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A 2-prover game is called unique if the answer of one prover uniquely determines the answer of the second prover and vice versa (we implicitly assume games to be one round games). The value of a 2-prover game is the maximum acceptance probability of the verifier over all the prover strategies. We make the following conjecture regarding the power of unique(More)
In this paper we disprove the following conjecture due to Goemans [17] and Linial [25] (also see [5, 27]): " Every negative type metric embeds into 1 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 1. Surprisingly,(More)
Based on a conjecture regarding the power of unique 2-prover-1-round games presented in [Khot02], we show that vertex cover is hard to approximate within any constant factor better than 2. We actually show a stronger result, namely, based on the same conjecture, vertex cover on k-uniform hypergraphs is hard to approximate within any constant factor better(More)
In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of α GW + , for all > 0; here α GW ≈ .878567 denotes the approximation ratio achieved by the Goemans-Williamson algorithm [26]. This implies that if the Unique Games Conjecture of Khot [37] holds then the Goemans-Williamson(More)
Given a <i>k</i>-uniform hyper-graph, the E<i>k</i>-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyper-edge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that E<i>k</i>-Vertex-Cover is NP-hard to approximate within factor <i>(k-1-&#949;)</i> for any <i>k(More)
We address well-studied problems concerning the learnability of parities and halfspaces in the presence of classification noise. Learning of parities under the uniform distribution with random classification noise, also called the noisy parity problem is a famous open problem in computational learning. We reduce a number of basic problems regarding learning(More)
We consider the following allocation problem arising in the setting of combinatorial auctions: a set of goods is to be allocated to a set of players so as to maximize the sum of the utilities of the players (i.e., the social welfare). In the case when the utility of each player is a monotone submodular function, we prove that there is no polynomial time(More)
Arora, Rao and Vazirani [2] showed that the standard semi-definite programming (SDP) relaxation of the Sparsest Cut problem with the <i>triangle inequality</i> constraints has an integrality gap of O(&#8730;log n). They conjectured that the gap is bounded from above by a constant. In this paper, we disprove this conjecture (referred to as the(More)
For arbitrarily small constants ε, δ > 0, we present a long code test with one free bit, completeness 1 − ε and soundness δ. Using the test, we prove the following two inapproximability results: 1. Assuming the Unique Games Conjecture of Khot [17], given an n-vertex graph that has two dis-joint independent sets of size (1 2 − ε)n each, it is NP-hard to find(More)