Johann A. Makowsky

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 denable 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 quantication. Such quantications are allowed in the case of graphs with bounded tree-width. As applications we discuss in detail how this aects the parametrized complexity of the permanent and the hamiltonian of a matrix, and more generally, various generating functions of MSOL denable graph properties. Finally, our results are also applicable to SAT and ]SAT. ? 2001 Elsevier Science B.V. All rights reserved.

Algorithmic uses of the Feferman-Vaught Theorem

The classical Feferman–Vaught Theorem for First Order Logic explains how to compute the truth value of a first order sentence in a generalized product of first order structures by reducing this computation to the computation of truth values of other first order sentences in the factors and evaluation of a monadic second order sentence in the index structure. This technique was later extended by Lauchli, Shelah and Gurevich to monadic second order logic. The technique has wide applications in decidability and definability theory. Here we give a unified presentation, including some new results, of how to use the Feferman–Vaught Theorem, and some new variations thereof, algorithmically in the case of Monadic Second Order Logic MSOL. We then extend the technique to graph polynomials where the range of the summation of the monomials is definable in MSOL. Here the Feferman–Vaught Theorem for these polynomials generalizes well known splitting theorems for graph polynomials. Again, these can be used algorithmically. Finally, we discuss extensions of MSOL for which the Feferman–Vaught Theorem holds as well.

Identifying extended entity-relationship object structures in relational schemas

Relational schemas consisting of relation-schemes, key dependencies and key-based inclusion dependencies (referential integrity constraints) are considered. Schemas of this form are said to be entity-relationship (EER)-convertible if they can be associated with an EER schema. A procedure that determines whether a relational schema is EER-convertible is developed. A normal form is proposed for relational schemas representing EER object structures. For EER-convertible relational schemas, the corresponding normalization procedure is presented. The procedures can be used for analyzing the semantics of existing relational databases and for converting relational database schemas into object-oriented database schemas. >

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.

Restrictions of minimum spanner problems

Abstract A t -spanner of a graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times their distance in G . Spanners arise in the context of approximating the original graph with a sparse subgraph (Peleg, D., and Schaffer, A. A. (1989), J. Graph. Theory 13 (1), 99–116). The MINIMUM t -SPANNER problem seeks to find a t -spanner with the minimum number of edges for the given graph. In this paper, we completely settle the complexity status of this problem for various values of t , on chordal graphs, split graphs, bipartite graphs and convex bipartite graphs. Our results settle an open question raised by L. Cai (1994, Discrete Appl. Math. 48 , 187–194) and also greatly simplify some of the proofs presented by Cai and by L. Cai and M. Keil (1994, Networks 24 , 233–249). We also give a factor 2 approximation algorithm for the MINIMUM 2-SPANNER problem on interval graphs. Finally, we provide approximation algorithms for the bandwidth minimization problem on convex bipartite graphs and split graphs using the notion of tree spanners.

Why Horn formulas matter in computer science: initial structures and generic examples

We introduce the notion of generic examples as a unifying principle for various phenomena in computer science such as initial structures in the area of abstract data types, and Armstrong relations in the area of data bases. Generic examples are also useful in defining the semantics of logic programming, in the formal theory of program testing and in complexity theory. We characterize initial structures in terms of their genericity properties and give a syntactic characterization of first-order theories admitting initial .structures. The latter can be used to explain why Horn formulas have gained such a predominant role in various areas of computer science.

Coloured Tutte polynomials and Kauffman brackets for graphs of bounded tree width

Jones polynomials and Kauffman polynomials are the most prominent invariants of knot theory. For alternating links, they are easily computable from the Tutte polynomials by a result of Thistlethwaite (1988), but in general one needs Kauffmans Tutte polynomials for signed graphs (1989), further generalized to colored Tutte polynomials, as introduced by Bollobas and Riordan (1999). Knots and links can be presented as labeled planar graphs. The tree width of a link L is defined as the minimal tree width of its graphical presentations D(L) as crossing diagrams. We show that the colored Tutte polynomial can be computed in polynomial time for graphs of tree width at most k. Hence, for (not necessarily alternating) knots and links of tree width at most k, even the Kauffman square bracket [L] introduced by Bollobas and Riordan can be computed in polynomial time. In particular, the classical Kauffman bracket (L) and the Jones polynomial of links of tree width at most k are computable in polynomial time.

From a Zoo to a Zoology: Towards a General Theory of Graph Polynomials

Abstract We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We introduce two classes of (hyper)graph polynomials definable in second order logic, and outline a research program for their classification in terms of definability and complexity considerations, and various notions of reducibilities.

Computing graph polynomials on graphs of bounded clique-width

We discuss the complexity of computing various graph polynomials of graphs of fixed clique-width. We show that the chromatic polynomial, the matching polynomial and the two-variable interlace polynomial of a graph G of clique-width at most k with n vertices can be computed in time O(nf( k)), where f(k) ≤3 for the inerlace polynomial, f(k) ≤2k+1 for the matching polynomial and f(k) ≤3 2k+2 for the chromatic polynomial.

Arity and alternation in second-order logic

Abstract We investigate the expressive power of second-order logic over finite structures, when two limitations are imposed. Let SAA(k, n)(AA(k, n)) be the set of second-order formulas such that the arity of the relation variables is bounded by k and the number of alternations of (both first-order and) second-order quantification is bounded by n. We show that this imposes a proper hierarchy on second-order logic, i.e. for every k, n there are problems not definable in AA(k, n) but definable in AA(k + c1, n + d1) for some c1, d1. The method to show this is to introduce the set AUTOSAT(F) of formulas in F which satisfy themselves. We study the complexity of this set for various fragments of second-order loeic. For first-order logic FOL with unbounded alternation of quantifiers AUTOSAT(FOL) is PSpacecomplete. For first-order logic FOLn with alternation of quantifiers bounded by n, AUTOSAT(FOLn) is definable in AA(3, n + 4). AUTOSAT(AA(k, n)) is definable in AA(k + c1,n + d1) for some c1, d1.