Inapproximability of Vertex Cover and Independent Set in Bounded Degree Graphs

We study the inapproximability of Vertex Cover and Independent Set on degree d graphs. We prove that: • Vertex Cover is Unique Games-hard to approximate to within a factor 2−(2+od(1))log log d log d . This exactly matches the algorithmic result of Halperin [7] up to the od(1) term. • Independent Set is Unique Games-hard to approximate to within a factor O( d log2 d ). This improves the d logO(1)(d) Unique Games hardness result of Samorodnitsky and Trevisan [15]. Additionally, our result does not rely on the construction of a query efficient PCP as in [15]. ∗ Work done while at KTH – Royal Institute of Technology in Stockholm, funded by ERC Advanced investigator grant 226203 and a grant from the Mittag-Leffler Institute. † Research funded by NSF CAREER and NSF Expedition grants. ‡ Research partially supported by a bi-national US-Israeli BSF grant, and by an ISF grant. 1 

• Type: Conference
• Year: 2009
• Where: COCO
• Authors: Per Austrin, Subhash Khot, Muli Safra
• Keywords: COCO 2009, Independent Set, Theoretical Computer Science, Unique Games-hard, Vertex Cover