Algorithmic meta-theorems

Kreutzer, Stephan

doi:10.1017/cbo9780511974960.006

Cited by 43 publications

(47 citation statements)

References 72 publications

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“…Split-width compares well to the usual measures of graph complexity: tree-width and clique-width [7,13,23]. This relation is stated in the following theorem.…”

Section: Split-width Tree-width and Clique-width Of Mnwsmentioning

confidence: 81%

“…If a class of MNWs is MSO definable and in addition has bounded tree-width (tree-width is a measure of the complexity or connectivity of a graph) then any MSO expressible property is decidable over such a class (As a matter of fact, this is true for any class of graphs. See [7,13,23]). They show that for bounded context, bounded phase and ordered multi-pushdown restrictions of MPDSs, the class of permissible MNWs is MSO definable and has bounded tree-width.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, split-width is a measure on MNWs which is similar to tree-width (or clique-width) [9,13]. This, calls for a comparison of split-width to tree-width.…”

Section: Introductionmentioning

confidence: 99%

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MSO Decidability of Multi-Pushdown Systems via Split-Width

Aiswarya

Gastin

Kumar

2012

Lecture Notes in Computer Science

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Abstract. Multi-threaded programs with recursion are naturally modeled as multi-pushdown systems. The behaviors are represented as multiply nested words (MNWs), which are words enriched with additional binary relations for each stack matching a push operation with the corresponding pop operation. Any MNW can be decomposed by two basic and natural operations: shuffle of two sequences of factors and merge of consecutive factors of a sequence. We say that the split-width of an MNW is k if it admits a decomposition where the number of factors in each sequence is at most k. The MSO theory of MNWs with split-width k is decidable. We introduce two very general classes of MNWs that strictly generalize known decidable classes and prove their MSO decidability via their split-width and obtain comparable or better bounds of tree-width of known classes.

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“…Split-width compares well to the usual measures of graph complexity: tree-width and clique-width [7,13,23]. This relation is stated in the following theorem.…”

Section: Split-width Tree-width and Clique-width Of Mnwsmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 99%

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MSO Decidability of Multi-Pushdown Systems via Split-Width

Aiswarya

Gastin

Kumar

2012

Lecture Notes in Computer Science

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“…The most general result of this form known to date is tractability of first-order model checking on classes of graphs of locally bounded expansion [15]. As far as first-order model checking is concerned, it was shown in [34] that FO model checking is essentially not FPT on any class of graphs which is somewhere dense, i.e., not nowhere dense. Hence, in this line of research, the main open problem is tractability of FO model checking on classes of graphs which are nowhere dense.…”

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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“…Examples of other metatheorems can be found in [148,21,71,64]. We also refer the reader to the surveys on algorithmic meta-theorems by Grohe and Kreutzer [92,119,93]. It is important to realize that while meta-theorems are immensely useful in quickly establishing whether a problem at hand admits an algorithm of a particular type, they are usually not of much use in providing algorithms that are usable.…”

Section: Introductionmentioning

confidence: 99%

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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Algorithmic meta-theorems

Cited by 43 publications

References 72 publications

MSO Decidability of Multi-Pushdown Systems via Split-Width

MSO Decidability of Multi-Pushdown Systems via Split-Width

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

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

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