New upper bounds for the problem of maximal satisfiability

Abstract

In this paper we present relatively simple proofs of the following new upper bounds:c N , where c < 2 is a constant and N is the number of variables, for MAX-SAT for formulas with constant clause density;2 K=6 , where K is the number of clauses, for MAX-2-SAT;2 N=6:7 for .n; 3/-MAX-2-SAT.All bounds are proved by the splitting method. The main two techniques are combined complexity measures and clause learning.

A new upper bound for Max-2-SAT: A graph-theoretic approach

“…This is currently the end of a sequence of polynomial-space algorithms each improving on the run time, strictly staying within the realm of Max-2-Sat: beginning with O * (2 K 2.88 ) which was achieved by [15], it was subsequently improved to O * (2 K 3.742 ) by [5], to O * (2 K 5 ) by [4], to O * (2 K 5.217 ) by [7], to O * (2 K 5.5 ) by [8], to O * (2 K 5.88 ) by [9] and finally to the hitherto fastest upper bound of O * (2 K 6 ) by [10]. Our improvement has been achieved due to heuristic priorities concerning the choice of the variable for branching in case of a maximum degree four variable graph.…”

“…Our improvement has been achieved due to heuristic priorities concerning the choice of the variable for branching in case of a maximum degree four variable graph. As [9] and [10] improved the case where the variable graph has maximum degree five, it seems that the only way to speed up the generic branching algorithm is to improve the maximum degree six case. Our analysis also implies that the situation when the variable graph is regular is not that harmful.…”

“…Choose any v with # 2 (v) = 5 and apply an appropriate branch as described by A.S. Kulikov and K. Kutzkov [10]; the branching can be also read off from the new run time analysis given below. 4.…”

“…3 ω 3 )-branch. The amount of 10 3 ω 3 has been found using a linear program, see Appendix A. If some v i has exactly three neighbors of weight less than five in F [v], w.l.o.g., i = 1, this entails a (5 5 + ω 5 , ω 5 + 5 5 + ω 4 + 3 4 + 5 , 6ω 5 + β)-branch.…”

“…Kulikov and K. Kutzkov [10] could prove a run time of O * (2 n 6.7 ) for the case of a 3-regular variable graph G var where n is the number of variables. We use this algorithm once we arrived at the point of a cubic variable graph.…”