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 $\Gamma$ of constraints, we denote by MinCSP($\Gamma$) the restriction of the problem where each constraint is from $\Gamma$. The polynomial-time solvability and the polynomial-time approximability of MinCSP($\Gamma$) were fully characterized by Khanna et al. [Siam J. Comput. '00]. 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 $f(k)n^{O(1)}$ if a solution of size at most $k$ exists. We especially focus on the case of constant-factor FP-approximability. We show the following dichotomy: for each finite constraint language $\Gamma$, either we exhibit a constant-factor FP-approximation for MinCSP($\Gamma$); or we prove that MinCSP($\Gamma$) has no constant-factor FP-approximation unless FPT$=$W[1]. In particular, we show that approximating the so-called Nearest Codeword within some constant factor is W[1]-hard. Recently, Arnab et al. [ICALP '18] showed that such a W[1]-hardness of approximation implies that Even Set is W[1]-hard under randomized reductions. Combining our results, we therefore settle the parameterized complexity of Even Set, a famous open question in the field.