Improved Non-Approximability Results for Minimum Vertex Cover with Density Constraints

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 suuciently large B, the recent 16/15 lower bound proved by Bellare et al. 5] 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-oo 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 for several Max SNP problems restricted to \dense" instances introduced by Arora et al. 2] cannot be applied.