Towards PTAS for Precedence Constrained Scheduling via Combinatorial Algorithms
Abstract
We study the classic problem of scheduling $n$ precedence constrained unitsize 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 SheraliAdams 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 quasipolynomial 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 LeveyRothvoss and Garg, which are based on linearprogramming 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.01231
 Bibcode:
 2020arXiv200401231L
 Keywords:

 Computer Science  Data Structures and Algorithms