Importance: Medium ✭✭
Author(s): Atserias, Albert
Recomm. for undergrads: no
Posted by: dberwanger
on: October 2nd, 2012
Conjecture   For every $ \varepsilon > 0 $ there is $ \delta > 0 $ such that, for every large $ n $, there are $ n $-vertex graphs $ G $ and $ H $ such that $ G \equiv_{\delta n}^{\mathrm{C}} H $ and $ \mathrm{vc}(G) \ge (2 - \varepsilon) \cdot \mathrm{vc}(H) $.

Here $ \equiv^{\mathrm{C}}_{k} $ denotes indistinguishability in $ k $-variable first-order logic with counting quantifiers, and $ \mathrm{vc}(G) $ denotes the cardinality of the minimum vertex-cover of $ G $. By~[1], $ G \equiv_{3}^{\mathrm{C}} H $ implies $ \mathrm{vc}(G) \leq 2 \cdot \mathrm{vc}(H) $. Also by~[1] a positive answer would imply that an integrality gap of $ 2-\varepsilon $ resists $ \delta n $ levels of Sherali-Adams linear programming relaxations of vertex-cover, on $ n $-vertex graphs. It is known that such a gap resists $ n^{\delta} $ levels~[2]. What we ask would let us replace $ n^{\delta} $ by $ \delta n $. If improving over $ n^{\delta} $ were not possible, then we could approximate vertex-cover by a factor better than~$ 2 $ in subexponential time (i.e. $ 2^{n^{o(1)}} $). 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]


[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.}

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