Sequential and Parallel Algorithms for Mixed Packing and Covering

TL;DR

This paper introduces approximation algorithms for mixed packing and covering problems, providing efficient sequential and parallel solutions that generalize previous algorithms for pure cases, applicable to various combinatorial optimization problems.

Contribution

It presents the first approximation algorithms for the general class of mixed packing and covering problems, extending prior work on pure cases with efficient sequential and parallel algorithms.

Findings

Sequential greedy algorithm finds epsilon-approximate solutions in O(epsilon^-2 log m) iterations.

Parallel algorithm achieves similar results with polylogarithmic runtime.

Generalizes previous algorithms for pure packing or covering problems.

Abstract

Mixed packing and covering problems are problems that can be formulated as linear programs using only non-negative coefficients. Examples include multicommodity network flow, the Held-Karp lower bound on TSP, fractional relaxations of set cover, bin-packing, knapsack, scheduling problems, minimum-weight triangulation, etc. This paper gives approximation algorithms for the general class of problems. The sequential algorithm is a simple greedy algorithm that can be implemented to find an epsilon-approximate solution in O(epsilon^-2 log m) linear-time iterations. The parallel algorithm does comparable work but finishes in polylogarithmic time. The results generalize previous work on pure packing and covering (the special case when the constraints are all "less-than" or all "greater-than") by Michael Luby and Noam Nisan (1993) and Naveen Garg and Jochen Konemann (1998).