B. Courcelle, J. A. Makowsky, and U. Rotics. On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic. Discrete Appl. Math., 108(1-2):23--52, 2001.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....problem parameterized by the treewidth of the input graph is a problem in FPT. For some counting problems on partial k trees, where k is a fixed constant, Courcelle [13] associated the existence of a polynomial time algorithm with their expressibility by Monadic Second Order Logic, see also [12]. As a consequence of these results, it is possible to construct a polynomialtime algorithm solving the #Hcoloring problem for partial k trees, when k and the size of H are fixed constants. However, the results of Courcelle do not provide implementable algorithms because of the very large hidden ....

B. Courcelle, J. Makowski, and U.Rotics. On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic. Discrete Applied Mathematics, 108(1-2):23--52, 2001.

....of labels, assuming that the graph is given in such a decomposed form. For example, decision, optimization, and enumeration problems expressible in MS1 logic, such as 3 Colorability, MaxClique and #MaxClique, can be solved in linear time on graphs given as clique decompositions of width at most k [4, 5]. And P recognizable problems, such as Hamiltonian Circuit (which is not MS1 expressible [4] can be solved in polynomial time on graphs given as NLC decompositions of width at most k [15] Note here that in theory it does not really matter which decomposition we have. For the transformations ....

.... and enumeration problems expressible in MS1 logic, such as 3 Colorability, MaxClique and #MaxClique, can be solved in linear time on graphs given as clique decompositions of width at most k [4, 5] And P recognizable problems, such as Hamiltonian Circuit (which is not MS1 expressible [4]) can be solved in polynomial time on graphs given as NLC decompositions of width at most k [15] Note here that in theory it does not really matter which decomposition we have. For the transformations between NLC decompositions and clique decompositions mentioned above can in fact be carried out ....

B. Courcelle, J. A. Makowsky and U. Rotics, "On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic." To appear in Discrete Appl. Math.

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B. Courcelle, J.A. Makowsky, and U. Rotics. On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic. Discrete and Applied Mathematics, xx:xx--yy, 2000.

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B. Courcelle, J. A. Makowsky, and U. Rotics. On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic. Discrete Applied Mathematics, 108:23--52, 2001.

....result for the Tutte polynomial for signed graphs introduced by L.H. Kauffman [Kau89] from which the Kauffman bracket and Jones polynomial can be easily computed. The result for the Tutte polynomials then uses a blend of techniques from logic and dynamic programming as previously developed in [CMR01] and [MM00,MM0x] Naturally, one would ask, whether the treewidth of S(L) has a natural interpretations in the language of knot theory. Our results show that this is indeed the case. First, there is a natural bijection oe : Diag(L) Sign(L) between the crossing diagrams of L and the signed ....

B. Courcelle, J.A. Makowsky, and U. Rotics. On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic. Discrete Applied Mathematics, 108(1-2):23--52, 2001.

....cliquewidth of a graph measures to what degree a graph is similar to an overlapping family of cliques. It was first introduced by Courcelle, Engelfriet and Rozenberg, CER93] We shall not need the technical definition of cliquewidth, and the interested reader is referred to, say, MR99,CO00,CMR00,CMR01] We collect here a few facts to allow comparison of our results on treewidth and possible extensions for cliquewidth. Fact 3 Cliques have cliquewidth 2. Golumbic and Rotics proved in [GR01] Proposition 4 The grids Grid n have cliquewidth n 1. From this, together with Fact 3 we see ....

....mixes logic and complexity. We therefore avoid this established terminology in the sequel. 6 Theorem 10 also holds for classes of cliquewidth at most 3, cf. CHL 00] but is open for cliquewidth at most 4 and higher. 5 Classes of graphs of clique width at most 3 are also MSOL polynomial, CMR01,CHL 00] So are their closures under induced subgraphs. It is therefore natural to ask whether an analogue of theorem 11 holds, i.e. whether an MSOL polynomial class of graphs which is I closed or S closed is necessarly of bounded clique width. The answer is negative. Proposition 13 There is ....

B. Courcelle, J.A. Makowsky, and U. Rotics. On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic. Discrete Applied Mathematics, 108(1-2):23--52, 2001.

....Noble, cf. And98,Nob98] have shown that the Tutte polynomial is computable in polynomial time on graphs of fixed bounded tree width. In [Mak00] this result is extended to the colored Tutte polynomials building on previous work by Arnborg, Courcelle, Lagergren, Makowsky, Rotics and Seese [ALS91,CMR00a,CMR00b]. Our proof had two ingredients, one logical and one combinatorial. The first consists in establishing that the polynomial in question can be expressed in a certain formalism of sums of products where the summation ranges over subsets of edges satisfying a condition in Monadic Second Order Logic ....

....a tree decomposition of the graph on which the polynomial is to be computed inductively. We produce here an abstract theorem (Theorem describing a very general situation where these ingredients yield polynomial time algorithms which extends the scope of applicability beyond the theorem given in [CMR00b,Mak01,Mak00]. However, it is sometimes not at all obvious how to recognize whether a graph polynomial is MSOL definable in the above sense. The purpose of this paper is to exhibit a wide range of graph polynomials which are hard to compute on arbitrary graphs but for which the existence of polynomial time ....

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B. Courcelle, J.A. Makowsky, and U. Rotics. On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic. Discrete Applied Mathematics, xx:xx--yy, 2000.

....where properties being expressible in a specific logical manner are considered. It uses methods from graph theory and model theoretic tools developed in the last 15 years and applies them to the algebraic setting. This work is an extension of work by B. Courcelle, J.A. Makowsky and U. Rotics [CMR00b], which extends [CM93] and [ALS91] The main new aspect with respect to those works is the ability to deal with a much larger class of algebraic properties captured by the logical framework we are going to define. This allows treatment of problems like the existence of zeros for polynomials, ....

....we are going to define. This allows treatment of problems like the existence of zeros for polynomials, linear programming and many more. The paper is organized as follows. In section 2 we introduce tree width of matrices, polynomials and systems of polynomials. In section 3 we state a result from [CMR00b] to illustrate the definition. Section 4 collects problems in relation with polynomial systems our approach applies to. The main results are then stated. The mathematical development begins in section 5. We define the logical framework in which we express our problems. This mainly refers to ....

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B. Courcelle, J.A. Makowsky, and U. Rotics. On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic. Discrete and Applied Mathematics, xx:xx--yy, 2000.

....Hence, theorem 3 also holds for the Kauffman square bracket [L] 2 Outline and discussion of the proof of theorems 2 and 3 We first sketch our new proof for the Tutte polynomials, as it contains all the essential ingredients which allow us to prove the additional results. The proof is based on [CMR00]. However, the presentation here does not depend on it. The main theorem of [CMR00] 3 combines methods from logic and graph theory. However, it cannot be applied directly to Tutte polynomials. We have to extend the framework of [CMR00] to allow order invariant definable polynomials with weight ....

....discussion of the proof of theorems 2 and 3 We first sketch our new proof for the Tutte polynomials, as it contains all the essential ingredients which allow us to prove the additional results. The proof is based on [CMR00] However, the presentation here does not depend on it. The main theorem of [CMR00] 3 combines methods from logic and graph theory. However, it cannot be applied directly to Tutte polynomials. We have to extend the framework of [CMR00] to allow order invariant definable polynomials with weight functions defined in arbitrary polynomial rings. Outline We assume our graphs have ....

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B. Courcelle, J.A. Makowsky, and U. Rotics. On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic. Discrete Applied Mathematics, xx:xx--yy, 2000.

....that can be linearly written or handled as trees in algorithms. Second, because they yield linear algorithms for the verification of MS properties (even NP complete, or on arbitrary levels of the polynomial hierarchy) and even for solving MS definable optimization or counting problems (see (Courcelle et al. 2000, DAM) A verification problem consists in deciding whether a certain property holds or not in a given graph or structure; an optimization problem consists in finding, typically, the largest set of edges or vertices satisfying a property ; a counting problem consists in counting the number of such ....

....algorithm taking as input structures that are not given by their syntax trees. However in concrete cases structures may be structured naturally, just because of the concrete situations they come from. Some structural properties may also help, this is the case for P 4 sparse graphs considered in (Courcelle et al. 2000, ToCS) It may happen that Question 3 has a yes answer whereas the answer to Question 2 is no. For a simple example consider a finite complete binary tree defined as the value of a term f(f(f( f(a) where f(x) stands for g(x; x) and a stands for itself. Every MS property of the binary tree ....

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B. Courcelle, J.A. Makowsky, and U. Rotics. (2000) On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic, Discrete and Applied Mathematics, xx, xx--yy.

....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 ....

B. Courcelle, J.A. Makowsky, and U. Rotics. On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic. in preparartion, 1998.

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B. Courcelle, J. A. Makowsky, and U. Rotics. On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic. Discrete Appl. Math., 108(1-2):23--52, 2001.

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B. Courcelle, J. Makowski, and U.Rotics. On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic. Discrete Applied Mathematics, 108(1-2):23--52, 2001.

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