Flow Time Scheduling and Prefix Beck-Fiala

TL;DR

This paper establishes a connection between discrepancy theory and scheduling problems, providing improved bounds on flow times through a novel reduction and deep geometric results, advancing theoretical guarantees in the field.

Contribution

It introduces a reduction from discrepancy bounds in the prefix Beck-Fiala setting to scheduling flow time bounds, leveraging convex geometry to improve guarantees.

Findings

Achieved $O(\sqrt{ ext{log } n})$ bounds for max flow time.

Achieved $O(\sqrt{ ext{log } n} ext{ log } P)$ bounds for total flow time.

Showed limitations of existing techniques for the 2-sparse case in discrepancy.

Abstract

We relate discrepancy theory with the classic scheduling problems of minimizing max flow time and total flow time on unrelated machines. Specifically, we give a general reduction that allows us to transfer discrepancy bounds in the prefix Beck-Fiala (bounded $\ell_1$-norm) setting to bounds on the flow time of an optimal schedule. Combining our reduction with a deep result proved by Banaszczyk via convex geometry, give guarantees of $O(\sqrt{\log n})$ and $O(\sqrt{\log n} \log P)$ for max flow time and total flow time, respectively, improving upon the previous best guarantees of $O(\log n)$ and $O(\log n \log P)$. Apart from the improved guarantees, the reduction motivates seemingly easy versions of prefix discrepancy questions: any constant bound on prefix Beck-Fiala where vectors have sparsity two (sparsity one being trivial) would already yield tight guarantees for both max flow time and total flow time. While known techniques solve this case when the entries take values in $\{-1,0,1\}$, we show that they are unlikely to transfer to the more general $2$-sparse case of bounded $\ell_1$-norm.