Courcelle’s theorem—A game-theoretic approach

Kneis, Joachim; Langer, Alexander; Rossmanith, Peter

doi:10.1016/j.disopt.2011.06.001

Cited by 37 publications

(50 citation statements)

References 27 publications

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“…While Courcelle's theorem has mostly been used as a quick way of proving that linear time algorithms exist, more recently it has been shown to be practically applicable and very promising implementations have emerged. See [32] and the project webpage [36].…”

Section: Introductionmentioning

confidence: 99%

Model Checking for Successor-Invariant First-Order Logic on Minor-Closed Graph Classes

Eickmeyer

Kawarabayashi

Kreutzer

2013

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science

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Abstract-Model checking problems for first-and monadic second-order logic on graphs have received considerable attention in the past, not the least due to their connections to problems in algorithmic graph structure theory. While the model checking problem for these logics on general graphs is computationally intractable, it becomes tractable on important classes of graphs such as those of bounded tree-width, planar graphs or more generally, classes of graphs excluding a fixed minor.It is well known that allowing an order relation or successor function can greatly increase the expressive power of the respective logics. This remains true even in cases where we require the formulas to be order-or successor-invariant, that is, while they can use an order relation, their truth in a given graph must not depend on the particular ordering or successor function chosen.Naturally, the question arises whether this increase in expressive power comes at a cost in terms of tractability on specific classes of graphs. In LICS 2012, Engelmann et al. studied this problem and showed that order-invariant monadic second-order logic (MSO) remains tractable on the same classes of graphs than MSO without an ordering. That is, adding order-invariance to MSO essentially comes at no extra cost in terms of model checking complexity. For successor-invariant first-order logic something similar should be true. However, they only managed to show that successor-invariant first-order logic is tractable on the class of planar graphs which is very far from the best tractability results currently known for first-order logic.In this paper we significantly improve the latter result and show that successor-invariant first-order logic is tractable on any class of graphs excluding a fixed minor. This is much closer to the best results known for FO without an ordering. The proof relies on the construction of k-walks in suitable supergraphs of the input graphs, i.e., walks which visit every vertex at least once and at most k times, for some k depending on the excluded minor H. The supergraphs may in general contain H minors, but they still exclude some possible larger minor H , so by results of Flum and Grohe [20] model checking on these graphs is still fixed-parameter tractable.

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Section: Introductionmentioning

confidence: 99%

Model Checking for Successor-Invariant First-Order Logic on Minor-Closed Graph Classes

Eickmeyer

Kawarabayashi

Kreutzer

2013

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science

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“…Rather, it is essentially a variant of Algorithm 2 that uses dynamic programming on the treedecomposition to compute the result. It is shown in [116] that the number of entries in the dynamic programming algorithm's tables is bounded by a constant, which then yields the desired linear running time. We will cover this approach in Section 5.6.…”

Section: Alternative Proofs Of Courcelle's Theoremmentioning

confidence: 99%

“…Another different approach is proposed by the authors of this survey [116,125]. It can be understood as a dynamic programming variant of Algorithm 2 and does not explicitly construct a tree automaton either.…”

Section: A Game-theoretic Approachmentioning

confidence: 99%

“…However, it was shown in [116] that one can use dynamic programming on the tree decomposition to simulate runs of Algorithm 2 in a way that for graphs of bounded treewidth the total running time remains linear in n. This is achieved by introducing a third possible return value to Algorithm 2 that we may call "don't know". This return value essentially means that the currently considered subgraph does not provide enough information to decide whether G |= ϕ[α] (true) or whether G |= ϕ[α] (false) on the input graph.…”

Section: A Game-theoretic Approachmentioning

confidence: 99%

“…Another approach [116] is based on model-checking games [101,89,90]. This gametheoretic approach does not rely on automata or the Splitting Theorem.…”

Section: Alternative Proofs Of Courcelle's Theoremmentioning

confidence: 99%

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Practical algorithms for MSO model-checking on tree-decomposable graphs

Langer

Reidl

Rossmanith

et al. 2014

Computer Science Review

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In this survey, we review practical algorithms for graph-theoretic problems that are expressible in monadic second-order logic. Monadic second-order (MSO) logic allows quantifications over unary relations (sets) and can be used to express a host of useful graph properties such as connectivity, c-colorability (for a fixed c), Hamiltonicity and minor inclusion. A celebrated theorem in this area by Courcelle states that any graph problem expressible in MSO can be solved in linear time on graphs that admit a tree-decomposition of constant width. Courcelle's Theorem has been used thus far as a theoretic tool to establish that linear-time algorithms exist for graph problems by demonstrating that the problem in question is expressible by an MSO formula. A straightforward implementation of the algorithm in the proof of Courcelle's Theorem is useless as it runs into space-explosion problems even for small values of treewidth. Of late, there have been several attempts to circumvent these problems and we review some of these in this survey. This survey also introduces the reader to the notions of tree-decompositions and the basics of monadic second order logic.

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