Complexity of Finding Embeddings in a <i>k</i>-Tree

Arnborg, Stefan; Corneil, Derek G.; Proskurowski, Andrzej

doi:10.1137/0608024

Cited by 1,008 publications

(817 citation statements)

References 14 publications

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“…The complexity of this algorithm is, of course, dependent on the variable elimination order and the problem structure. Computing the optimal elimination order is an NP-hard problem (Arnborg, Corneil, & Proskurowski, 1987) and elimination orders yielding low induced tree width do not exist for some problems. These issues have been confronted successfully for a large variety of practical problems in the Bayesian network community, which has benefited from a large variety of good heuristics which have been developed for the variable elimination ordering problem (Bertele & Brioschi, 1972;Kjaerulff, 1990;Reed, 1992;Becker & Geiger, 2001).…”

Section: Example 42 Assumementioning

confidence: 99%

Efficient Solution Algorithms for Factored MDPs

Guestrin¹,

Koller²,

Parr³

et al. 2003

jair

280

268

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This paper addresses the problem of planning under uncertainty in large Markov Decision Processes (MDPs). Factored MDPs represent a complex state space using state variables and the transition model using a dynamic Bayesian network. This representation often allows an exponential reduction in the representation size of structured MDPs, but the complexity of exact solution algorithms for such MDPs can grow exponentially in the representation size. In this paper, we present two approximate solution algorithms that exploit structure in factored MDPs. Both use an approximate value function represented as a linear combination of basis functions, where each basis function involves only a small subset of the domain variables. A key contribution of this paper is that it shows how the basic operations of both algorithms can be performed efficiently in closed form, by exploiting both additive and context-specific structure in a factored MDP. A central element of our algorithms is a novel linear program decomposition technique, analogous to variable elimination in Bayesian networks, which reduces an exponentially large LP to a provably equivalent, polynomial-sized one. One algorithm uses approximate linear programming, and the second approximate dynamic programming. Our dynamic programming algorithm is novel in that it uses an approximation based on max-norm, a technique that more directly minimizes the terms that appear in error bounds for approximate MDP algorithms. We provide experimental results on problems with over 10 40 states, demonstrating a promising indication of the scalability of our approach, and compare our algorithm to an existing state-of-the-art approach, showing, in some problems, exponential gains in computation time.

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Section: Example 42 Assumementioning

confidence: 99%

Efficient Solution Algorithms for Factored MDPs

Guestrin¹,

Koller²,

Parr³

et al. 2003

jair

280

268

View full text Add to dashboard Cite

show abstract

“…, P n 1 contain vertices corresponding to opposite literals. But this is impossible, because no P i 1 contains opposite literals, since no clause C i contains opposite literals and because there are no connections between P 1 1 , . .…”

Section: Theorem 1 the Top Problem Is Np-completementioning

confidence: 99%

“…It uses a dynamic programming technique and it is similar to the algorithm given in [1] for the recognition of partial k-trees. Its idea is based on the following lemmas:…”

Section: Remarkmentioning

confidence: 99%

NP-Completeness of Some Tree-Clustering Problems

Schreiber

Skodinis

1998

Graph Drawing

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“…Hence we can determine the hardness of a given instance with respect to our algorithm in advance. This is not possible for treewidth and related parameters: computation of tree-width or branch-width is NP-hard [3,26], and it is not known whether graphs with fixed cliquewidth ≥ 4 can be recognized in polynomial time [5]. (3) Franco, et al [13] show that satisfiability of certain propositional formulas whose only connective is the implication is fixed-parameter tractable with respect to the number of occurrences of the always-false constant f (this result is listed in the appendix of [10] as pure implicational satisfiability of fixed f-depth); an improved algorithm is presented in [16].…”

Section: Fixed-parameter Tractable Parameterizations Of Satmentioning

confidence: 99%

Minimal Unsatisfiable Formulas with Bounded Clause-Variable Difference are Fixed-Parameter Tractable

Szeider

2003

Lecture Notes in Computer Science

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Complexity of Finding Embeddings in a k-Tree

Cited by 1,008 publications

References 14 publications

Efficient Solution Algorithms for Factored MDPs

Efficient Solution Algorithms for Factored MDPs

NP-Completeness of Some Tree-Clustering Problems

Minimal Unsatisfiable Formulas with Bounded Clause-Variable Difference are Fixed-Parameter Tractable

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