Improved non-approximability results for vertex cover with density constraints

Abstract

We provide new non-approximability results for the restrictions of the Min Vertex Cover problem to bounded-degree, sparse and dense graphs. We show that, for a sufficiently large B, the recent 16/15 lower bound proved by Bellare et al. [3] extends with negligible loss to graphs with bounded degree B. Then, we consider sparse graphs with no dense components (i.e. everywhere sparse graphs), and we show a similar result but with a better trade-off between non-approximability and sparsity. Finally we observe that the Min Vertex Cover problem remains APX-complete when restricted to dense graph and thus recent techniques developed by Arora et al. [1] for several Max SNP problems restricted to “dense” instances cannot be applied.