Sequential and Parallel Algorithms for Mixed Packing and Covering

We describe sequential and parallel algorithms that approximately solve linear programs with no negative coefficients (a.k.a. mixed packing and covering problems). For explicitly given problems, our fastest sequential algorithm returns a solution satisfying all constraints within a 1 ± ǫ factor in O(md log(m)/ǫ) 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 ǫ 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 “≤” or only “≥” inequalities, but not both) and run in time independent of the so-called width of the problem.