Exact Algorithms With Worst-case Guarantee For Scheduling: From Theory to Practice

TL;DR

This thesis develops and improves exact algorithms with worst-case guarantees for NP-hard scheduling problems, combining theoretical analysis with practical enhancements like Memorization to solve larger instances efficiently.

Contribution

It introduces a dynamic programming algorithm for F3Cmax, a Branch & Merge method for 1||SumTi, and a novel Memorization technique that significantly enhances practical performance.

Findings

The DP algorithm solves F3Cmax in O*(3^n) time and space.

Branch & Memorize solves 1||SumTi with instances of 300 more jobs.

Memorization improves efficiency across multiple scheduling problems.

Abstract

This PhD thesis summarizes research works on the design of exact algorithms that provide a worst-case (time or space) guarantee for NP-hard scheduling problems. Both theoretical and practical aspects are considered with three main results reported. The first one is about a Dynamic Programming algorithm which solves the F3Cmax problem in O*(3^n) time and space. The algorithm is easily generalized to other flowshop problems and single machine scheduling problems. The second contribution is about a search tree method called Branch & Merge which solves the 1||SumTi problem with the time complexity converging to O*(2^n) and in polynomial space. Our third contribution aims to improve the practical efficiency of exact search tree algorithms solving scheduling problems. First we realized that a better way to implement the idea of Branch & Merge is to use a technique called Memorization. By the finding of a new algorithmic paradox and the implementation of a memory cleaning strategy, the method succeeded to solve instances with 300 more jobs with respect to the state-of-the-art algorithm for the 1||SumTi problem. Then the treatment is extended to another three problems 1|ri|SumCi, 1|dtilde|SumwiCi and F2||SumCi. The results of the four problems all together show the power of the Memorization paradigm when applied on sequencing problems. We name it Branch & Memorize to promote a systematic consideration of Memorization as an essential building block in branching algorithms like Branch and Bound. The method can surely also be used to solve other problems, which are not necessarily scheduling problems.