Weighted NP optimization problems: logical definability and approximation properties

Abstract

It is shown that all NP-optimization problems can be characterized by /spl Pi//sub 2/ first-order formulae and that /spl Pi//sub 1/ formulae are not sufficient for this goal. This generalizes the result of Kolaitis and Thakur (1990). The approximation properties of the weighted versions of MAX NP, MAX SNP, MAX SNP(/spl pi/), MIN F/sup +//spl Pi//sub 1/ and MIN F/sup +//spl Pi//sub 2/(1) are analyzed and dramatic changes are observed when negative weights are allowed.