Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families

Abstract

Abstract. This paper describes a predicate calculus in which graph problems can be expressed. Any problem possessing such an expression can be solved in linear time on any recursively constructed graph, once its decomposition tree is known. Moreover, the linear-time algorithm can be generated automatically from the expression, because all our theorems are proved constructively. The calculus is founded upon a short list of particularly primitive predicates, which in turn are combined by fundamental logical operations. This framework is rich enough to include the vast majority of known linear-time solvable problems.We have obtained these results independently of similar results by Courcelle [11], [12], through utilization of the framework of Bern et al. [6]. We believe our formalism is more practical for programmers who would implement the automatic generation machinery, and more readily understood by many theorists.

Probabilistic Inference and Monadic Second Order Logic

“…Many graph properties can be expressed in it. E.g, Borie et al [8] show how many graph properties can be expressed in MSOL. For example, the following property expresses that directed graph…”

“…In this paper, we use the notion of Counting Monadic Second Order Logicand the notion of regularity. The result of Courcelle has been extended a number of times (e.g., [1,8,12], see also [9]. )…”

Optimization Problems in Dotted Interval Graphs

“…We note that for Minimum Vertex Cover and many other problems for this class, our algorithm also works for the general weighted case using the local ratio technique [3] for the clique-deletion process in the proof of Theorem 3. However since the Borie et al [7] extension of Courcelle's …”

“…First, notice that for any positive integer w, the Minimum G-Deletion problem can be solved in linear time when restricted to graphs of treewidth w; this is due to an extension of Courcelle's Theorem [10] due to Borie et al [7]. Second, notice that the clique-deletion technique that is applied in the proof of Theorem 3 can be extended to Minimum G-Deletion.…”

Graph Minors and Parameterized Algorithm Design

“…The importance of treewidth for algorithmic graph theory is illustrated by the celebrated Courcelle's theorem stating that if Π k is a parameterized property of graphs expressible by a CMSOL formula φ k , then there is an algorithm that, given as input a graph G, can check whether G satisfies property Π k (i.e., whether [8] and Borie, Parker, and Tovey in [17]. An alternative game-theoretic proof has appeared recently in [81,82].…”