Exact Algorithms for MAX-SAT

The maximum satisfiability problem (MAX-SAT) is stated as follows: Given a boolean formula in CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX-SAT is MAX-SNP-complete and received much attention recently. One of the challenges posed by Alber, Gramm and Niedermeier in a recent survey paper asks: Can MAX-SAT be solved in less than 2 n “steps”? Here, n is the number of distinct variables in the formula and a step may take polynomial time of the input. We answered this challenge positively by showing that a popular algorithm based on branch-and-bound is bounded by O(b2 n ) in time, where b is the maximum number of occurrences of any variable in the input. When the input formula is in 2-CNF, that is, each clause has at most two literals, MAX-SAT becomes MAX-2-SAT and the decision version of MAX-2-SAT is still NP-complete. The best bound of the known algorithms for MAX-2-SAT is O(m2 m/5 ), where m is the number of clauses. We propose an ecient decision algorithm for MAX-2-SAT whose time complexity is bound by O(n2 n ). This result is substantially better than the previously known results. Experimental results also show that our algorithm outperforms any algorithm we know on MAX-2-SAT.

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