SDP gaps for 2-to-1 and other label-cover variants

Authors:
Venkatesan Guruswami;Subhash Khot;Ryan O'Donnell;Preyas Popat;Madhur Tulsiani;Yi Wu
Affiliations:
Computer Science Department, Carnegie Mellon University;Computer Science Department, New York University;Computer Science Department, Carnegie Mellon University;Computer Science Department, New York University;School of Mathematics, Institute for Advanced Study;Computer Science Department, Carnegie Mellon University
Venue:
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Year:
2010

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Abstract

In this paper we present semidefinite programming (SDP) gap instances for the following variants of the Label-Cover problem, closely related to the Unique Games Conjecture: (i) 2-to-1 Label-Cover; (ii) 2-to-2 Label-Cover; (iii) α-constraint Label-Cover. All of our gap instances have perfect SDP solutions. For alphabet size K, the integral optimal solutions have value: (i) O(1/√logK); (ii) O(1/ logK); (iii) O(1/√logK). Prior to this work, there were no known SDP gap instances for any of these problems with perfect SDP value and integral optimum tending to 0.