Graph Scheduling with Group Completion Times

2026-06-30

AI Summary

This work addresses the problem of minimizing the weighted sum of group completion times on general graphs, where each group completes when its last edge is scheduled. The authors extend the iterative rounding framework—previously limited to bipartite graphs—to general graphs by integrating odd-set inequalities from matching polyhedra with multigraph edge-coloring theory, yielding a novel approximation algorithm. This approach overcomes the longstanding restriction to bipartite settings and achieves a nearly tight (2 + ε)-approximation ratio when the optimal solution value is sufficiently large. Moreover, it uniformly improves upon the best-known approximation guarantees for the Data Migration problem across all scales of the optimum.

Abstract

In the Graph Scheduling problem we schedule a given multiset of edges on discrete time steps, such that at each step the set of edges forms a matching. The goal is to minimize the sum of weighted group completion times, where a group is a set of edges and it completes when the last edge has been scheduled. Two popular variants of this problem are Coflow Scheduling and Data Migration. Our main result is extending a recent iterated rounding approach from Coflow Scheduling, roughly corresponding to the bipartite case, to the general Graph Scheduling problem. This yields an essentially tight $(2+ε)$-approximation for the asymptotic setting where OPT is assumed to be large. For this we rely on polyhedral techniques from general matching, namely odd-set inequalities, and graph theoretical results on edge colorings in multigraphs. The state-of-the-art approximation algorithm for Data Migration is a $(1 + φ)$-approximation that improves when OPT is small. Taking the best of this and our main result, we obtain an improvement of the approximation rate for Data Migration in any regime.

Problem

• Graph Scheduling
• Group Completion Times
• Coflow Scheduling
• Data Migration
• Matching

Innovation

• iterated rounding
• odd-set inequalities
• edge coloring
• graph scheduling
• approximation algorithm