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By Thomas Erlebach, Giuseppe Persiano

This ebook constitutes the completely refereed submit court cases of the 3rd overseas Workshop on Approximation and on-line Algorithms, WAOA 2005, held in Palma de Mallorca, Spain in October 2005 as a part of the ALGO 2005 event.

The 26 revised complete papers provided have been conscientiously reviewed and chosen from sixty eight submissions. themes addressed by means of the workshop are algorithmic online game thought, approximation sessions, coloring and partitioning, aggressive research, computational finance, cuts and connectivity, geometric difficulties, inapproximability effects, mechanism layout, community layout, packing and overlaying, paradigms, randomization thoughts, real-world functions, and scheduling problems.

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2 A Semidefinite Programming Relaxation for MAX SAT The semidefinite programming relaxation of MAX SAT is shown in Figure 3. As in the semidefinite programming relaxation of MAX NAE-SAT, each Boolean variable xi corresponds to a unit vector v i . Here, the additional vector v 0 corresponds to the value FALSE and the vector −v 0 corresponds to the value T RUE. We use Sˆk to denote the set of all permutation of {0, 1, . . , k}, i0 to denote the index 0 and π(k +1) to denote π(0). As before to ensure a program of polynomial size we should take kmax = O( logloglogn n ), but eventually we take kmax to be some constant.

To ensure that xn+i = x¯i , we require v i · v n+i = −1, for 1 ≤ i ≤ n. To check that this is indeed a relaxation of the MAX NAE-SAT instance, note that for every Boolean assignment α1 , . . , αn ∈ {0, 1} to the variables x1 , . . , xn , the vectors 30 A. Avidor, I. Berkovitch, and U. t. zj ≤ v i1 · v i2 kj − È kj l=1 vi π(l) 4 Cj = NAE(xi1 , . . , xikj ) π ∈ Skj , kj ≤ kmax 1≤j≤m ·v i π(l+1) zj ≤ 1 v i · v n+i = −1 + v i1 · v i3 + v i2 · v i3 ≥ −1 v i ∈ S n−1 1≤j≤m 1≤i≤n 1 ≤ i1 , i2 , i3 ≤ 2n 1 ≤ i ≤ 2n Fig.

P1 αk+1 (fSAT ) + p2 ηka ≥ β p√1 + p2 = 1 e 2 ≤ a≤1 0 ≤ p1 , p2 ≤ 1 k≥1 We conjecture that conjecture 1 holds for fSAT as well. Based on the conjecture we calculated lower bounds on α ˆk (fSAT ). 5, for a proper choice of kmax these bounds are also bounds on αk (fSAT ). 8434. , our algorithm achieves its worst case ratio on instances in which all clauses are of size 1, 2 or 5. Improved Approximation Algorithms for MAX NAE-SAT and MAX SAT 39 We note that the addition of any combination of Goemans and √ Williamson algorithm [GW94] with the rounding function f3a (for any 1/2 ≤ a ≤ e/2), Lewin, Livnat and Zwick algorithm [LLZ02], Halperin and Zwick algorithms [HZ01] with or without perturbation does not yield an improved approximation ratio.

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