Prophet Upper Bounds for Online Matching and Auctions

TL;DR

This paper establishes new upper bounds on the competitiveness of algorithms for online 2-bounded auctions and prophet matching problems, improving understanding of their theoretical limits under various arrival models.

Contribution

It provides the first known upper bounds for online 2-bounded auctions and prophet matching, including adversarial, random, IID, and prophet-secretary models.

Findings

Adversarial model bound: no algorithm exceeds 4/11 competitiveness.

Prophet matching bound: no algorithm exceeds approximately 0.4189 competitiveness.

Improved bounds for random order, IID, and prophet-secretary models using continuous-time analysis.

Abstract

In the online 2-bounded auction problem, we have a collection of items represented as nodes in a graph and bundles of size two represented by edges. Agents are presented sequentially, each with a random weight function over the bundles. The goal of the decision-maker is to find an allocation of bundles to agents of maximum weight so that every item is assigned at most once, i.e., the solution is a matching in the graph. When the agents are single-minded (i.e., put all the weight in a single bundle), we recover the maximum weight prophet matching problem under edge arrivals (a.k.a. prophet matching). In this work, we provide new and improved upper bounds on the competitiveness achievable by an algorithm for the general online 2-bounded auction and the (single-minded) prophet matching problems. For adversarial arrival order of the agents, we show that no algorithm for the online 2-bounded auction problem achieves a competitiveness larger than $4/11$, while no algorithm for prophet matching achieves a competitiveness larger than $\approx 0.4189$. Using a continuous-time analysis, we also improve the known bounds for online 2-bounded auctions for random order arrivals to $\approx 0.5968$ in the general case, a bound of $\approx 0.6867$ in the IID model, and $\approx 0.6714$ in prophet-secretary model.