An Optimal Truthful Mechanism for the Online Weighted Bipartite Matching Problem

TL;DR

This paper presents an optimal truthful mechanism for online weighted bipartite matching under the secretary model, achieving the e-competitive ratio and surpassing previous mechanisms with logarithmic bounds.

Contribution

It introduces a new, optimal truthful mechanism for online weighted bipartite matching that matches the e-competitive ratio, closing the gap with non-truthful algorithms.

Findings

Achieves e-competitive ratio for the problem

Demonstrates truthfulness does not reduce competitiveness

Introduces a novel proof technique based on mechanism independence

Abstract

In the weighted bipartite matching problem, the goal is to find a maximum-weight matching in a bipartite graph with nonnegative edge weights. We consider its online version where the first vertex set is known beforehand, but vertices of the second set appear one after another. Vertices of the first set are interpreted as items, and those of the second set as bidders. On arrival, each bidder vertex reveals the weights of all adjacent edges and the algorithm has to decide which of those to add to the matching. We introduce an optimal, $e$-competitive truthful mechanism under the assumption that bidders arrive in random order (secretary model). It has been shown that the upper and lower bound of e for the original secretary problem extends to various other problems even with rich combinatorial structure, one of them being weighted bipartite matching. But truthful mechanisms so far fall short of reasonable competitive ratios once respective algorithms deviate from the original, simple threshold form. The best known mechanism for weighted bipartite matching by Krysta and V\"ocking (ICALP 2012) offers only a ratio logarithmic in the number of online vertices. We close this gap, showing that truthfulness does not impose any additional bounds. The proof technique is new in this surrounding, and based on the observation of an independency inherent to the mechanism. The insights provided hereby are interesting in their own right and appear to offer promising tools for other problems, with or without truthfulness.