Logspace Versions of the Theorems of Bodlaender and Courcelle

Elberfeld, Michael; Jakoby, Andreas; Tantau, Till

doi:10.1109/focs.2010.21

Cited by 101 publications

(101 citation statements)

References 39 publications

(51 reference statements)

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“…Hence, by the log-space version of Courcelle's Theorem [11] parity games whose tree-width and maximal priority are bounded by constants can be solved in log-space. For our purposes it is important that the maximal priority of a parity game can be assumed to be linear in the number of vertices by a compression of the priority function, see [12].…”

Section: Parity Gamesmentioning

confidence: 99%

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Parity Games of Bounded Tree- and Clique-Width

Ganardi

2015

Lecture Notes in Computer Science

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Abstract. In this paper it is shown that deciding the winner of a parity game is in LogCFL, if the underlying graph has bounded tree-width, and in LogDCFL, if the tree-width is at most 2. It is also proven that parity games of bounded clique-width can be solved in LogCFL via a log-space reduction to the bounded tree-width case, assuming that a k-expression for the parity game is part of the input.

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Section: Parity Gamesmentioning

confidence: 99%

“…Many NP-complete problems become solvable in log-space or linear time on classes of bounded treeor clique-width, see [11,13,14].…”

Section: Tree-widthmentioning

confidence: 99%

Parity Games of Bounded Tree- and Clique-Width

Ganardi

2015

Lecture Notes in Computer Science

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“…(For simplicity, we just write the index i for variable X i .) The scope of C is the list of variables corresponding to the white boxes of the sequence D; the relation of C contains the legal words for D. For the example in Figure 1, we have C 1H = ((1, 2, 3, 4, 5), r 1H ), C 8H = ((8, 9, 10), r 8H ), C 11H = ( (11,12,13), r 11H ), C 20H = ( (20,21,22,23,24,25,26), r 20H ), C 1V = ( (1,7,11,16,20), r 1V ), C 5V = ( (5, 8, 14, 18, 24), r 5V ), C 6V = ((6, 10, 15, 19, 26), r 6V ), C 13V = ( (13,17,22), r 13V ). Subscripts H and V stand for "Horizontal" and "Vertical," respectively, resembling the usual naming of definitions in crossword puzzles.…”

Section: Application To Constraint Satisfactionmentioning

confidence: 99%

“…Let φ be a fixed MSO sentence, let k be a fixed constant, and let C k be a class of finite structures having treewidth bounded by k. Then, for each finite structure A ∈ C k , deciding whether A |= φ holds is feasible in linear time [18] and logarithmic space [26] (w.r.t. ||A||).…”

Section: Msomentioning

confidence: 99%

“…To determine the treewidth of a graph G is NP-hard. However, for each fixed natural number k, checking whether tw(G) ≤ k, and if so, computing a tree decomposition for G of optimal width, is achievable in linear time [9], and was recently shown to be achievable in logarithmic space [26]. Note that the multiplicative constant factor of Bodlaenders linear algorithm [9] is exponential in k. However, there are algorithms that find exact tree decompositions in reasonable time or good upper approximations in many cases of practical relevance-see, for example, [10,11] and the references therein.…”

Section: Definition 21 ([72])mentioning

confidence: 99%

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Treewidth and Hypertree Width

Gottlob¹,

Greco²,

Scarcello³

2014

Tractability

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The chapter covers methods for identifying islands of tractability for NP-hard combinatorial problems by exploiting suitable properties of their graphical structure. Acyclic structures are considered, as well as nearly acyclic ones identified by means of so-called structural decomposition methods. In particular, the chapter focuses on the tree decomposition method, which is the most powerful decomposition method for graphs, and on the hypertree decomposition method, which is its natural counterpart for hypergraphs. These problem-decomposition methods give rise to corresponding notions of width of an instance, namely, treewidth and hypertree width. It turns out that many NP-hard problems can be solved efficiently over classes of instances of bounded treewidth or hypertree width: deciding whether a solution exists, computing a solution, and even computing an optimal solution (if some cost function over solutions is specified) are all polynomial-time tasks. Example applications include problems from artificial intelligence, databases, game theory, and combinatorial auctions.

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Parameterised Complexity of Model Checking and Satisfiability in Propositional Dependence Logic

Mahmood

Meier

2020

Lecture Notes in Computer Science

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In this paper, we initiate a systematic study of the parametrised complexity in the field of Dependence Logics which finds its origin in the Dependence Logic of Väänänen from 2007. We study a propositional variant of this logic (PDL) and investigate a variety of parametrisations with respect to the central decision problems. The model checking problem (MC) of PDL is NP-complete. The subject of this research is to identify a list of parametrisations (formula-size, treewidth, treedepth, team-size, number of variables) under which MC becomes fixed-parameter tractable. Furthermore, we show that the number of disjunctions or the arity of dependence atoms (dep-arity) as a parameter both yield a paraNP-completeness result. Then, we consider the satisfiability problem (SAT) showing a different picture: under team-size, or dep-arity SAT is paraNP-complete whereas under all other mentioned parameters the problem is in FPT. Finally, we introduce a variant of the satisfiability problem, asking for teams of a given size, and show for this problem an almost complete picture. ACM Subject Classification Theory of computation → Complexity theory and logic; Theory of computation → Parameterized complexity and exact algorithmsContributions In Table 1, we give an overview of our results. We study eight different parametrisations for each of the model checking problem, the satisfiability problem, as well as a variant of satisfiability (this problem asks for a team of a given size). Thereby, we prove dichotomies for MC and SAT: depending on the parameter the problem is either fixed-parameter tractable or paraNP-complete. Only the satisfiability variant and the parameters treewidth and dep-arity resist a complete classification and are left for further research. Related workThe technique of Courcelle's theorem [4] (see Prop. 6) has been used in different contexts: temporal logic [23], knowledge representation [13], and nonmonotonic logic [25]. Elberfeld et al. [10] enriched Courcelle's theorem to also yield results for the complexity class XL. This improvement applies to our results utilising this theorem as well and affects Theorem 12, 16, and 25.Organisation of the article At first, we introduce some required notions and definitions in (parametrised) complexity theory, dependence logic, and first-order logic. Then we study the parametrised complexity of the model checking problem. Proceed with the satisfiability problem and study a variant of it. Finally, we conclude and discuss open questions. Proofs of results marked with a ( ) can be found in the appendix.

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Logspace Versions of the Theorems of Bodlaender and Courcelle

Cited by 101 publications

References 39 publications

Parity Games of Bounded Tree- and Clique-Width

Parity Games of Bounded Tree- and Clique-Width

Treewidth and Hypertree Width

Parameterised Complexity of Model Checking and Satisfiability in Propositional Dependence Logic

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