Algorithmic Meta-theorems for Restrictions of Treewidth

Possibly the most famous algorithmic meta-theorem is Courcelle’s theorem, which states that all MSO-expressible graph properties are decidable in linear time for graphs of bounded treewidth. Unfortunately, the running time’s dependence on the formula describing the problem is in general a tower of e...

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Veröffentlicht in:Algorithmica Jg. 64; H. 1; S. 19 - 37
1. Verfasser: Lampis, Michael
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer-Verlag 01.09.2012
Schlagworte:
Algorithm Analysis and Problem Complexity
Algorithms
Computer Science
Computer Systems Organization and Communication Networks
Courcelle's theorem
Data Structures and Information Theory
Mathematics of Computing
Max-leaf number
MSO logic
Theory of Computation
Vertex cover
Vertex cover
Max-leaf number
MSO logic
Courcelle’s theorem
ISSN:0178-4617, 1432-0541
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Zusammenfassung:Possibly the most famous algorithmic meta-theorem is Courcelle’s theorem, which states that all MSO-expressible graph properties are decidable in linear time for graphs of bounded treewidth. Unfortunately, the running time’s dependence on the formula describing the problem is in general a tower of exponentials of unbounded height, and there exist lower bounds proving that this cannot be improved even if we restrict ourselves to deciding FO logic on trees. We investigate whether this parameter dependence can be improved by focusing on two proper subclasses of the class of bounded treewidth graphs: graphs of bounded vertex cover and graphs of bounded max-leaf number. We prove stronger algorithmic meta-theorems for these more restricted classes of graphs. More specifically, we show it is possible to decide any FO property in both of these classes with a singly exponential parameter dependence and that it is possible to decide MSO logic on graphs of bounded vertex cover with a doubly exponential parameter dependence. We also prove lower bound results which show that our upper bounds cannot be improved significantly, under widely believed complexity assumptions. Our work addresses an open problem posed by Michael Fellows.
ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-011-9554-x