We study optimization problems that may be expressed as \Boolean constraint satisfaction problems". An instance of a Boolean constraint satisfaction problem is given by m constraints applied to n Boolean variables. Di erent computational problems arise from constraint satisfaction problems depending on the nature of the \underlying" constraints as well as on the goal of the optimization task. Here we consider four possible goals: Max CSP (Min CSP) is the class of problems where the goal is to nd an assignment maximizing the number of satis ed constraints (minimizing the number of unsatis ed constraints). Max Ones (Min Ones) is the class of optimization problems where the goal is to nd an assignment satisfying all constraints with maximum (minimum) number of variables set to 1. Each class consists of in nitely many problems and a problem within a class is speci ed by a nite collection of nite Boolean functions that describe the possible constraints that may be used.Tight bounds on the approximability of every problem in Max CSP were obtained byCreignou 11]. In this work we determine tight bounds on the \approximability" (i.e., the ratio to within which each problem may be approximated in polynomial time) of every problem in Max Ones, Min CSP and Min Ones. Combined with the result of Creignou, this completely classi es all optimization problems derived from Boolean constraint satisfaction. Our results capture a diverse collection of optimization problems such as MAX 3-SAT, Max Cut, Max Clique, Min Cut, Nearest Codeword etc. Our results unify recent results on the (in)approximability of these optimization problems and yield a compact presentation of most known results. Moreover, these results provide a formal basis to many statements on the behavior of natural optimization problems, that have so far only been observed empirically.Preliminary versions of parts of this paper appeared in

Abstract. Starting with the seminal paper of Impagliazzo and Rudich [17], there has been a large body of work showing that various cryptographic primitives cannot be reduced to each other via "black-box" reductions. The common interpretation of these results is that there are inherent limitations in using a primitive as a black box, and that these impossibility results can be overcome only by explicitly using the code of the primitive in the construction. In this paper we revisit these negative results, give a more careful taxonomy of the ways in which "black-box reductions" can be formalized, strengthen some previous results (in particular giving unconditional impossibility results for reductions that were previously only shown to imply P = NP ), and offer a new interpretation of them: in many cases, there is no limitation in using a primitive as a black box, but there is a limitation in treating adversaries as such. In particular, these negative results may be overcome by using the code of the adversary in the analysis.

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