A FPTAS for Minimizing Positive Half-Product

The problem of minimizing a quadratic function of boolean variables, which we call PHP (positive half-product), is formulated. A fully polynomial time approximation scheme (FPTAS) for PHP is derived. Several NP-hard scheduling problems can be formulated in terms of PHP. In this presentation, we concentrate on a single machine scheduling problem with controllable job processing times to minimize a linear combination of the total weighted job completion time and the total weighted processing time compression. 1. Problems HP (half-product) and PHP (positive half-product) Denote ( ) n x x x , , 1 K = . The problem of minimizing the half-product, which we denote as HP, can be formulated as follows: min, ) ( 1 1 → − = ∑ ∑ = ≤ < ≤ n j j j n j i j i j i x h x x b a x HP subject to { } . , , 1 , 0 , , , 1 , 0 n j h b a x j j j j K = ≥ ∈ Badics and Boros [1] have proved that problem HP is NP-hard by using a reduction from PARTITION. They also derived a FPTAS for problem HP. Let us remind a definition of a FPTAS for a minimization problem. Denote by F* and F the values of an optimal and approximate solutions for this problem, respectively. Given 0 > ε , an approximate solution is called an ε-approximate one, if * * 0 F F F ε ≤ . A family of approximation algorithms { } ε G forms a FPTAS if ε G delivers an ε-approximate solution in time polynomial in the problem input length in binary encoding and 1/ε . There is an equivalence between problem HP and some scheduling problems in a sense that their exact solutions coincide. However, ε-approximate solution to problem HP is not an εapproximate solution to a scheduling problem in most of the cases. We now formulate a problem which we call PHP (positive half-product). Several scheduling problems can be formulated in terms of problem PHP and exact and ε-approximate solutions of problem PHP and these scheduling problems coincide. Problem PHP: min, ) 1 ( ) ( 1 1 1 → + − − + = ∑ ∑ ∑ = = ≤ < ≤ D x g x h x x b a x F n