Optimization, Randomized Approximability, and Boolean Constraint Satisfaction Problems

We give a unified treatment to optimization problems that can be expressed in the form of nonnegative-real-weighted Boolean constraint satisfaction problems. Creignou, Khanna, Sudan, Trevisan, and Williamson studied the complexity of approximating their optimal solutions whose optimality is measured by the sums of outcomes of constraints. To explore a wider range of optimization constraint satisfaction problems, following an early work of Marchetti-Spaccamela and Romano, we study the case where the optimality is measured by products of constraints’ outcomes. We completely classify those problems into three categories: PO problems, NPO-hard problems, and intermediate problems that lie between the former two categories. To prove this trichotomy theorem, we analyze characteristics of nonnegative-real-weighted constraints using a variant of the notion of T-constructibility developed earlier for complex-weighted counting constraint satisfaction problems. keywords: optimization problem, approximation algorithm, constraint satisfaction problem, PO, APX, approximation-preserving reducibility 1 Maximization by Multiplicative Measure In the 1980s started extensive studies that have greatly improved our understandings of the exotic behaviors of various optimization problems within a scope of computational complexity theory. These studies have brought us deep insights into the approximability and inapproximability of optimization problems; however, many studies have targeted individual problems by cultivating different and independent methods for them. To push our insights deeper, we are focused on a collection of “unified” optimization problems, whose foundations are all formed in terms of Boolean constraint satisfaction problems (or CSPs, in short). Creignou is the first to have given a formal treatment to maximization problems derived from CSPs [3]. The maximization constraint satisfaction problems (or MAX-CSPs for succinctness) are, in general, optimization problems in which we seek a truth assignment σ of Boolean variables that maximizes an objective value† (or a measure) of σ, which equals the number of constraints being satisfied at once. Creignou presented three criteria (which are 0-validity, 1-validity, and 2-monotonicity) under which we can solve the MAX-CSPs in polynomial time; that is, the problems belong to PO. Creignou’s result was later reinforced by Khanna, Sudan, Trevisan, and Williamson [7], who gave a unified treatment to several types of CSP-based optimization problems, including MAX-CSP, MIN-CSP, and MAX-ONE-CSP. With constraints limited to “nonnegative” integer values, Khanna et al. defined MAX-CSP(F) as the maximization problem in which constraints are all taken from constraint set F and the maximization is measured by the “sum” of the objective values of Present Affiliation: Department of Information Science, University of Fukui, 3-9-1 Bunkyo, Fukui 910-8507, Japan A function that associates an objective value (or a measure) to each solution is called an objective function or (a measure function).