Branch and Recharge: Exact Algorithms for Generalized Domination
Abstract
In this paper we present branching algorithms for infinite classes of problems. The novelty in the design and analysis of our branching algorithms lies in the fact that the weights are redistributed over the graph by the algorithms. Our particular setting to make this idea work is a combination of a branching approach with a recharging mechanism. We call it Branch \& Recharge. To demonstrate this approach we consider a generalized domination problem. Let $\sigma$ and $\varrho$ be two nonempty sets of nonnegative integers. A vertex subset $S\subseteq V$ of an undirected graph $G=(V(G),E(G))$ is called a $(\sigma,\varrho)$-dominating set of $G$ if $|N(v)\cap S| \in \sigma$ for all $v\in S$ and $|N(v)\cap S| \in \varrho$ for all $v\in V\setminus S$. This notion generalizes many domination-type graph invariants. We present Branch \& Recharge algorithms enumerating all $(\sigma,\varrho)$-dominating sets of an input graph $G$ in time $O^*(c^n)$ for some $c<2$, if $\sigma$ is successor-free, i.e., it does not contain two consecutive integers, and either both $\sigma$ and $\varrho$ are finite, or one of them is finite and $\sigma \cap \varrho = \emptyset$. Our algorithm implies a non trivial upper bound of $O^*(c^n)$ on the number of $(\sigma,\varrho)$-dominating sets in an $n$-vertex graph under the above conditions on $\sigma$ and $\varrho$.