Relating Partial and Complete Solutions and the Complexity of Computing Smallest Solutions

Große, André; Rothe, Jörg; Wechsung, Gerd

doi:10.1007/3-540-45446-2_22

Cited by 7 publications

(3 citation statements)

References 14 publications

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“…We let R ⊆ Σ * × Σ * be any p-relation with |y| = p(|x|) for all (x, y) ∈ R for some polynomial p(n). Now we define the following problem (see also [52,Definition 3.2]):…”

Section: Algorithm B2 Setlevelseriespartitionsmentioning

confidence: 99%

Complexity and Geometry of Sampling Connected Graph Partitions

Najt¹,

DeFord²,

Solomon³

2019

Preprint

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In this paper, we prove intractability results about sampling from the set of partitions of a planar graph into connected components. Our proofs are motivated by a technique introduced by Jerrum, Valiant, and Vazirani. Moreover, we use gadgets inspired by their technique to provide families of graphs where the "flip walk" Markov chain used in practice for this sampling task exhibits exponentially slow mixing. Supporting our theoretical results we present some empirical evidence demonstrating the slow mixing of the flip walk on grid graphs and on real data. Inspired by connections to the statistical physics of self-avoiding walks, we investigate the sensitivity of certain popular sampling algorithms to the graph topology. Finally, we discuss a few cases where the sampling problem is tractable. Applications to political redistricting have recently brought increased attention to this problem, and we articulate open questions about this application that are highlighted by our results.

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“…We let R ⊆ Σ * × Σ * be any p-relation with |y| = p(|x|) for all (x, y) ∈ R for some polynomial p(n). Now we define the following problem (see also [52,Definition 3.2]):…”

Section: Algorithm B2 Setlevelseriespartitionsmentioning

confidence: 99%

Complexity and Geometry of Sampling Connected Graph Partitions

Najt¹,

DeFord²,

Solomon³

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…Kadin [Ka2] proved that if NP has sparse Turing-hard sets, then the polynomial hierarchy collapses to P NP . Krentel [Kr] studied P NP and other levels of the polynomial hierarchy that are relevant for certain optimization problems, see also [GRW1] and [GRW2]. Ogihara studied the truth-table and log-Turing reducibilities in a general setting; his results in particular apply to P NP and related classes [Og1].…”

Section: Outline and Context Of Our Resultsmentioning

confidence: 99%

“…and other levels of the polynomial hierarchy that are relevant for certain optimization problems, see also [GRW01,GRW02]. Ogihara studied the truth-table and log-Turing reducibilities in a general setting; his results in particular apply to P NP || and related classes [Ogi94].…”

Section: |mentioning

confidence: 99%

Complexity of the exact domatic number problem and of the exact conveyor flow shop problem

Riege

Rothe

Proceedings. 2004 International Conference on Information and Communication Technologies: From Theory to Applications, 2004.

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Abstract. We prove that the exact versions of the domatic number problem are complete for the levels of the boolean hierarchy over NP. The domatic number problem, which arises in the area of computer networks, is the problem of partitioning a given graph into a maximum number of disjoint dominating sets. This number is called the domatic number of the graph. We prove that the problem of determining whether or not the domatic number of a given graph is exactly one of k given values is complete for BH 2k (NP), the 2kth level of the boolean hierarchy over NP. In particular, for k = 1, it is DP-complete to determine whether or not the domatic number of a given graph equals exactly a given integer. Note that DP = BH 2 (NP). We obtain similar results for the exact versions of generalized dominating set problems and of the conveyor flow shop problem. Our reductions apply Wagner's conditions sufficient to prove hardness for the levels of the boolean hierarchy over NP.

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Hierarchies Based on NP

Texts in Theoretical Computer Science an EATCS Series

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Relating Partial and Complete Solutions and the Complexity of Computing Smallest Solutions

Cited by 7 publications

References 14 publications

Complexity and Geometry of Sampling Connected Graph Partitions

Complexity and Geometry of Sampling Connected Graph Partitions

Complexity of the exact domatic number problem and of the exact conveyor flow shop problem

Hierarchies Based on NP

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