In this paper we disprove the following conjecture due to Goemans [16] and Linial [24] (also see [5, 26]): “Every negative type metric embeds into ℓ1 with constant distortion.” We show that for every δ>0, and for large enough n, there is an n-point negative type metric which requires distortion at-least (log log n) 1/6−δ to embed into ℓ1. Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot [19], establishing a previously unsuspected connection between PCPs and the theory of metric embeddings. We first prove that the UGC implies super-constant hardness results for (non-uniform) SPARSEST CUT and MINIMUM UNCUT problems. It is already known that the UGC also implies an optimal hardness result for MAXIMUM CUT [20]. Though these hardness results depend on the UGC, the integrality gap instances rely “only ” on the PCP reductions for the respective problems. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of UNIQUE GAMES. Then, we “simulate ” the PCP reduction and “translate ” the integrality gap instance of UNIQUE GAMES to integrality gap instances for the respective cut problems! This enables us to prove a (log log n) 1/6−δ integrality gap for (non-uniform) SPARSEST CUT and MIN-IMUM UNCUT, and an optimal integrality gap for MAX-IMUM CUT. All our SDP solutions satisfy the so-called “triangle inequality ” constraints. This also shows, for the first time, that the triangle inequality constraints do not add any power to the Goemans-Williamson’s SDP relaxation of MAXIMUM CUT. The integrality gap for SPARSEST CUT immediately implies a lower bound for embedding negative type metrics into ℓ1. It also disproves the non-uniform version of Arora, Rao and Vazirani’s Conjecture [5], asserting that the integrality gap of the SPARSEST CUT SDP, with the triangle inequality constraints, is bounded from above by a constant.

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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.

Semidefinite Programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of Semidefinite Programming. Making this connection precise, we show the following result: If UGC is true, then for every constraint satisfaction problem(CSP) the best approximation ratio is given by a certain simple SDP. Specifically, we show a generic conversion from SDP integrality gaps to UGC hardness results for every CSP. This result holds both for maximization and minimization problems over arbitrary finite domains. Using this connection between integrality gaps and hardness results we obtain a generic polynomial-time algorithm for all CSPs. Assuming the Unique Games Conjecture, this algorithm achieves the optimal approximation ratio for every CSP. Unconditionally, for all 2-CSPs the algorithm achieves an approximation ratio equal to the integrality gap of a natural SDP used in literature. Further the algorithm achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut

Submodular maximization generalizes many important problems including Max Cut in directed/undirected graphs and hypergraphs, certain constraint satisfaction problems and maximum facility location problems. Unlike the problem of minimizing submodular functions, the problem of maximizing submodular functions is NP-hard. In this paper, we design the first constant-factor approximation algorithms for maximizing nonnegative submodular functions. In particular, we give a deterministic local search 1 2-approximation and a randomized-approximation algo-

Gowers [Gow98, Gow01] introduced, for d ≥ 1, the notion of dimension-d uniformity U d (f) of a function f: G → C, where G is a finite abelian group. Roughly speaking, if a function has small Gowers uniformity of dimension d, then it “looks random ” on certain structured subsets of the inputs. We prove the following inverse theorem. Write G = G1 × · · · × Gn as a product of groups. If a bounded balanced function f: G1 × · · · Gn → C is such that U d (f) ≥ ε, then one of the coordinates of f has influence at least ε/2 O(d). Other inverse theorems are known [Gow98, Gow01, GT05, Sam05], and U 3 is especially well understood, but the properties of functions f with large U d (f), d ≥ 4, are not yet well characterized. The dimension-d Gowers inner product 〈{fS} 〉 U d of a collection {fS} S⊆[d] of functions is a related measure of pseudorandomness. The definition is such that if all the functions fS are equal to the same fixed function f, then 〈{fS} 〉 U d = U d (f). We prove that if fS: G1 × · · · × Gn → C is a collection of bounded functions such that |〈{fS} 〉 U d | ≥ ε and at least one of the fS is balanced, then there is a variable that has influence at least ε 2 /2 O(d) for at least four functions in the collection. Finally, we relate the acceptance probability of the “hypergraph long-code test ” proposed by Samorodnitsky and Trevisan to the Gowers inner product of the functions being tested and we deduce the following result: if the Unique Games Conjecture is true, then for every q ≥ 3 there is a PCP characterization of NP where the verifier makes q queries, has almost perfect completeness, and soundness at most 2q/2 q. For infinitely many q, the soundness is (q + 1)/2 q, which might be a tight result. Two applications of this results are that, assuming that the unique games conjecture is true, it is hard to approximate Max kCSP within a factor 2k/2 k ((k + 1)/2 k for infinitely many k), and it is hard to approximate Independent Set in graphs of degree D within a factor (log D) O(1) /D. 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

In this paper we derive tight bounds on the expected value of products of low influence functions defined on correlated probability spaces. The proofs are based on extending Fourier theory to an arbitrary number of correlated probability spaces, on a generalization of an invariance principle recently obtained with O’Donnell and Oleszkiewicz for multilinear polynomials with low influences and bounded degree and on properties of multi-dimensional Gaussian distributions. We present two applications of the new bounds to the theory of social choice. We show that Majority is asymptotically the most predictable function among all low influence functions given a random sample of the voters. Moreover, we derive an almost tight bound in the context of Condorcet aggregation and low influence voting schemes on a large number of candidates. In particular, we show that for every low influence aggregation function, the probability that Condorcet voting on k candidates will result in a unique candidate that is preferable to all others is k−1+o(1). This matches the asymptotic behavior of the majority function for which the probability is k−1−o(1). A number of applications in hardness of approximation in theoretical computer science were

We study the APPROXIMATE-COLORING(q, Q) problem: Given a graph G, decide whether χ(G) ≤ q or χ(G) ≥ Q (where χ(G) is the chromatic number of G). We derive conditional hardness for this problem for any constant 3 ≤ q < Q. For q ≥ 4, our result is based on Khot’s 2-to-1 conjecture [Khot’02]. For q = 3, we base our hardness result on a certain ‘⊲< shaped ’ variant of his conjecture. We also prove that the problem ALMOST-3-COLORINGε is hard for any constant ε> 0, assuming Khot’s Unique Games conjecture. This is the problem of deciding for a given graph, between the case where one can 3-color all but a ε fraction of the vertices without monochromatic edges, and the case where the graph contains no independent set of relative size at least ε. Our result is based on bounding various generalized noise-stability quantities using the invariance principle of Mossel et al [MOO’05].

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