Approximation Algorithms for Scheduling and Fair Allocations

Abstract

In this thesis, we will have discussions on two main topics, max-min allocation and schedulingjobs with precedent constraints on machines with communication delays. New approximation algorithms are given in Chapter 2, 4 and 5, where linear programming plays a fairly important role on algorithm designs, while Chapter 3 contains partial results on the general max-min allocation. The Santa Claus problem is also known as the restricted max-min fair allocation. In this problem, Santa Claus has a set of gifts, and he wants to distribute them among a set of children so that the least happy child is made as happy as possible. Here, the value that a child i has for a present j is of the form pij ∈ {0, pj}. Based on a modification of Haxell’s hypergraph matching argument, a polynomial time algorithm by Annamalai et al. gives a 12.33-approximation. In joint work with Sami Davies and Thomas Rothvoss, a matroid version of the Santa Claus problem is introduced. The algorithm is based on Haxell’s augmenting tree, but with the introduction of the matroid structure. Our result can then be used as a blackbox to obtain a (4 + ε)-approximation for Santa Claus, comparing against a natural, compact LP. A recent work of Cheng and Mao [CM19] also gets the factor (4 + ε). On the second half, we first consider the classic problem of scheduling jobs with precedence constraints on identical machines to minimize makespan, in the presence of communication delays. In this setting, denoted by P | prec, c | Cmax, if two dependent jobs are scheduled on different machines, then at least c units of time must pass between their executions. Despite its relevance to many applications, the best known approximation ratio was O(c), whereas Graham’s greedy list scheduling algorithm already gives a (c + 1)-approximation in that setting. 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 Sami Davies, Janardhan Kulkarni, Thomas Rothvoss and Jakub Tarnawski, we give a polynomial-time O(log c ·logm)-approximation algorithm for this problem, where m is the number of machines and c is the communication delay. Our approach is based on a Sherali-Adams lift of a linear programming relaxation and a randomized clustering of the semimetric space induced by this lift. Finally, a more general version of this problem is considered in Chapter 5, to minimize the weighted sum of completion times on related machines, denoted by Q | prec, c | wjCj . Our main result is an O(log4 n)-approximation algorithm for the problem. As a byproduct of our result, we also obtain an O(log3 n)-approximation algorithm for the problem of minimizing makespan Q | prec, c | Cmax, which improves upon the O(log5 n/ log log n)-approximation algorithm due to a recent work of Maiti et al. [MRS+20].