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

We study the parameterized complexity of domination-type problems. (σ, ρ)-domination is a general and unifying framework introduced by Telle: 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. We mainly show that for any σ and ρ the problem of (σ, ρ)-domination is W[2] when parameterized by the size of the dominating set. 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 also prove that (σ, ρ)-domination is W[2] for the dual parameterization, i.e. when parameterized by the size of the dominated set. 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 Fq is W[2] for both standard and dual parameterizations, and W[1]-hard for the dual parameterization. To prove 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 NonDeterministic 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. 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes, G.2.2 Graph Theory