Nearly-Linear Time LP Solvers and Rounding Algorithms for Scheduling Problems

TL;DR

This paper develops nearly-linear time approximation algorithms for complex scheduling problems using linear programming relaxations and advanced rounding techniques, achieving near-optimal ratios efficiently.

Contribution

It introduces nearly-linear time algorithms for scheduling problems with LP relaxations and novel rounding methods, matching best polynomial-time approximation ratios.

Findings

Nearly-linear time algorithms match best polynomial-time approximation ratios.

LP relaxations are formulated for unrelated machine scheduling and solved efficiently.

A new rounding algorithm achieves a (2+ε)-approximation for makespan.

Abstract

We study nearly-linear time approximation algorithms for non-preemptive scheduling problems in two settings: the unrelated machine setting, and the identical machine with job precedence constraints setting, under the well-studied objectives such as makespan and weighted completion time. For many problems, we develop nearly-linear time approximation algorithms with approximation ratios matching the current best ones achieved in polynomial time. Our main technique is linear programming relaxation. For the unrelated machine setting, we formulate mixed packing and covering LP relaxations of nearly-linear size, and solve them approximately using the nearly-linear time solver of Young. For the makespan objective, we develop a rounding algorithm with $(2+\epsilon)$-approximation ratio. For the weighted completion time objective, we prove the LP is as strong as the rectangle LP used by Im and Li, leading to a nearly-linear time $(1.45 + \epsilon)$-approximation for the problem. For problems in the identical machine with precedence constraints setting, the precedence constraints can not be formulated as packing or covering constraints. To achieve the nearly-linear running time, we define a polytope for the constraints, and leverage the multiplicative weight update (MWU) method with an oracle which always returns solutions in the polytope.