Exact Algorithms for Weak Roman Domination

We investigate a domination-like problem from the exact exponential algorithms viewpoint. The classical Dominating Set problem ranges among one of the most famous and studied NP -complete covering problems [6]. In particular, the trivial enumeration algorithm of runtime O∗(2n) 4 has been improved to O∗(1.4864n) in polynomial space, and O∗(1.4689n) with exponential space [9]. Many variants of the Dominating Set problem have been introduced and studied extensively both from structural and algorithmic viewpoints (for a survey see, e.g., [7]). One of those variants called Roman Domination was introduced in [4] and motivated by the articles Defend the Roman Empire!" of I. Stewart [15] and Defendens Imperium Romanum: a classical problem in military strategy of C.S. ReVelle and K.E. Rosing [14]. In general, the aim is to protect a set of locations (vertices of a graph) by using a smallest possible amount of legions (to be placed on those vertices). Motivated by a decree of the Emperor Constantine the Great in the fourth century A.D., Roman Domination uses the following rules for protecting a graph: a vertex can protect itself if it has one legion, and protect all its neighbors if it owns two legions, since Constantine decreed that two legions must be placed at a location before one may move to a nearby location (adjacent vertex) to defend it. The Roman Domination problem asks to minimize the number of legions used to defend all vertices. Since then, numerous articles have been published around this problem (see, e.g., [1, 3, 5, 12]). In particular, this NP -complete problem has been tackled using exact exponential algorithms. The rst non-trivial one achieved had running time O∗(1.6183n) and used polynomial space [10]. This result has recently been improved toO∗(1.5673n) [16], which can be lowered toO∗(1.5014n) at the cost of exponential space [16]. Moreover, the Roman Domination problem can be related to several other variants of defenselike domination, such as secure domination, or eternal domination. We focus our attention on yet another variant of the Roman Domination problem. In 2003, Henning et al. [8] considered the following idea: location t can also be protected if one of its neighbors possesses one legion that can be moved to t in such a way that the whole collection of locations (set of vertices) remains protected. This variation adds some kind of dynamics to the problem and gives rise to theWeak Roman Domination problem. Formally, it can be de ned as follows: