Linear Optimization over a Polymatroid with Side Constraints -- Scheduling Queues and Minimizing Submodular Functions

TL;DR

This paper introduces a grouping algorithm that efficiently solves queueing scheduling problems with side constraints and applies it to minimize submodular functions, revealing a deep connection between these problems via polymatroids.

Contribution

The paper develops a novel grouping algorithm for scheduling with side constraints and demonstrates its application to submodular function minimization, achieving polynomial computational complexity.

Findings

The algorithm solves queueing scheduling with side constraints in O(n^3LP(n)) time.

It minimizes submodular functions with an overall effort of O(n^4LP(n)).

The approach links queueing theory and submodular optimization through polymatroids.

Abstract

Two seemingly unrelated problems, scheduling a multiclass queueing system and minimizing a submodular function, share a rather deep connection via the polymatroid that is characterized by a submodular set function on the one hand and represents the performance polytope of the queueing system on the other hand. We first develop what we call a {\it grouping} algorithm that solves the queueing scheduling problem under side constraints, with a computational effort of $O(n^3LP(n))$, $n$ being the number of job classes, and LP(n) being the computational efforts of solving a linear program with no more than $n$ variables and $n$ constraints. The algorithm organizes the job classes into groups, and identifies the optimal policy to be a priority rule across the groups and a randomized rule within each group (to enforce the side constraints). We then apply the grouping algorithm to the submodular function minimization, mapping the latter to a queueing scheduling problem with side constraints. %Each time the algorithm is applied, it identifies a subset; and We show the minimizing subset can be identified by applying the grouping algorithm $n$ times. Hence, this results in a algorithm that minimizes a submodular function with an effort of $O(n^4LP(n))$.