# Vertex Cover Integrality Gap

**Conjecture**For every there is such that, for every large , there are -vertex graphs and such that and .

Here denotes indistinguishability in -variable first-order logic with counting quantifiers, and denotes the cardinality of the minimum vertex-cover of . By~[1], implies . Also by~[1] a positive answer would imply that an integrality gap of resists levels of Sherali-Adams linear programming relaxations of vertex-cover, on -vertex graphs. It is known that such a gap resists levels~[2]. What we ask would let us replace by . If improving over were not possible, then we could approximate vertex-cover by a factor better than~ in subexponential time (i.e. ). Approximating vertex-cover by a factor better than~1.36 is NP-hard~[3], and approximating vertex-cover by factor better than~2 is UG-hard~[4], where UG stands for Unique Games (from the Unique Games Conjecture); but note that UG-hardness does not rule out subexponential-time algorithms because UG itself is solvable in subexponential time~[5]

## Bibliography

[1] A. Atserias and E. Maneva. *Sherali-Adams Relaxations and Indistinguishability in Counting Logics*, in Proc. 3rd ACM ITCS, pp. 367-379, 2012.

[2] M. Charikar, K. Makarychev and Y. Makarychev. *Integrality Gaps for Sherali-Adams Relaxations*, in Proc. 41st ACM STOC, pp. 283-292, 2009.

[3] I. Dinur and S. Safra. *On the Hardness of Approximating Minimum Vertex-Cover*, Annals of Mathematics, 162(1):439-485, 2005.

[4] S. Khot and O. Regev. *Vertex cover might be hard to approximate to within 2-epsilon*, J. Comput. Syst. Sci. 74(3):335-349, 2008.

[5] S. Arora, B. Barak, and D. Steurer. *Subexponential Algorithms for Unique Games and Related problems*, in Proc. 51th IEEE FOCS, pp. 563-572, 2010.}

* indicates original appearance(s) of problem.