Exact Algorithms for Weak Roman Domination

Abstract

We consider the Weak Roman Domination problem. Given an undirected graph G = (V,E), the aim is to find a weak roman domination function (wrd-function for short) of minimum cost, i.e. a function f: V → {0,1,2} such that every vertex v ∈ V is defended (i.e. there exists a neighbor u of v, possibly u = v, such that $f(u) \geqslant 1$) and for every vertex v ∈ V with f(v) = 0 there exists a neighbor u of v such that $f(u) \geqslant 1$ and the function f _u → v defined by:

$$ f_{u \rightarrow v}(x) = \left\{ \begin{array} {l l} 1 ~~~~~~~~~~~~~\!~ \textrm{if~} x = v\\ f(u) - 1 ~~~~\textrm{if~} x = u\\ f(x) ~~~~~~~~~\textrm{if~} x \notin \{u,v\}\\ \end{array} \right. $$

does not contain any undefended vertex. The cost of a wrd-function f is defined by cost(f) = ∑  _v ∈ V f(v). The trivial enumeration algorithm runs in time $\mathcal{O}^*(3^n)$ and polynomial space and is the best one known for the problem so far. We are breaking the trivial enumeration barrier by providing two faster algorithms: we first prove that the problem can be solved in $\mathcal{O}^*(2^n)$ time needing exponential space, and then describe an $\mathcal{O}^*(2.2279^n)$ algorithm using polynomial space. Our results rely on structural properties of a wrd-function, as well as on the best polynomial space algorithm for the Red-Blue Dominating Set problem.

This work was supported by the French AGAPE project (ANR-09-BLAN-0159).