The Parameterized Complexity of Domination-Type Problems and Application to Linear Codes

Abstract

We study the parameterized complexity of domination-type problems. (σ,ρ)-domination is a general and unifying framework introduced by Telle: given σ, ρ ⊆ ℕ, a set D of vertices of a graph G is (σ,ρ)-dominating if for any v ∈ D, |N(v) ∩ D| ∈ σ and for any v ∉ D, |N(v) ∩ D| ∈ ρ. Our main result is that for any σ and ρ recursive sets, deciding whether there exists a (σ,ρ)-dominating set of size k, or of size at most k, are both in W[2]. This general statement is optimal in the sense that several particular instances of (σ,ρ)-domination are W[2]-complete (e.g. Dominating Set). We prove the W[2]-membership for the dual parameterization too, i.e. deciding whether there exists a (σ,ρ)-dominating set of size n − k (or at least n − k) is in W[2], where n is the order of the input graph. We extend this result to a class of domination-type problems which do not fall into the (σ,ρ)-domination framework, including Connected Dominating Set. We also consider problems of coding theory which are related to domination-type problems with parity constraints. In particular, we prove that the problem of the minimal distance of a linear code over ${\mathbb{F}}_{q}$ is in W[2] when q is a power of prime, for both standard and dual parameterizations, and W[1]-hard for the dual parameterization.

To prove the W[2]-membership of the domination-type problems we extend the Turing-way to parameterized complexity by introducing a new kind of non-deterministic Turing machine with the ability to perform ‘blind’ transitions, i.e. transitions which do not depend on the content of the tapes. We prove that the corresponding problem Short Blind Multi-Tape Non-Deterministic Turing Machine is W[2]-complete. We believe that this new machine can be used to prove W[2]-membership of other problems, not necessarily related to domination.