On the approximability of the vertex cover and related problems

In this paper we show that the problem of identifying an edge $(i,j)$ in a graph $G$ such that there exists an optimal vertex cover $S$ of $G$ containing exactly one of the nodes $i$ and $j$ is NP-hard. Such an edge is called a weak edge. We then develop a polynomial time approximation algorithm for the vertex cover problem with performance guarantee $2-\frac{1}{1+\sigma}$, where $\sigma$ is an upper bound on a measure related to a weak edge of a graph. Further, we discuss a new relaxation of the vertex cover problem which is used in our approximation algorithm to obtain smaller values of $\sigma$. We also obtain linear programming representations of the vertex cover problem for special graphs. Our results provide new insights into the approximability of the vertex cover problem - a long standing open problem.

💡 Research Summary

The paper investigates a previously unexplored structural element of the Vertex Cover problem – a “weak edge”. An edge (i, j) in a graph G is defined as weak if there exists at least one optimal vertex cover S that contains exactly one of the two endpoints. The authors first prove that deciding whether a weak edge exists in a given graph is NP‑hard. The reduction is from 3‑SAT: each variable and clause is encoded as vertices and edges such that a weak edge corresponds to a satisfying assignment. Consequently, any algorithm that could reliably locate a weak edge would solve an NP‑complete problem, establishing the intrinsic difficulty of the task.

Building on this hardness result, the authors design a new approximation algorithm for Vertex Cover whose performance guarantee depends on a parameter σ that measures the “weakness” of the edges selected during the algorithm’s execution. In each iteration the algorithm picks an edge that is (or is treated as) weak, adds one of its endpoints to the cover, and removes all incident edges. For the chosen edge (i, j) a non‑negative value σ(i, j) is defined; it quantifies the extra cost incurred relative to the optimal solution when the algorithm decides which endpoint to include. Let σmax be the maximum σ(i, j) observed over all iterations. The algorithm then achieves an approximation ratio of

  2 − 1⁄(1 + σmax).

If σmax = 0 the bound collapses to the classic 2‑approximation, while any σmax > 0 yields a strictly better guarantee.

A central technical contribution is a novel linear‑programming relaxation that drives σ down. The standard LP for Vertex Cover uses variables xv∈