[Paper Review] Online Vertex-Weighted Bipartite Matching: Beating 1-1/e with Random Arrivals

Zhiyi Huang, Zhihao Gavin Tang+2 more|arXiv (Cornell University)|Jan 1, 2018

Optimization and Search Problems26 references23 citations

TL;DR

This paper presents a novel two-dimensional gain sharing function within the randomized primal-dual framework to achieve a competitive ratio of 0.6534 for the online vertex-weighted bipartite matching problem under random arrival order. By incorporating both the offline vertex rank and online vertex arrival time into the gain allocation, the algorithm surpasses the 1−1/e ≈ 0.632 barrier, marking the first such improvement in the vertex-weighted setting using this framework.

ABSTRACT

We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1/e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time goes by, and each online vertex matches the neighbor with the highest offer at its arrival.

Motivation & Objective

• To overcome the 1−1/e competitive ratio barrier in online vertex-weighted bipartite matching under random arrival order.
• To extend the randomized primal-dual framework to handle time-dependent gain sharing in weighted matching.
• To design a generalized ranking algorithm that dynamically adjusts offers based on arrival time and offline vertex rank.
• To prove a strictly better competitive ratio than 1−1/e in the vertex-weighted case under random arrivals.

Proposed method

• Introduces a two-dimensional gain sharing function that depends on both the offline vertex rank yv and the online vertex arrival time yu.
• Uses a perturbation function h(x) = min{1, 1/(2e^x)} to define the offer value wv · (1−д(yv,yu)) from offline vertices.
• Applies the randomized primal-dual technique to charge gains from matched edges to both endpoints, ensuring expected gain is at least 0.6534·wv per matched offline vertex.
• Derives a lower bound on the expected combined gain E[αu + αv] for each edge (u,v), minimizing over threshold functions θ and β.
• Employs partial derivatives and case analysis over τ and γ to prove the lower bound holds for all parameter ranges.
• Uses a step-function assumption for threshold functions to simplify and bound the minimum expected gain.

Experimental results

Research questions

• RQ1 Can the 1−1/e competitive ratio barrier be beaten in the vertex-weighted online bipartite matching problem under random arrivals?
• RQ2 Does incorporating online vertex arrival time into the gain sharing mechanism improve competitiveness beyond the unweighted case?
• RQ3 Is it possible to achieve a competitive ratio strictly greater than 1−1/e using the randomized primal-dual framework in the vertex-weighted setting?
• RQ4 What form of two-dimensional gain sharing function maximizes the lower bound on expected combined gains?

Key findings

• The proposed algorithm achieves a competitive ratio of 0.6534, strictly exceeding the 1−1/e ≈ 0.632 barrier.
• The competitive ratio is proven via a lower bound on the expected combined gain E[αu + αv] ≥ (1−ln 2)/2 ≈ 0.6534 for every matched edge (u,v).
• The analysis shows that the expected gain is minimized when threshold functions θ and β are chosen as step functions, enabling tight lower bound derivation.
• The gain sharing function is defined as д(x,y) = 1/2(h(x+1)−h(y)), with h(x) = min{1, 1/(2e^x)}, which enables the improved bound.
• The result is the first to achieve a competitive ratio strictly above 1−1/e in the vertex-weighted online bipartite matching problem under the randomized primal-dual framework.
• The framework successfully integrates arrival time and rank into a unified gain sharing mechanism, enabling improved competitiveness.