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

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## 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 approximability of Max PB 0 – 1 Programming, the problem of finding a binary vector *x* that satisfies a set of linear relations such that the objective value Σ*c*_{ i }*x*_{ i } is maximized, where *c*_{ i } are binary numbers. We show that, unless P = NP, Max PB 0 – 1 Programming is not approximable within the factor *n*^{1−ε} for any ε>0, where *n* is the number of inequalities, and is not approximable within \(m^{1/2 - \varepsilon }\)for any 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.

## Keywords

Maximization Problem Performance Ratio Truth Assignment Boolean Formula Strong Lower Bound## Preview

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