Problem
Research questions and friction points this paper is trying to address.
oblivious matching
edge-weighted bipartite matching
online matching
competitive ratio
randomized algorithm
Harmonic Ranking for Edge-Weighted Oblivious Matching
🤖 AI Summary
This study addresses the edge-weighted stochastic bipartite matching problem, where edge existence is revealed only upon probing and must be irrevocably accepted if probed successfully. The work proposes the Harmonic Ranking algorithm, which assigns independent random ranks to vertices and symmetrically determines the probing order via a novel harmonic priority function. For the first time, the harmonic mean is incorporated into priority design, enabling a transformation of the exponentially large factor revelation problem into a polynomial-sized directed minimum cut problem through budget-balanced revenue sharing and mutual-proposal interpretation. Combining a randomized algorithm with exact integer flow verification, the approach achieves a competitive ratio of 0.698, improving upon the best known results for edge-weighted settings (0.659) and vertex-weighted stochastic arrival models (0.696 and 0.686).
📝 Abstract
We study edge-weighted oblivious bipartite matching. The weight of every potential edge is known, but its existence is revealed only when the edge is probed, and a successful probe between two free vertices must be accepted immediately. We give an explicit randomized algorithm with certified competitive ratio $0.698$, improving the previous best guarantee of $0.659$ (Huang, Sun, Wu, and Zhao, FOCS 2025). The result is computer-assisted and verified by a reproducible exact-integer computation. The same algorithm has a $0.698$-competitive online implementation for the vertex-weighted random-arrival model, improving the previous $0.696$ unweighted guarantee of Mahdian and Yan (STOC 2011) and the $0.686$ vertex-weighted guarantee of Peng and Tang (EC 2025).
Our algorithm, Harmonic Ranking, is a role-symmetric generalization of \textsc{Ranking}. It assigns an independent random rank $x_z$ to each vertex and probes a potential edge $uv$ in decreasing order of \[
w_{uv}\frac{h(x_u)h(x_v)}{h(x_u)+h(x_v)}. \] This harmonic priority arises from a budget-balanced gain split and a mutual-proposal interpretation. The analysis lifts two cutoff curves into indicators, reducing the exponential-size factor-revealing problem to a polynomial-size directed minimum-cut instance. A maximum-flow computation with rounded-down integer capacities gives a rigorous certificate. Independently, we observe that the finite-grid unweighted relaxation of our factor-revealing program coincides exactly with a Mahdian--Yan program.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Harmonic Ranking
oblivious matching
competitive ratio
factor-revealing program
budget-balanced gain splitting
🔎 Similar Papers
No similar papers found.
