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.
On leave from the Centre Universitaire d'Informatique of the University of Geneve. Partially supported by the Swiss National Science Foundation, grant 21-43309.95.
Part of this research has been done while this author was visiting the Centre Universitaire d'Informatique of the University of Geneve partially supported by the HCM project SCOOP of the European Union.
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Clementi, A.E.F., Trevisan, L. (1996). Improved non-approximability results for vertex cover with density constraints. In: Cai, JY., Wong, C.K. (eds) Computing and Combinatorics. COCOON 1996. Lecture Notes in Computer Science, vol 1090. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61332-3_167
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DOI: https://doi.org/10.1007/3-540-61332-3_167
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