Optimal broadcast domination of arbitrary graphs in polynomial time ⋆ Pinar Heggernes and Daniel Lokshtanov Department of Informatics, University of Bergen, N-5020 Bergen, Norway. pinar.heggernes@ii.uib.no, daniel.lokshtanov@student.uib.no Abstract. Broadcast domination was introduced by Erwin in 2002, and it is a variant of the standard dominating set problem, such that vertices can be assigned various domination powers. Broadcast domination as- signs a power f (v) ≥ 0 to each vertex v of a given graph, such that every vertex of the graph is within distance f (v) from some vertex v having f (v) ≥ 1. The optimal broadcast domination problem seeks to minimize the sum of the powers assigned to the vertices of the graph. Since the presentation of this problem its computational complexity has been open, and the general belief has been that it might be NP -hard. In this paper, we show that optimal broadcast domination is actually in P , and we give a polynomial time algorithm for solving the problem on arbitrary graphs, using a non standard approach. 1 Introduction A dominating set in a graph is a subset of the vertices of the graph such that every vertex of the graph either belongs to the dominating set or has a neighbor in the dominating set. A vertex outside of the dominating set is said to be dominated by one of its neighbors in the dominating set. The standard optimal domination problem seeks to find a dominating set of minimum cardinality. Since the introduction of this problem [2], [12], many domination related graph parameters have been introduced and studied, and domination in graphs is one of the most well known and widely studied subjects within graph algorithms [7], [8]. The standard dominating set problem can be seen as to represent a set of cities having broadcast stations, where every city can hear a broadcast station placed in it or in a neighboring city [11]. In 2002 Erwin [5] introduced the broad- cast domination problem, which is more realistic in the sense that the various broadcast stations are allowed to transmit at different powers. FM radio sta- tions are distinguished both by their transmission frequency and by their ERP (Effective Radiated Power). A transmitter with a higher ERP can transmit fur- ther, but it is more expensive to build and to operate. Consequently, the optimal broadcast domination problem asks to compute an integer valued power function ⋆ This work is supported by the Research Council of Norway through the SPECTRUM project grant.