A Tight Bound on Online Vertex Cover under Edge Arrivals

2026-08-05

AI Summary

This work addresses the online vertex cover problem under the edge-arrival model, focusing on establishing tight lower bounds on its competitive ratio. Specifically, it resolves the long-standing open question of whether any algorithm can achieve a competitive ratio strictly better than 2. Building upon the blueprint framework introduced by Assadi et al., the authors employ a direct reduction to establish, for the first time, a tight lower bound of 2 against oblivious adversaries on bipartite graphs. This result demonstrates that no randomized integral or fractional algorithm can attain a competitive ratio strictly below 2, thereby proving that the classic greedy algorithm—which selects both endpoints of each uncovered edge—achieves the optimal competitive ratio of 2 in this model.

Abstract

We prove a tight impossibility result for online vertex cover under edge arrivals. No randomized integral or fractional algorithm achieves a competitive ratio strictly below $2$ against an oblivious adversary, even on bipartite graphs. Since the standard algorithm that takes both endpoints of every uncovered edge is $2$-competitive, this settles the optimal ratio. Our proof is a direct reduction from the recent breakthrough blueprint framework of Assadi, Jiang, and Xiang.

Problem

• online vertex cover
• edge arrivals
• competitive ratio
• oblivious adversary
• bipartite graphs

Innovation

• online vertex cover
• competitive ratio
• edge arrivals
• impossibility result
• oblivious adversary