Designing Scheduling Algorithms via a Mathematical Perspective

Abstract

This document will discuss three problems that I worked on during my Ph.D. Chapter \ref{chapter: SC} contains my work on the Santa Claus problem, and Chapters \ref{chapter: S1} and \ref{chapter: S2} contain my work on scheduling with precedence constraints and communication delays. New algorithms for scheduling and resource allocation problems have far reaching implications, as problems in scheduling and resource allocation are a foundational playground for studying computational hardness and are practically relevant. In the Santa Claus problem, Santa has a set of gifts, and he wants to distribute them among a set of children so that the least happy child is as happy as possible. Child $i$ has value $p_{ij}$ for present $j$, where $p_{ij}$ is in $ \{ 0,p_j\}$. A modification of Haxell's hypergraph matching argument by Annamalai, Kalaitzis, and Svensson gives a $12.33$-approximation algorithm for the problem. In joint work with Thomas Rothvoss and Yihao Zhang, we introduce a matroid version of the Santa Claus problem. While our algorithm is also based on the augmenting tree by Haxell, the introduction of the matroid structure allows us to solve a more general problem with cleaner methods. Using our result from the matroid version of the problem, we obtain a $(4+\varepsilon)$-approximation algorithm for Santa Claus. In scheduling theory, one of the most poorly understood, yet practically interesting, models is scheduling in the presence of communication delays. Here, if two jobs are dependent and scheduled on different machines, then at least $c$ units of time must pass between their executions. Even for the special case where an unlimited number of identical machines are available, the best known approximation ratio for minimizing makespan is $O(c)$. An outstanding open problem in the top-10 list by Schuurman and Woeginger (and its recent update by Bansal) asks whether there exists a constant-factor approximation algorithm. In joint work with Janardhan Kulkarni, Thomas Rothvoss, Jakub Tarnawski, and Yihao Zhang, we prove a $O(\log c \cdot \log m)$-approximation algorithm for the problem of minimizing makespan on $m$ identical machines; this work is presented in Chapter \ref{chapter: S1}. Our approach is based on a Sherali-Adams lift of a linear programming relaxationand a randomized clustering of the semimetric space induced by this lift. We extend our work to the related machines setting and study the objectives of minimizing makespan and minimizing the weighted sum of completion times. Here, we also obtain polylogarithmic approximation algorithms, and these results are presented in Chapter \ref{chapter: S2}.