In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the Goemans-Williamson algorithm [25]. This implies that if the Unique Games

We show that the MULTICUT, SPARSEST-CUT, and MIN-2CNF ≡ DELETION problems are NP-hard to approximate within every constant factor, assuming the Unique Games Conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of Ω(log log n). 1.

Unique games are constraint satisfaction problems that can be viewed as a generalization of Max-Cut to a larger domain size. The Unique Games Conjecture states that it is hard to distinguish between instances of unique games where almost all constraints are satisfiable and those where almost none are satisfiable. It has been shown to imply a number of inapproximability results for fundamental problems that seem difficult to obtain by more standard complexity assumptions. Thus, proving or refuting this conjecture is an important goal. We present significantly improved approximation algorithms for unique games. For instances with domain size k where the optimal solution satisfies 1 − ε fraction of all constraints, our algorithms satisfy roughly k −ε/(2−ε) and 1 − O ( √ ε log k) fraction of all constraints. Our algorithms are based on rounding a natural semidefinite programming relaxation for the problem and their performance almost matches the integrality gap of this relaxation. Our results are near optimal if the Unique Games Conjecture is true, i.e. any improvement (beyond low order terms) would refute the conjecture. 1

We present a polynomial time algorithm based on semidefinite programming that, given a unique game of value 1 − O(1 / log n), satisfies a constant fraction of constraints, where n is the number of variables. For sufficiently large alphabets, it improves an algorithm of Khot (STOC’02) that satisfies a constant fraction of constraints in unique games of value 1 − c/(k 10 (log k) 5), where k is the size of the alphabet. We also present a simpler algorithm for the special case of unique games with linear constraints, and a combinatorial algorithm for the general case. Finally, we present a simple approximation algorithm for 2-to-1 games. 1

The Unique Games problem is the following: we are given a graph G = (V, E), with each edge e = (u, v) having a weight we and a permutation πuv on [k]. The objective is to find a labeling of each vertex u with a label fu ∈ [k] to minimize the weight of unsatisfied edges—where an edge (u, v) is satisfied if fv = πuv(fu). The Unique Games Conjecture of Khot [8] essentially says that for each ε> 0, there is a k such that it is NP-hard to distinguish instances of Unique games with (1−ε) satisfiable edges from those with only ε satisfiable edges. Several hardness results have recently been proved based on this assumption, including optimal ones for Max-Cut, Vertex-Cover and other problems, making it an important challenge to prove or refute the conjecture. In this paper, we give an O(log n)-approximation algorithm for the problem of minimizing the number of unsatisfied edges in any Unique game. Previous results of Khot [8] and Trevisan [12] imply that if the optimal solution has OPT = εm unsatisfied edges, semidefinite relaxations of the problem could give labelings with min{k2ε1/5, (ε log n) 1/2}m unsatisfied edges. In this paper we show how to round a LP relaxation to get an O(log n)-approximation to the problem; i.e., to find a labeling with only O(εm log n) = O(OPT log n) unsatisfied edges. 1

We use semidefinite programming to prove that any constraint satisfaction problem in two variables over any domain allows an efficient approximation algorithm that does better than picking a random assignment. Specifically we consider the case when each variable can take values in [d] and that each constraint rejects t out of the d2 possible input pairs. Then, for some universal constant c, wecan,in probabilistic polynomial time, find an assignment whose objective value is, in expectation, within a factor 1 − t d2 + ct d4 log d of optimal, improving on the trivial bound of 1 − t/d².

Max-Satisfy is the problem of finding an assignment that satisfies the maximum number of equations in a system of linear equations over Q. We prove that unless NP⊂BPP Max-Satisfy cannot be efficiently approximated within an approximation ratio of 1/n 1−ɛ, if we consider systems of n linear equations with at most n variables and ɛ> 0 is an arbitrarily small constant. Previously, it was known that the problem is NP-hard to approximate within a ratio of 1/n α, but 0 < α < 1 was some specific constant that could not be taken to be arbitrarily close to 1.

In this report, we study the Unique Games conjecture of Khot [32] and its implications on the hardness of approximating some important optimization problems. The conjecture states that it is NP-hard to determine whether the value of a unique 1-round game between two provers and a verifier is close to 1 or negligible. It gives rise to PCP systems where the verifier needs to query only 2 bits from the provers (in contrast, Håstad’s verifier queries 3 bits [44]). We start by investigating the conjecture through the lens of H˚astad’s 3-bit PCP. We then discuss in detail two results that are consequences of the conjecture. The first states that Min-2SAT-Deletion is NP-hard to approximate within any constant factor [32]. The second result shows that minimum vertex cover is NP-hard to approximate within a factor of 2 − ɛ for every ɛ> 0 [34]. We display the use of Fourier techniques for analyzing the soundness of the PCP used to prove the first result, and we display the use of techniques from extremal combinatorics for analyzing the soundness of the PCP used to prove the second result. Finally, we present Khot’s algorithm which shows that for the conjecture to be true, the domain of answers of the two provers must be large, and we survey some recent results examining the