A Tight Competitive Ratio for Online Submodular Welfare Maximization

TL;DR

This paper introduces a new randomized algorithm for online submodular welfare maximization that achieves a tight competitive ratio of 1/4 against an adversarial order and improves to approximately 0.275 under random item arrivals, surpassing previous bounds.

Contribution

It presents a simplified algorithm with a tight competitive ratio for adversarial order and an improved ratio for random order, based on refined analysis of existing algorithms.

Findings

Achieves a tight 1/4 competitive ratio against adversarial item order.

Improves the competitive ratio to approximately 0.275 for random item order.

Provides a new analysis of the Residual Random Greedy algorithm that may be of independent interest.

Abstract

In this paper we consider the online Submodular Welfare (SW) problem. In this problem we are given $n$ bidders each equipped with a general (not necessarily monotone) submodular utility and $m$ items that arrive online. The goal is to assign each item, once it arrives, to a bidder or discard it, while maximizing the sum of utilities. When an adversary determines the items' arrival order we present a simple randomized algorithm that achieves a tight competitive ratio of $\nicefrac{1}{4}$. The algorithm is a specialization of an algorithm due to [Harshaw-Kazemi-Feldman-Karbasi MOR`22], who presented the previously best known competitive ratio of $3-2\sqrt{2}\approx 0.171573 $ to the problem. When the items' arrival order is uniformly random, we present a competitive ratio of $\approx 0.27493$, improving the previously known $\nicefrac{1}{4}$ guarantee. Our approach for the latter result is based on a better analysis of the (offline) Residual Random Greedy (RRG) algorithm of [Buchbinder-Feldman-Naor-Schwartz SODA`14], which we believe might be of independent interest.