Theoretical Computer Science pp 187-197 | Cite as

# Analysis of polynomial approximation algorithms for constraint expressions

## Abstract

The generalized maximum satisfiability problem contains a large class of interesting combinatorial optimization problems. Since most of them are NP-complete we analyze fast approximation algorithms.

Every generalized ψ-satisfiability problem has a polynomial ɛ_{ψ}-approximate algorithm for a naturally defined constant ɛ_{ψ}, 0≤ɛ_{ψ}>1 which is determined here explicitly for several ψ. It is shown that ɛ_{ψ} can be approximated by the Soviet Ellipsoid Algorithm. The fraction ɛ_{ψ}is known to be best-possible in the sense that the following set is NP-complete: The ψ-formulas S which have an assignment satisfying the fraction τ' <1−ɛ_{ψ}(τ' rational) of all clauses in S.

Among other results we also show that for many ψ, local search algorithms fail to be ɛ_{ψ}-approximate algorithms. In some cases, local search algorithms can be arbitrarily far from optimal.

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