Parameterized Approximability for Modular Linear Equations

We consider the Min-$r$-Lin$(Z_m)$ problem: given a system $S$ of length-$r$ linear equations modulo $m$, find $Z \subseteq S$ of minimum cardinality such that $S-Z$ is satisfiable. The problem is NP-hard and UGC-hard to approximate in polynomial time within any constant factor even when $r = m = 2$. We focus on parameterized approximation with solution size as the parameter. Dabrowski et al. showed that Min-$2$-Lin$(Z_m)$ is in FPT if $m$ is prime (i.e. $Z_m$ is a field), and it is W[1]-hard if $m$ is not a prime power. We show that Min-$2$-Lin$(Z_{p^n})$ is FPT-approximable within a factor of $2$ for every prime $p$ and integer $n \geq 2$. This implies that Min-$2$-Lin$(Z_m)$, $m \in Z^+$, is FPT-approximable within a factor of $2\omega(m)$ where $\omega(m)$ counts the number of distinct prime divisors of $m$. The idea behind the algorithm is to solve ever tighter relaxations of the problem, decreasing the set of possible values for the variables at each step. Working over $Z_{p^n}$ and viewing the values in base-$p$, one can roughly think of a relaxation as fixing the number of trailing zeros and the least significant nonzero digits of the values assigned to the variables. To solve the relaxed problem, we construct a certain graph where solutions can be identified with a particular collection of cuts. The relaxation may hide obstructions that will only become visible in the next iteration of the algorithm, which makes it difficult to find optimal solutions. To deal with this, we use a strategy based on shadow removal to compute solutions that (1) cost at most twice as much as the optimum and (2) allow us to reduce the set of values for all variables simultaneously. We complement the algorithmic result with two lower bounds, ruling out constant-factor FPT-approximation for Min-$3$-Lin$(R)$ over any nontrivial ring $R$ and for Min-$2$-Lin$(R)$ over some finite commutative rings $R$.