Sequential and parallel algorithms for mixed packing and covering

Abstract

We describe sequential and parallel algorithms that approximately solve linear programs with no negative coefficients (aka mixed packing and covering problems). For explicitly given problems, our fastest sequential algorithm returns a solution satisfying all constraints within a 1/spl plusmn//spl epsi/ factor in O(mdlog(m)//spl epsi//sup 2/) time, where m is the number of constraints and d is the maximum number of constraints any variable appears in. Our parallel algorithm runs in time polylogarithmic in the input size times /spl epsi//sup -4/ and uses a total number of operations comparable to the sequential algorithm. The main contribution is that the algorithms solve mixed packing and covering problems (in contrast to pure packing or pure covering problems, which have only "/spl les/" or only "/spl ges/" inequalities, but not both) and run in time independent of the so-called width of the problem.