A Constant-Factor Approximation for Generalized Malleable Scheduling under $M^\natural$-Concave Processing Speeds

TL;DR

This paper presents a constant-factor approximation algorithm for generalized malleable scheduling with $M^ atural$-concave processing speeds, extending previous results and connecting it to fair resource allocation problems.

Contribution

It introduces a novel approximation approach for malleable scheduling with $M^ atural$-concave speeds and links it to fair item allocation, broadening the scope of applicable functions.

Findings

Achieves a constant-factor approximation for the scheduling problem.

Establishes a reduction connecting malleable scheduling to fair resource allocation.

Provides resource-augmented algorithms for the related allocation problem.

Abstract

In generalized malleable scheduling, jobs can be allocated and processed simultaneously on multiple machines so as to reduce the overall makespan of the schedule. The required processing time for each job is determined by the joint processing speed of the allocated machines. We study the case that processing speeds are job-dependent $M^\natural$-concave functions and provide a constant-factor approximation for this setting, significantly expanding the realm of functions for which such an approximation is possible. Further, we explore the connection between malleable scheduling and the problem of fairly allocating items to a set of agents with distinct utility functions, devising a black-box reduction that allows to obtain resource-augmented approximation algorithms for the latter.