Strong lower bounds on the approximability of some NPO PB-complete maximization problems

Abstract

The approximability of several NP maximization problems is investigated and strong lower bounds for the studied problems are proved. For some of the problems the bounds are the best that can be achieved, unless P = NP. For example we investigate the approximabflity of MAX PB 0 - 1 PROGRA1VIMING, the problem of finding a binary vector x that satisfies a set of linear relations such that the objective value ∑cixi is maximized, where cl are binary numbers. We show that, unless P = NP, MAX PB 0 - 1 PROGRAMMING is not approximable within the factor n1-ε for any ε > 0, where n is the number of inequalities, and is not approximable within m1/2-ε for amy ε > 0, where m is the number of variables. Similar hardness results are shown for other problems on binary linear systems, some problems on the satisfiability of boolean formulas and the longest induced circuit problem.