Complexity of approximating bounded variants of optimization problems

Chlebík, Miroslav; Chlebíková, Janka

doi:10.1016/j.tcs.2005.11.029

“…Theorem 5 (Chlebík and Chlebíková [5]). Given an instance G of Min-3-VC with n vertices, it is, for every sufficiently small > 0, NP-hard to decide whether the size of a minimum vertex cover of G is at most (…”

Section: Hardnessmentioning

confidence: 99%

Approximability of Minimum AND-Circuits

Arpe

¹

,

Manthey

²

2007

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Given a set of monomials, the Minimum AND-Circuit problem asks for a circuit that computes these monomials using AND-gates of fan-in two and being of minimum size. We prove that the problem is not polynomial time approximable within a factor of less than 1.0051 unless P = NP, even if the monomials are restricted to be of degree at most three. For the latter case, we devise several efficient approximation algorithms, yielding an approximation ratio of 1.278. For the general problem, we achieve an approximation ratio of d − 3/2, where d is the degree of the largest monomial. In addition, we prove that the problem is fixed parameter tractable with the number of monomials as parameter. Finally, we reveal connections between the Minimum ANDCircuit problem and several problems from different areas.

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“…Theorem 3 (Chlebík and Chlebíková [5]). Given an instance G of Min-6-VC with n vertices, it is, for every sufficiently small > 0, NP-hard to decide whether the size of a minimum vertex cover of G is at most ( 474 494 + ) · n or at least ( 484 494 − ) · n. Thus, it is NP-hard to decide whether the instance of Min-AC corresponding to the graph can be computed by a circuit of size at most |E| + ( 474 494 + )|V | or if every circuit for this instance has a size of at least |E| + ( 484 494 − )|V | for sufficiently small > 0.…”

Section: Hardnessmentioning

confidence: 99%

“…Theorem 5 (Chlebík and Chlebíková [5]). Given an instance G of Min-3-VC with n vertices, it is, for every sufficiently small > 0, NP-hard to decide whether the size of a minimum vertex cover of G is at most ( 494 564 + ) · n or at least ( 499 564 − ) · n. Analogously to the proof of Theorem 2, the inapproximability result for Min-3-AC follows from plugging in the inequality |E| ≤ (3/2) · |V |.…”

Section: Hardnessmentioning

confidence: 99%

Approximability of Minimum AND-Circuits

Arpe

¹

,

Manthey

²

2006

Algorithm Theory – SWAT 2006

1

0

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Given a set of monomials, the Minimum AND-Circuit problem asks for a circuit that computes these monomials using AND-gates of fan-in two and being of minimum size. We prove that the problem is not polynomial time approximable within a factor of less than 1.0051 unless P = NP, even if the monomials are restricted to be of degree at most three. For the latter case, we devise several efficient approximation algorithms, yielding an approximation ratio of 1.278. For the general problem, we achieve an approximation ratio of d − 3/2, where d is the degree of the largest monomial. In addition, we prove that the problem is fixed parameter tractable with the number of monomials as parameter. Finally, we reveal connections between the Minimum ANDCircuit problem and several problems from different areas.

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“…Min-VC-3 is APX-hard; see [4]. Moreover, Chlebík and Chlebíková [12] showed that Min-VC-3 and Min-VC-4 are NP-hard to approximate within 1.0101215 and 1.0202429, respectively, and, for any ∆ ≥ 228, Min-VC-∆ is NP-hard to approximate within − O(log ∆/∆). Dinur and Safra [14] showed that Minimum Vertex Cover is NP-hard to approximate within any constant less than 10 √ 5 − 21 = 1.3606 .…”

Section: Preliminariesmentioning

confidence: 99%

“…We have obtained an L-reduction from Max-IS-3 to MSR-4 with αβ = 1. Chlebík and Chlebíková [12] showed that Max-IS-3 is NP-hard to approximate within 1.010661. It follows that MSR-4 is also NP-hard to approximate within 1.010661.…”

Section: L-reduction From Max-is-3 To Msr-4mentioning

confidence: 99%

Inapproximability of Maximal Strip Recovery

Jiang

¹

2009

Algorithms and Computation

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In comparative genomic, the first step of sequence analysis is usually to decompose two or more genomes into syntenic blocks that are segments of homologous chromosomes. For the reliable recovery of syntenic blocks, noise and ambiguities in the genomic maps need to be removed first. Maximal Strip Recovery (MSR) is an optimization problem proposed by Zheng, Zhu, and Sankoff for reliably recovering syntenic blocks from genomic maps in the midst of noise and ambiguities. Given d genomic maps as sequences of gene markers, the objective of MSR-d is to find d subsequences, one subsequence of each genomic map, such that the total length of syntenic blocks in these subsequences is maximized. For any constant d ≥ 2, a polynomial-time 2d-approximation for MSR-d was previously known. In this paper, we show that for any d ≥ 2, MSR-d is APX-hard, even for the most basic version of the problem in which all gene markers are distinct and appear in positive orientation in each genomic map. Moreover, we provide the first explicit lower bounds on approximating MSR-d for all d ≥ 2. In particular, we show that MSR-d is NP-hard to approximate within Ω(d/ log d). From the other direction, we show that the previous 2d-approximation for MSR-d can be optimized into a polynomial-time algorithm even if d is not a constant but is part of the input. We then extend our inapproximability results to several related problems including CMSR-d, δ-gap-MSR-d, and δ-gap-CMSR-d.

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Complexity of approximating bounded variants of optimization problems

Cited by 99 publications

References 19 publications

Approximability of Minimum AND-Circuits

Approximability of Minimum AND-Circuits

Approximability of Minimum AND-Circuits

Inapproximability of Maximal Strip Recovery

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