Fixed-Parameter Approximability of Boolean MinCSPs

The minimum unsatisfiability version of a constraint satisfaction problem (MinCSP) asks for an assignment where the number of unsatisfied constraints is minimum possible, or equivalently, asks for a minimum-size set of constraints whose deletion makes the instance satisfiable. For a finite set Γ of constraints, we denote by MinCSP(Γ) the restriction of the problem where each constraint is from Γ. The polynomial-time solvability and the polynomial-time approximability of MinCSP(Γ) were fully characterized by Khanna et al. [33]. Here we study the fixed-parameter (FP-) approximability of the problem: given an instance and an integer k, one has to find a solution of size at most g(k) in time f(k) · nO(1) if a solution of size at most k exists. We especially focus on the case of constant-factor FP-approximability. Our main result classifies each finite constraint language Γ into one of three classes: (1) MinCSP(Γ) has a constant-factor FP-approximation; (2) MinCSP(Γ) has a (constant-factor) FP-approximation if and only if Nearest Codeword has a (constant-factor) FP-approximation; (3) MinCSP(Γ) has no FPapproximation, unless FPT = W[P]. We show that problems in the second class do not have constant-factor FP-approximations if both the Exponential-Time Hypothesis (ETH) and the Linear PCP Conjecture (LPC) hold. We also show that such an approximation would imply the existence of an FP-approximation for the k-Densest Subgraph problem with ratio 1 − for any > 0. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems