Fast algorithms for decomposable graphs
Year: 2013
Type: article
Abstract: A celebrated theorem by Courcelle states that every problem definable in monadic second-order logic (MSO) can be solved in linear time on graphs of bounded treewidth. This meta-theorem along with its extensions by Arnborg, Lagergren, and Seese as well as by Courcelle and Mosbah explains, why many important graph problems that are NP-hard on general graphs can be solved efficiently on tree-decomposable graphs. Such problems include, for instance, Vertex Cover, Dominating Set, 3-Colorability, Steiner Tree, Max-Cut, and Hamiltonian Cycle. The standard proof of Courcelle's Theorem is to translate the MSO-formula into a finite-state tree automaton that accepts a tree decomposition of the input graph iff the graph is a model for the MSO-formula. Existing, optimized software such as MONA can be used to construct the corresponding tree automaton, which for bounded treewidth is of constant size. Unfortunately, the constants involved can become extremely large - every quantifier alternation in the MSO formula requires a power set construction for the automaton. Here, the required space causes severe problems in practical applications. In this thesis, we develop a novel approach based on model checking games, also known as Hintikka games. We show that one can construct the model checking game via a linear-time dynamic programming algorithm on a tree decomposition of the input graph. To make the size of the games manageable in practical settings, we introduce a three-valued extension of the classical model checking game and a technique that we call "early determinization." Furthermore, we describe our implementation of the resulting MSO model checking algorithm for graphs of bounded treewidth and present experimental results. These indicate that for some natural optimization problems our approach is a suitable alternative to Integer Linear Optimization.
Authors: Alexander Langer, Peter Rossmanith
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2275184243