The Online Best Reply Algorithm for Resource Allocation Problems

TL;DR

This paper analyzes the performance of a greedy online resource allocation algorithm with diseconomy of scale, providing new upper bounds on its competitive ratio for polynomial cost functions of arbitrary degree.

Contribution

It derives general upper bounds on the competitive ratio of the greedy algorithm for unweighted and weighted resource allocation problems with polynomial costs of any degree.

Findings

Upper bound for unweighted case: $d(d / W((1.2d-1)/(d+1)))^{d+1}$

Upper bound for weighted case: $(d/W(d/(d+1)))^{d+1}$

Extends known bounds to arbitrary polynomial degrees

Abstract

We study the performance of a best reply algorithm for online resource allocation problems with a diseconomy of scale. In an online resource allocation problem, we are given a set of resources and a set of requests that arrive in an online manner. Each request consists of a set of feasible allocations and an allocation is a set of resources. The total cost of an allocation vector is given by the sum of the resources' costs, where each resource's cost depends on the total load on the resource under the allocation vector. We analyze the natural online procedure where each request is allocated greedily to a feasible set of resources that minimizes the individual cost of that particular request. In the literature, this algorithm is also known as a one-round walk-in congestion games starting from the empty state. For unweighted resource allocation problems with polynomial cost functions with maximum degree $d$, upper bounds on the competitive ratio of this greedy algorithm were known only for the special cases $d\in\{1, 2, 3\}$. In this paper, we show a general upper bound on the competitive ratio of $d(d / W(\frac{1.2d-1}{d+1}))^{d+1}$ for the unweighted case where $W$ denotes the Lambert-W function on $\mathbb{R}_{\geq 0}$. For the weighted case, we show that the competitive ratio of the greedy algorithm is bounded from above by $(d/W(\frac{d}{d+1}))^{d+1}$.