The Minimum Vertex Cover problem is the problem of, given a graph, finding a smallest set of vertices that touches all edges. We show that it is NP-hard to approximate this problem 1.36067, improving on the previously known hardness result for a 6 factor. 1

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 under the uniform distribution to learning of noisy parities. We show that under the uniform distribution, learning parities with adversarial classification noise reduces to learning parities with random classification noise. Together with the parity learning algorithm of Blum et al. [5], this gives the first nontrivial algorithm for learning parities with adversarial noise. We show that learning of DNF expressions reduces to learning noisy parities of just logarithmic number of variables. We show that learning of k-juntas reduces to learning noisy parities of k variables. These reductions work even in the presence of random classification noise in the original DNF or junta. We then consider the problem of learning halfspaces over Qn with adversarial noise or finding a halfspace that maximizes the agreement rate with a given set of examples. We prove an essentially optimal hardness factor of 2 − ɛ, improving the factor of 85 84 − ɛ due to Bshouty and Burroughs [8]. Finally, we show that majorities of halfspaces are hard to PAC-learn using any representation, based on the cryptographic assumption underlying the Ajtai-Dwork cryptosystem.

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].

Abstract Assuming that NP 6 ` &quot;ffl?0 BPTIME(2nffl), we show that Graph Min-Bisection, Dense kSubgraph and Bipartite Clique have no Polynomial Time Approximation Scheme (PTAS). We give a reduction from the Minimum Distance of Code Problem (MDC). Starting with an instance of MDC, we build a Quasi-random PCP that suffices to prove the desired inapproximability results. In a Quasi-random PCP, the query pattern of the verifier looks random in certain precise sense. Among the several new techniques we introduce, the most interesting one gives a way of certifying that a given polynomial belongs to a given linear subspace of polynomials. As is important for our purpose, the certificate itself happens to be another polynomial and it can be checked probabilistically by reading a constant number of its values.

In this paper, we consider the problem of coloring a 3-colorable 3-uniform hypergraph. In the min-imization version of this problem, given a 3-colorable 3-uniform hypergraph, one seeks an algorithm tocolor the hypergraph with as few colors as possible. We show that it is NP-hard to color a 3-colorable 3-uniform hypergraph with constantly many colors. In fact, we show a stronger result that it is NP-hard to distinguish whether a 3-uniform hypergraph with n vertices is 3-colorable or it contains no independentset of size ffin for an arbitrarily small constant ffi? 0. In the maximization version of the problem, givena 3-uniform hypergraph, the goal is to color the vertices with 3 colors so as to maximize the number ofnon-monochromatic edges. We show that it is NP-hard to distinguish whether a 3-uniform hypergraphis 3-colorable or any coloring of the vertices with 3 colors has at most 89 + ffl fraction of the edges non-monochromatic where ffl? 0 is an arbitrarily small constant. This result is tight since assigning a randomcolor independently to every vertex makes 8 9 fraction of the edges non-monochromatic.These results are obtained via a new construction of a probabilistically checkable proof system (PCP) for NP. We develop a new construction of the PCP Outer Verifier. An important feature of this construc-tion is smoothening of the projection maps. We believe that the techniques in this paper would be quite useful in future. As an application of ourtechniques, we give a simpler proof of H*astad's result [11] that for every constant ffl? 0, it is NP-hardto distinguish satisfiable instances of Max-3SAT from instances where no assignment satisfies more that 78 + ffl fraction of the clauses. Dinur, Regev and Smyth [6] independently showed that it is NP-hard to color a 2-colorable 3-uniformhypergraph with constantly many colors. In the &quot;good case&quot;, the hypergraph they construct is 2-colorableand hence their result is stronger. In the &quot;bad case &quot; however, the hypergraph we construct has a stronger property, namely, it does not even contain an independent set of size ffin.

A new simple generalization of the Motzkin-Straus theorem for the maximum weight clique problem is formulated and directly proved. Within this framework a new trust region heuristic is developed. In contrast to usual trust region methods, it regards not only the global optimum of a quadratic objective over a sphere, but also a set of other stationary points of the program. We formulate and prove a condition when a Motzkin-Straus optimum coincides with such a point. The developed method has complexity O(n ), where n is the number of graph vertices. It was implemented in a publicly available software package QUALEX-MS.

For every integer k > 0, we present a PCP characterization of NP where the verifier uses logarithmic randomness, queries 4k + k 2 bits in the proof, accepts a correct proof with probability 1 (i.e. it is has perfect completeness) and accepts any supposed proof of a false statement with probability at most 2 k 2 +1 . In particular, the verifier achieves optimal amortized query complexity of 1 + for arbitrarily small constant > 0. Such a characterization was already proved by Samorodnitsky and Trevisan [15], but their verifier loses perfect completeness and their proof makes an essential use of this feature. By using an adaptive verifier we can decrease the number of query bits to 2k + k 2 , the same number obtained in [15]. Finally we extend some of the results to larger domains. Royal Institute of Technology, Stockholm, work done while visiting Institute for Advanced Study, supported by NSF grant CCR-9987077.

We present two new algorithms for contention management in transactional memory, the deterministic algorithm CommitRounds and the randomized algorithm RandomizedRounds. Our randomized algorithm is efficient: in some notorious problem instances (e.g., dining philosophers) it is exponentially faster than prior work from a worst case perspective. Both algorithms are (i) local and (ii) starvation-free. Our algorithms are local because they do not use global synchronization data structures (e.g., a shared counter), hence they do not introduce additional resource conflicts which eventually might limit scalability. Our algorithms are starvation-free because each transaction is guaranteed to complete. Prior work sometimes features either (i) or (ii), but not both. To analyze our algorithms (from a worst case perspective) we introduce a new measure of complexity that depends on the number of actual conflicts only. In addition, we show that even a non-constant approximation of the length of an optimal (shortest) schedule of a set of transactions is NP-hard – even if all transactions are known in advance and do not alter their resource requirements. Furthermore, in case the needed resources of a transaction varies over time, such that for a transaction the number of conflicting transactions increases by a factor k, the competitive ratio of any contention manager is Ω(k) for k < √ m, where m denotes the number of cores. 1

We study the inapproximability of Vertex Cover and Independent Set on degree d graphs. We prove that: • Vertex Cover is Unique Games-hard to approximate log log d to within a factor 2−(2+od(1)). This exactly log d matches the algorithmic result of Halperin [1] up to the od(1) term. • Independent Set is Unique Games-hard to approxi-d mate to within a factor O( log2). This improves the d d logO(1) Unique Games hardness result of Samorod-

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