We prove a number of improved inaproximability results, including the best up to date explicit approximation thresholds for MIS problem of bounded degree, bounded occurrences MAX-2SAT, and bounded degree Node Cover. We prove also for the first time inapproximability of the problem of Sorting

The Traveling Salesman Problem is one of the most studied problems in the theory of algorithms and its approximability is a long-standing open question. Prior to the present work, the best known inapproximability threshold was 220/219, due to Papadimitriou and Vempala. Here, using an essentially

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A two-state spin system is specified by a matrix A =

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Many recent results show the hardness of approximating NP-hard functions. We formalize, in a very simple way, what these results involve: a code-like Levin reduction. Assuming a well-known complexity assumption, we show that such reductions cannot prove the NPhardness of the following problems, where ffl is any positive fraction: (i) achieving an approximation ratio

Understanding the theoretical limitations of efficient computation is one of the most fundamental open problems of modern mathematics. This thesis studies the approximability of intractable optimization problems. In particular, we study so-called Max CSP problems. These are problems in which we are given a set of constraints, each constraint acting on some k variables, and are asked to find an assignment to the variables satisfying as many of the constraints as possible. A predicate P:[q] k →{0, 1} is said to be approximation resistant if it is intractable to approximate the corresponding CSP problem to within a factor which is better than what is expected from a completely random assignment to the variables. We prove that if the Unique Games Conjecture is true, then a sufficient condition for a predicate P: [q] k →{0, 1} to be approximation resistant is that there exists a pairwise independent distribution over [q] k which is supported on the set of satisfying assignments P −1 (1) of P. We also study predicates P: {0, 1} 2 →{0, 1} on two boolean variables. The corresponding CSP problems include fundamental computational problems such as Max Cut

inapproximability of the problem of Sorting by Reversals and display an explicit approximation threshold for this problem.

good inapproximability result to a certain numeric minimization problem. The key objects in our analysis are the vector triples arising when doing clause-by-clause analysis of algorithms based on semidefinite programming. Given a weighted set of such triples of a certain restricted type, which