Approximation and Online Algorithms: 6th International by Evripidis Bampis, Martin Skutella

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By Evripidis Bampis, Martin Skutella

This booklet constitutes the completely refereed publish workshop complaints of the sixth foreign Workshop on Approximation and on-line Algorithms, WAOA 2008, held in Karlsruhe, Germany, in September 2008 as a part of the ALGO 2008 convention occasion. The 22 revised complete papers provided have been rigorously reviewed and chosen from fifty six submissions. The workshop lined parts reminiscent of algorithmic online game conception, approximation sessions, coloring and partitioning, aggressive research, computational finance, cuts and connectivity, geometric difficulties, inapproximability effects, mechanism layout, community layout, packing and masking, paradigms for layout and research of approximation and on-line algorithms, randomization suggestions, real-world functions, and scheduling difficulties.

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Extra resources for Approximation and Online Algorithms: 6th International Workshop, WAOA 2008, Karlsruhe, Germany, September 18-19, 2008, Revised Papers (Lecture Notes in Computer Science)

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We begin by showing that max size min bp (3, 3)-smi is NP-hard and not approximable within some δ > 1 unless P=NP. To prove this, we give a reduction from a restricted version of sat. Given a Boolean formula B in CNF and a truth assignment f , let t(f ) denote the number of clauses of B satisfied simultaneously by f , and let t(B) denote the maximum value of t(f ), taken over all truth assignments f of B. Let max (2,2)-e3-sat [4] denote the problem of finding, given a Boolean formula B in CNF in which each clause contains exactly 3 literals and each variable occurs exactly twice as an unnegated literal in B and exactly twice as a negated literal in B, a truth assignment f such that t(f ) = t(B).

To prove this, we give a reduction from a restricted version of sat. Given a Boolean formula B in CNF and a truth assignment f , let t(f ) denote the number of clauses of B satisfied simultaneously by f , and let t(B) denote the maximum value of t(f ), taken over all truth assignments f of B. Let max (2,2)-e3-sat [4] denote the problem of finding, given a Boolean formula B in CNF in which each clause contains exactly 3 literals and each variable occurs exactly twice as an unnegated literal in B and exactly twice as a negated literal in B, a truth assignment f such that t(f ) = t(B).

For the formal argument showing the correctness of the reduction, we claim (see [5] for the proof) that t(B) + bp+ (I) = n + 2m = 11 4 m, since 3m = 4n. Berman et al. [4] show that it is NP-hard to distinguish between instances B of max (2,2)-e3-sat for which (i) t(B) ≥ (1 − ε)m and (ii) t(B) ≤ 1015 1016 + ε m. By our construction, it follows that in case (i), bp+ (I) ≤ 3556 + ε m, whilst in 2032 − ε m. Hence an approximation algorithm for max size case (ii), bp+ (I) ≥ 3558 2032 3557 min bp (3, 3)-smi with performance guarantee r, for any r ≤ 3556+2032ε , could be used to decide between cases (i) and (ii) for max (2,2)-e3-sat in polynomial time, which is a contradiction unless P=NP.

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