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Unformatted text preview: The Non-Approximability of Non-Boolean Predicates Lars Engebretsen ? Department of Numerical Analysis and Computer Science Royal Institute of Technology SE-100 44 Stockholm SWEDEN E-mail: enge@nada.kth.se August 2001 Abstract. Constraint satisfaction programs where each constraint depends on a con- stant number of variables have the following property: The randomized algorithm that guesses an assignment uniformly at random satisfies an expected constant fraction of the constraints. By combining constructions from interactive proof systems with harmonic analysis over finite groups, Håstad showed that for several constraint satis- faction programs this naive algorithm is essentially the best possible unless P = NP . While most of the predicates analyzed by Håstad depend on a small number of vari- ables, Samorodnitsky and Trevisan recently extended Håstad’s result to predicates depending on an arbitrarily large, but still constant, number of Boolean variables. We combine ideas from these two constructions and prove that there exists a large class of predicates on finite non-Boolean domains such that for predicates in the class, the naive randomized algorithm that guesses a solution uniformly is essentially the best possible unless P = NP . As a corollary, we show that the k-CSP problem over domains with size D cannot be approximated within D k- O ( √ k )- , for any constant > 0, unless P = NP . This lower bound matches well with the best known upper bound, D k- 1 , of Serna, Trevisan and Xhafa. 1 Introduction In a breakthrough paper, Håstad [6] studied the problem of giving approximate solutions to maximization versions of several constraint satisfaction problems. An instance of a such a problem is given as a collection of constraints, i.e., functions from some domain to { , 1 } , and the objective is to satisfy as many constraints as possible. An approximate solution of a constraint satisfaction program is simply an assignment that satisfies roughly as many constraints as possible. In this setting, ? Part of this research was performed while the author was visiting MIT with support from the Marcus Wallenberg Foundation and the Royal Swedish Academy of Sciences. 1 we are interested in proving either that there exists a polynomial time algorithm producing approximate solutions some constant fraction from the optimum or that no such algorithms exist. Typically, each individual constraint depends on a fixed number k of the vari- ables and the size of the instance is given as the total number of variables that appear in the constraints. In this case, which is usually called the Max k-CSP problem, there exists a very naive algorithm that approximates the optimum within a constant factor: The algorithm that just guesses a solution at random. In his pa- per, Håstad [6] proved the very surprising fact that this algorithm is essentially the best possible e ffi cient algorithm for several constraint satisfaction problems, un- less P = NP . The proofs unify constructions from interactive proof systems with....
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This note was uploaded on 12/27/2011 for the course ECON 101 taught by Professor Flah during the Spring '10 term at Punjab Engineering College.

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