Improved Approximation for Ranking on General Graphs

2025-11-09

AI Summary

This paper investigates the approximation ratio of the Ranking algorithm for online matching on general graphs, aiming to break its long-standing theoretical lower-bound stagnation. Addressing the bottleneck where the best-known lower bound (0.526, Chan et al.) and upper bound remain unseparated, we introduce the novel concept of “vertex backup” to characterize the rank distribution structure among matched vertices, thereby circumventing limitations imposed by adverse events in prior analyses. Integrating the primal-dual framework, random permutation modeling, and refined probabilistic analysis, we optimize the derivation of the approximation ratio. Our result improves the lower bound for Ranking on general graphs to 0.5469—surpassing, for the first time, the best-known approximation ratio (0.531) achievable by any non-adaptive algorithm. This advancement significantly deepens the theoretical understanding of Ranking in non-bipartite settings.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Search and Optimization: Algorithm Configuration
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Personalized content ranking and presentation

Abstract

In this paper, we study Ranking, a well-known randomized greedy matching algorithm, for general graphs. The algorithm was originally introduced by Karp, Vazirani, and Vazirani [STOC 1990] for the online bipartite matching problem with one-sided vertex arrivals, where it achieves a tight approximation ratio of 1 - 1/e. It was later extended to general graphs by Goel and Tripathi [FOCS 2012]. The Ranking algorithm for general graphs is as follows: a permutation $sigma$ over the vertices is chosen uniformly at random. The vertices are then processed sequentially according to this order, with each vertex being matched to the first available neighbor (if any) according to the same permutation $sigma$. While the algorithm is quite well-understood for bipartite graphs-with the approximation ratio lying between 0.696 and 0.727, its approximation ratio for general graphs remains less well characterized despite extensive efforts. Prior to this work, the best known lower bound for general graphs was 0.526 by Chan et al. [TALG 2018], improving on the approximation ratio of 0.523 by Chan et al. [SICOMP 2018]. The upper bound, however, remains the same as that for bipartite graphs. In this work, we improve the approximation ratio of extsc{Ranking} for general graphs to 0.5469, up from 0.526. This also surpasses the best-known approximation ratio of $0.531$ by Tang et al. [JACM 2023] for the oblivious matching problem. Our approach builds on the standard primal-dual analysis. The novelty of our work lies in proving new structural properties of Ranking by introducing the notion of the backup for vertices matched by the algorithm. For a fixed permutation, a vertex's backup is its potential match if its current match is removed. This concept helps characterize the rank distribution of the match of each vertex, enabling us to eliminate certain bad events that constrained previous work.

Problem

• Improving approximation ratio of Ranking algorithm for general graphs
• Characterizing matching performance beyond bipartite graph limitations
• Establishing better lower bounds using novel backup vertex analysis

Innovation

• Introduces backup vertices concept for potential matches
• Uses primal-dual analysis to prove structural properties
• Improves ranking algorithm approximation ratio to 0.5469