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.

翻译：本文研究在线次模福利（SW）问题。在此问题中，给定$n$个投标人，每个投标人具有一般（不一定是单调的）次模效用函数，以及$m$个在线到达的物品。目标是在每个物品到达时将其分配给某个投标人或丢弃，同时最大化效用之和。当对手决定物品到达顺序时，我们提出一个简单的随机化算法，实现了紧致竞争比$\nicefrac{1}{4}$。该算法是[Harshaw-Kazemi-Feldman-Karbasi MOR`22]所提出算法的特化，该团队此前为该问题给出了最佳已知竞争比$3-2\sqrt{2}\approx 0.171573$。当物品到达顺序为均匀随机时，我们提出竞争比约为$0.27493$，改进了此前$\nicefrac{1}{4}$的保证。后一结果的方法基于对[Buchbinder-Feldman-Naor-Schwartz SODA`14]的（离线）残差随机贪心（RRG）算法的更好分析，我们相信这一分析可能具有独立的研究价值。