# Sum edge coloring of multigraphs via configuration LP

## Abstract

We consider the scheduling of biprocessor jobs under sum objective (BPSMS). Given a collection of unitlength jobs where each job requires the use of two processors, find a schedule such that no two jobs involving the same processor run concurrently. The objective is to minimize the sum of the completion times of the jobs. Equivalently, we would like to find a sum edge coloring of a given multigraph, that is, a partition of its edge set into matchings M1, ⋯ ,Mt minimizing ∑ti=1 i|M i|. This problem is APX-hard, even in the case of bipartite graphs [Marx 2009]. This special case is closely related to the classic open shop scheduling problem. We give a 1.8298-approximation algorithm for BPSMS improving the previously best ratio known of 2 [Bar-Noy et al. 1998]. The algorithm combines a configuration LP with greedy methods, using nonstandard randomized rounding on the LP fractions. We also give an efficient combinatorial 1.8886-approximation algorithm for the case of simple graphs, which gives an improved 1.79568 + O(log d̄/d̄)-approximation in graphs of large average degree d̄. © 2011 ACM.