Improved Algorithmic Results for Unsplittable Stable Allocation Problems

TL;DR

This paper introduces a polynomial-time algorithm for a variant of the unsplittable stable allocation problem, providing solutions or minimal congestion when feasible solutions do not exist, under a new stability model.

Contribution

It presents a novel polynomial-time solvable model for unsplittable stable allocations and a rounding technique for near-feasible solutions, advancing understanding of these complex problems.

Findings

Polynomial-time solvability under the new stability model

Algorithm computes minimal congestion solutions when no feasible solution exists

Rounding technique produces mildly infeasible solutions with limited overcongestion

Abstract

The stable allocation problem is a many-to-many generalization of the well-known stable marriage problem, where we seek a bipartite assignment between, say, jobs (of varying sizes) and machines (of varying capacities) that is "stable" based on a set of underlying preference lists submitted by the jobs and machines. We study a natural "unsplittable" variant of this problem, where each assigned job must be fully assigned to a single machine. Such unsplittable bipartite assignment problems generally tend to be NP-hard, including previously-proposed variants of the unsplittable stable allocation problem. Our main result is to show that under an alternative model of stability, the unsplittable stable allocation problem becomes solvable in polynomial time; although this model is less likely to admit feasible solutions than the model proposed iby McDermid and Manlove, we show that in the event there is no feasible solution, our approach computes a solution of minimal total congestion (overfilling of all machines collectively beyond their capacities). We also describe a technique for rounding the solution of a stable allocation problem to produce "relaxed" unsplit solutions that are only mildly infeasible, where each machine is overcongested by at most a single job.