twin, 5-wheel)-free graph is perfect or has bounded clique width. As consequence we obtain a polynomial time algorithm for VERTEX COLORING for (4K1, hole-twin, 5-wheel)-free graphs.

Parameterized complexity of distance labeling and uniform channel assignment problems

We rephrase the Distance labeling problem as a specific uniform variant of the Channel Assignment problem and show that the latter one is fixed parameter tractable when parameterized by the neighborhood diversity together with the largest weight. Consequently, the Distance labeling problem is FPT when parameterized by the neighborhood diversity, the maximum pi and k. This is indeed a more general answer to an open question of Fiala et al.: Parameterized complexity of coloring problems: Treewidth versus vertex cover.Finally, we show that the uniform variant of the Channel Assignment problem becomes NP-complete when generalized to graphs of bounded clique width.

Characterizations of ([formula omitted], [formula omitted], [formula omitted])-free graphs

Given a set L of graphs, a graph G is L-free if G does not contain any graph in L as an induced subgraph. There has been keen interest in colouring graphs whose forbidden list L contains graphs with four vertices. A recent paper of Lozin and Malyshev (in press) discusses the state of the art on this problem, identifying three outstanding classes: L=(4K1,claw), L=(4K1,claw, co-diamond), and L=(4K1,C4). In this paper we investigate the class of (4K1, C4, C5)-free graphs and show that if G is a (4K1, C4, C5)-free graph, then G either has bounded clique width or is perfect.

Scalable Node-Level Computation Kernels for Parallel Exact Inference

In this paper, we investigate data parallelism in exact inference with respect to arbitrary junction trees. Exact inference is a key problem in exploring probabilistic graphical models, where the computation complexity increases dramatically with clique width and the number of states of random variables. We study potential table representation and scalable algorithms for node-level primitives. Based on such node-level primitives, we propose computation kernels for evidence collection and evidence distribution. A data parallel algorithm for exact inference is presented using the proposed computation kernels. We analyze the scalability of node-level primitives, computation kernels, and the exact inference algorithm using the coarse-grained multicomputer (CGM) model. According to the analysis, we achieve O(Nd <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> w <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> Pi <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j=1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">wc</sup> r <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C,j</sub> /P) local computation time and O(N) global communication rounds using P processors, 1 les P les max <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> PiPi <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">wc</sup> r <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C,j,</sub> where N is the number of cliques in the junction tree; d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> is the clique degree; r <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C,j</sub> is the number of states of the jth random variable in C; wc is the clique width; and w <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> is the separator width. We implemented the proposed algorithm on state-of-the-art clusters. Experimental results show that the proposed algorithm exhibits almost linear scalability over a wide range.

Exact Algorithms for a Loading Problem with Bounded Clique Width

In this paper we discuss a special pallet-loading problem, which we encountered at a manufacturing company. In graph-theoretical terms, the problem is equivalent to partitioning a permutation graph into bounded-size cliques. We formulate the problem as an integer program, and present two exact algorithms for solving it. The first algorithm is a branch-and-price algorithm based on the integer-programming formulation; the second one is an algorithm based on the concept of bounded clique width. The latter algorithm was motivated by the structure present in the real-world instances. Test results are given, both for real-world instances and randomly generated instances. As far as we are aware, this is the first implementation of an algorithm based on bounded clique width.

A model-theoretic characterisation of clique width

We generalise the concept of clique width to structures of arbitrary signature and cardinality. We present characterisations of clique width in terms of decompositions of a structure and via interpretations in trees. Several model-theoretic properties of clique width are investigated including VC-dimension and preservation of finite clique width under elementary extensions and compactness.

On the structure of ( [formula omitted],gem)-free graphs

We give a complete structure description of ( P 5 ,gem)-free graphs. By the results of a related paper, this implies bounded clique width for this graph class. Hereby, as usual, the P 5 is the induced path with five vertices a , b , c , d , e and four edges ab , bc , cd , de , and the gem consists of a P 4 a , b , c , d with edges ab , bc , cd plus a universal vertex e adjacent to a , b , c , d .

On spectra of sentences of monadic second order logic with counting

Abstract.We show that the spectrum of a sentence ϕ in Counting Monadic Second Order Logic (CMSOL) using one binary relation symbol and finitely many unary relation symbols, is ultimately periodic, provided all the models of ϕ are of clique width at most k, for some fixed k. We prove a similar statement for arbitrary finite relational vocabularies τ and a variant of clique width for τ-structures. This includes the cases where the models of ϕ are of tree width at most k. For the case of bounded tree-width, the ultimate periodicity is even proved for Guarded Second Order Logic GSOL. We also generalize this result to many-sorted spectra, which can be viewed as an analogue of Parikh's Theorem on context-free languages, and its analogues for context-free graph grammars due to Habel and Courcelle.Our work was inspired by Gurevich and Shelah (2003), who showed ultimate periodicity of the spectrum for sentences of Monadic Second Order Logic where only finitely many unary predicates and one unary function are allowed. This restriction implies that the models are all of tree width at most 2, and hence it follows from our result.

On the structure and stability number of P5- and co-chair-free graphs

We give a O(nm) time algorithm for the maximum weight stable set (MWS) problem on P 5- and co-chair-free graphs without recognizing whether the (arbitrary) input graph is P 5- and co-chair-free. This algorithm is based on the fact that prime P 5- and co-chair-free graphs containing 2 K 2 are matched co-bipartite graphs and thus have very simple structure, and for 2 K 2-free graphs, there is a polynomial time algorithm for the MWS problem due to a result of Farber saying that 2 K 2-free graphs contain at most O(n 2) maximal stable sets. A similar argument can be used for ( P 5,co-P)-free graphs; their prime graphs are 2 K 2-free. Moreover, we give a complete classification of ( P 5,co-chair, H)-free graphs with respect to their clique width when H is a one-vertex P 4 extension; this extends the characterization of ( P 5, P 5 ,co-chair)-free graphs called semi- P 4- sparse by Fouquet and Giakoumakis. For H being a house, P, bull or gem, the class of ( P 5,co-chair, H)-free graphs has bounded clique width and very simple structure whereas for the other four cases, namely H being a co-gem, chair, co- P or C 5, the class has unbounded clique width due to a result of Makowsky and Rotics. Bounded clique width implies linear time algorithms for all algorithmic problems expressible in LinEMSOL—a variant of Monadic Second Order Logic including the MWS Problem.

( P5,diamond)-free graphs revisited: structure and linear time optimization

Using the concept of prime graphs and modular decomposition of graphs, we give a complete structure description of ( P 5,diamond)-free graphs implying that these graphs have bounded clique width (the P 5 is the induced path with five vertices a, b, c, d, e and four edges ab, bc, cd, de, and the diamond consists of four vertices a, b, c, d such that a, b, c form an induced path with edges ab, bc, and vertex d is adjacent to a, b and c). The structure and bounded clique width of this graph class allows to solve several algorithmic problems on this class in linear time, among them the problems Maximum Weight Stable Set (MWS), Maximum Weight Clique, Domination, Steiner Tree and in general every algorithmic problem which is, roughly speaking, expressible in a certain kind of Monadic Second-Order Logic using quantification only over vertex but not over edge set predicates. This improves previous results on ( P 5,diamond)-free graphs in several ways: We give a complete structure description of prime ( P 5,diamond)-free graphs, we do not only solve the MWS problem on this class, we achieve linear time algorithms (instead of a recent time bound O ( n m ) ), and we can do all this on a larger graph class containing ( P 5,diamond)-free graphs which admits linear time recognition.

Structure and stability number of chair-, co-P- and gem-free graphs revisited

The P 4 is the induced path with vertices a, b, c, d and edges ab, bc, cd. The chair (co-P, gem) has a fifth vertex adjacent to b ( a and b, a, b, c and d, respectively). We give a complete structure description of prime chair-, co-P- and gem-free graphs which implies bounded clique width for this graph class. It is known that this has some nice consequences; very roughly speaking, every problem expressible in a certain kind of Monadic Second Order Logic (quantifying only over vertex set predicates) can be solved efficiently for graphs of bounded clique width. In particular, we obtain linear time for the problems Vertex Cover, Maximum Weight Stable Set (MWS), Maximum Weight Clique, Steiner Tree, Domination and Maximum Induced Matching on chair-, co-P- and gem-free graphs and a slightly larger class of graphs. This drastically improves a recently published O( n 4) time bound for Maximum Stable Set on butterfly-, chair-, co-P- and gem-free graphs.

Stability number of bull- and chair-free graphs revisited

De Simone showed that prime bull- and chair-free graphs containing a co-diamond are either bipartite or an induced cycle of odd length at least five. Based on this result, we give a complete structural characterization of prime (bull,chair)-free graphs having stability number at least four as well as of (bull,chair,co-chair)-free graphs. This implies constant-bounded clique width for these graph classes which leads to linear time algorithms for some algorithmic problems. Moreover, we obtain a robust O ( n m ) time algorithm for the maximum weight stable set problem on bull- and chair-free graphs without testing whether the (arbitrary) input graph is bull- and chair-free. This improves previous results with respect to structural insight, robustness and time bounds.

Maximum Weight Stable Set on graphs without claw and co-claw (and similar graph classes) can be solved in linear time

Minty has shown that the Maximum Weight Stable Set (MWS) Problem can be solved in polynomial time when restricted to claw-free graphs. We show that the structure of graphs being both claw-free and co-claw-free is very simple which implies bounded clique width for this graph class. It is known that for graph classes of bounded clique width (assuming that k-expressions can be determined in linear time), there are linear time algorithms for all problems expressible in Monadic Second Order Logic with quantification only over vertex sets. The problems MWS, Maximum Clique, Domination and Steiner Tree, for example, are expressible in this way. Moreover, we describe the structure of prime graphs being both H- and H -free for any four-vertex graph H and obtain bounded clique width for these graph classes except the 2K 2 - and C 4 -free graphs.

On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic

We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is definable in monadic second-order logic (MSOL). We show that for bounded tree-width these problems are solvable in polynomial time. The same holds for bounded clique width in the cases, where the decomposition, which establishes the bound on the clique-width, can be computed in polynomial time and for problems expressible by monadic second-order formulas without edge set quantification. Such quantifications are allowed in the case of graphs with bounded tree-width. As applications we discuss in detail how this affects the parametrized complexity of the permanent and the hamiltonian of a matrix, and more generally, various generating functions of MSOL definable graph properties. Finally, our results are also applicable to SAT and ♯ SAT.

ON THE CLIQUE-WIDTH OF SOME PERFECT GRAPH CLASSES

Graphs of clique–width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique–width of perfect graph classes. On one hand, we show that every distance–hereditary graph, has clique–width at most 3, and a 3–expression defining it can be obtained in linear time. On the other hand, we show that the classes of unit interval and permutation graphs are not of bounded clique–width. More precisely, we show that for every [Formula: see text] there is a unit interval graph In and a permutation graph Hn having n2 vertices, each of whose clique–width is at least n. These results allow us to see the border within the hierarchy of perfect graphs between classes whose clique–width is bounded and classes whose clique–width is unbounded. Finally we show that every n×n square grid, [Formula: see text], n ≥ 3, has clique–width exactly n+1.

Clique width of Partner limited graphs

Abstract The clique-width of a graph G is the minimum number of labels that are required for dening G by an expression based on graph operations using vertex labels. The Partner limited graphs ( PL -graphs for short) are defined to be graphs with a limited number of P 4 's. We prove that PL -graphs are of bounded clique-width. It follows that a large number of optimization problems have polynomial solutions for this family of graphs

Upper bounds to the clique width of graphs

Hierarchical decompositions of graphs are interesting for algorithmic purposes. Many NP complete problems have linear complexity on graphs with tree-decompositions of bounded width. We investigate alternate hierarchical decompositions that apply to wider classes of graphs and still enjoy good algorithmic properties. These decompositions are motivated and inspired by the study of vertex-replacement context-free graph grammars. The complexity measure of graphs associated with these decompositions is called clique width. In this paper we bound the clique width of a graph in terms of its tree width on the one hand, and of the clique width of its edge complement on the other.

ON THE CLIQUE–WIDTH OF GRAPH WITH FEW P4'S

Babel and Olariu (1995) introduced the class of (q, t) graphs in which every set of q vertices has at most t distinct induced P4s. Graphs of clique-width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique–width of the (q, t) graphs, for almost all possible combinations of q and t. On one hand we show that every (q, q - 3) graph for q ≥ 7, has clique–width ≤ q and a q–expression defining it can be obtained in linear time. On the other hand we show that the class of (q, q - 3) graphs for 4 ≤ q ≤ 6 and the class of (q, q - 1) graphs for q ≥ 4 are not of bounded clique-width.