B. Courcelle, J.A. Makowsky, and U. Rotics. Linear time solvable optimization problems on certain structured graph families. In Graph Theoretic Concepts in Computer Science - WG'98, Lecture Notes in Computer Science 1517, pages 1-16. Springer-Verlag, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of bounded treewidth graphs is the class of bounded cliquewidth graphs. It is shown in [CO98] than if a graph has treewidth k, then its cliquewidth is bounded from above by 2 k 1 1. Thus, a class of graphs that has bounded treewidth also has bounded cliquewidth. Courcelle, Makowsky, and Rotics [CMR98] showed that if a class C is e ectively of bounded cliquewidth, then every monadic second order 27 property on C is polynomial. It follows that CSP(A; B) is in PTIME for each B if A is of bounded cliquewidth. On the other hand, every clique has cliquewidth 2 [CO98] and we have observed above ....

B. Courcelle, J.A. Makowsky, and U. Rotics. Linear time solvable optimization problems on certain structured graph families. In Graph Theoretic Concepts in Computer Science - WG'98, Lecture Notes in Computer Science 1517, pages 1-16. Springer-Verlag, 1998.

....comments The techniques used in this paper to solve the simple max cut problem on compositions of graphs as in Section 5 can be used also for many other graph problems. In fact, recent results on the notion of clique width give general results that also imply our earlier result on simple max cut [8]. We conclude this paper with some small observations on the weighted variant of the problem, called here max cut instead of simple max cut. First, observe that max cut is NP complete, when restricted to cliques and when only edge weights 0 and 1 are allowed. The problem in this form is ....

Courcelle, B., Makowsky, J. A., and Rotics, U. 1998. Linear time solvable optimization problems on certain structured graph families, extended abstract. In Proc. 24th International Workshop on Graph Theoretic Concepts in Computer Science, Springer Verlag Lecture Notes in Computer Science vol. 1517, 1--16.

....of bounded treewidth graphs is the class of bounded cliquewidth graphs. It is shown in [CO98] than if a graph has treewidth k, then its cliquewidth is bounded from above by 2 k 1 1. Thus, a class of graphs that has bounded treewidth also has bounded cliquewidth. Courcelle, Makowsky, and Rotics [CMR98] showed that if a class C is effectively of bounded cliquewidth, then every monadic second order property on C is polynomial. It follows that CSP(A;B) is in PTIME for each B if A is of bounded cliquewidth. On the other hand, every clique has cliquewidth 2 [CO98] and we have observed above that ....

B. Courcelle, J.A. Makowsky, and U. Rotics. Linear time solvable optimization problems on certain structured graph families. Technical report, Technion, 1998.

....over one letter (also called tally languages) which are in P (NP) In Section 5 we show that: Theorem 6. If P 1 6= NP 1 then there is an MSOL( 2 ) definable decision problem over the class of cliques which is not solvable in polynomial time. An extended abstract of this paper was presented in [CMR98]. 2 Background 2.1 Graphs as logical structures In what follows, we will use the term graph for finite nonempty undirected graphs without self loops or multiple edges. We will use the term labeled graph for graphs having labels which are associated with their vertices such that each vertex has ....

B. Courcelle, J.A. Makowsky, and U. Rotics. Linear time solvable optimization problems on certain structured graph families, extended abstract. In Graph Theoretic Concepts in Computer Science, 24th International Workshop, WG'98, volume 1517 of Lecture Notes in Computer Science, pages 1--16. Springer Verlag, 1998.

....be done in linear time as well, Bod96,BK96] All this together makes it attractive to use the treewidth as the parameter for fixed parameter complexity considerations. 1. 4 Main result Theorem 1, as stated, is new, but it can be proved using the automata theoretic methods from [ALS91,CM93] In [CMR98], those results were extended to optimization problems on graphs of fixed cliquewidth, a notion introduced by Courcelle, Engelfriet and Rozenberg [CER93] We shall give the detailed definitions in section 3. The proofs in [CMR98] are model theoretic rather than automata theoretic. The purpose of ....

....be proved using the automata theoretic methods from [ALS91,CM93] In [CMR98] those results were extended to optimization problems on graphs of fixed cliquewidth, a notion introduced by Courcelle, Engelfriet and Rozenberg [CER93] We shall give the detailed definitions in section 3. The proofs in [CMR98] are model theoretic rather than automata theoretic. The purpose of this paper is to state and prove an extension of theorem 1 (Theorem 9 of section 4) to a wide class of generating functions of graph properties which are definable in Monadic Second Order Logic. We also show that many variations ....

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B. Courcelle, J.A. Makowsky, and U. Rotics. Linear time solvable optimization problems on certain structured graph families, extended abstract. To appear in the proceedings of WG98, 1998.

....then every decision, optimization, enumeration or evaluation problem on C which can be defined by a Monadic Second Order formula can be solved in time c k Delta O(jV j) T (jV j) where c k is a constant which depends only on and k and v is the number of vertices of the input. For details, cf. [CMR98a, CMR98b, CMR99]. In this paper we study the clique width of the (q; t) graphs for almost all combinations of q and t. We first show that: Theorem 1 For every (q; q Gamma 3) graph G such that q 7, G has clique width q, and a q expression defining it can be constructed in time O(jV j jEj) The proof of ....

....the other vertices occuring in t 1 ) is well defined. The label of v at a is defined as the label that v has when the operation a is applied on the subtree of tree(t) rooted at a. 2. 2 Clique width and the modular decomposition of graphs In this section we recall the connection established in [CMR98a] between the well known concept of the modular decomposition of graphs and the clique width property of graphs. The modular decomposition of a graph G, is tree denoted as T (G) together with a set of prime graphs associated with the internal nodes of the tree labeled by N . We start by ....

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B. Courcelle, J.A. Makowsky, and U. Rotics. Linear time solvable optimization problems on certain structured graph families, extended abstract. To appear in the proceedings of WG98, 1998.

....done in linear time as well, Bod96, BK96] All this together makes it attractive to use the treewidth as the parameter for fixed parameter complexity considerations. 1. 4 Main result Theorem 1, as stated, is new, but it can be proved using the automata theoretic methods from [ALS91, CM93] In [CMR98], those results were extended to optimization problems on graphs of fixed cliquewidth, a notion introduced by Courcelle, Engelfriet and Rozenberg [CER93] We shall give the detailed definitions in section 4. The proofs in [CMR98] are model theoretic rather than automata theoretic. The purpose of ....

....be proved using the automata theoretic methods from [ALS91, CM93] In [CMR98] those results were extended to optimization problems on graphs of fixed cliquewidth, a notion introduced by Courcelle, Engelfriet and Rozenberg [CER93] We shall give the detailed definitions in section 4. The proofs in [CMR98] are model theoretic rather than automata theoretic. The purpose of this paper is to state and prove an extension of theorem 1 to a wide class of generating functions of graph properties which are definable in Monadic Second Order Logic. We shall again use a model theoretic proof similar to that ....

[Article contains additional citation context not shown here]

B. Courcelle, J.A. Makowsky, and U. Rotics. Linear time solvable optimization problems on certain structured graph families, extended abstract. To appear in the proceedings of WG98, 1998.

....formula f(y) such that for every H in C, f(y) holds in H 1 iff y holds in H 2 . For such a class, MS2 and MS1 are equally expressive. However, the drawback of this formulation is that it says nothing on formulas with free variables. And these formulas are useful for algorithmic applications [CM, CMR]. We will use an alternative one, based on transformations of relational structures, called definable transductions of relational structures. See [CouT] Let R and Q be two finite ranked sets of relation symbols. Let W be a finite set of set variables, called here the set of parameters. A ....

....L also has a decidable MS 1 theory (because the transformation g : G 1 G 1 is MS definable) hence L 1 i o t (B) by the first part, for some t . Hence L 1 g o i o t (B) and g o i o t is an MS definable transduction. 4. 5) Algorithmic consequences Theorems 6 and 5 of [CMR] say respectively that the decision or counting graph problems expressible in MS 1 (resp. MS2 ) logic can be solved in polynomial time (resp. linear time) on classes of graphs for which a certain hierarchical decomposition (considered in [CO] resp. a tree decomposition of bounded width k, k ....

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B. Courcelle, J. Makowsky U. Rotics, Linear time solvable optimization problems on certain structured graphs, to appear in Proceedings of WG'98, Lec. Notes Comput. Science.

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