A (1.999999)-approximation ratio for vertex cover problem

TL;DR

This work tackles the vertex cover problem by introducing a semidefinite programming formulation that enables a 1.999999-approximation on arbitrary graphs, advancing beyond the longstanding 2-approximation barrier. It develops a framework rooted in LP relaxations, vector-based SDP relaxations, and a graph augmentation technique to either exploit a bipartite substructure or derive strong lower bounds, yielding the near-optimal ratio. The proposed Mahdis Algorithm orchestrates LP extreme solutions, SDP (4) on an augmented graph $G2$, and a bipartite completion on a large induced subgraph $G_\varepsilon$, to guarantee the claimed approximation bound. The claimed result has implications for the Unique Games Conjecture, arguing against its hardness for achieving constants better than 2 in vertex cover approximations.

Abstract

The vertex cover problem is a famous combinatorial problem, and its complexity has been heavily studied. While a 2-approximation can be trivially obtained for it, researchers have not been able to approximate it better than 2-\textit{o}(1). In this paper, by introducing a new semidefinite programming formulation that satisfies new properties, we introduce an approximation algorithm for the vertex cover problem with a performance ratio of 1.999999 on arbitrary graphs, en route to answering an open question about the correctness of the unique games conjecture.