Towards PTAS for Precedence Constrained Scheduling via Combinatorial Algorithms

TL;DR

This paper introduces a nearly polynomial-time combinatorial approximation algorithm for precedence constrained scheduling, improving upon prior LP-based methods by avoiding liftings and hierarchies.

Contribution

It presents a new combinatorial algorithm achieving a (1+ε)-approximation with near-polynomial runtime, bypassing complex LP hierarchies used in previous approaches.

Findings

Achieves (1+ε)-approximation in near-polynomial time for fixed m and ε.

Replaces LP hierarchy conditioning with schedule guessing.

Improves computational efficiency over prior LP-based algorithms.

Abstract

We study the classic problem of scheduling $n$ precedence constrained unit-size jobs on $m = O(1)$ machines so as to minimize the makespan. In a recent breakthrough, Levey and Rothvoss \cite{LR16} developed a $(1+\epsilon)$-approximation for the problem with running time $\exp\Big(\exp\Big(O\big(\frac{m^2}{\epsilon^2}\log^2\log n\big)\Big)\Big)$, via the Sherali-Adams lift of the basic linear programming relaxation for the problem by $\exp\Big(O\big(\frac{m^2}{\epsilon^2}\log^2\log n\big)\Big)$ levels. Garg \cite{Garg18} recently improved the number of levels to $\log ^{O(m^2/\epsilon^2)}n$, and thus the running time to $\exp\big(\log ^{O(m^2/\epsilon^2)}n\big)$, which is quasi-polynomial for constant $m$ and $\epsilon$. In this paper we present an algorithm that achieves $(1+\epsilon)$-approximation for the problem with running time $n^{O\left(\frac{m^4}{\epsilon^3}\log^3\log n\right)}$, which is very close to a polynomial for constant $m$ and $\epsilon$. Unlike the algorithms of Levey-Rothvoss and Garg, which are based on linear-programming hierarchy, our algorithm is purely combinatorial. For this problem, we show that the conditioning operations on the lifted LP solution can be replaced by making guesses about the optimum schedule.