Weighted total acquisition

In the Weighted Total Acquisition problem (WTA) on a weighted graph G (only positive non-zero weights), a vertex v can acquire the total weight of a neighbour u if and only if the current weight of v is at least that of u. The new weight of u is then zero, and the new weight of v is then the sum of the weights at u and v just before the acquisition. Over all possible acquisition sequences in G with weight function w, the minimum number of vertices with a non-zero weight at the end is denoted by at(G,w). Given a graph G, a weighting w, and an integer k≥1, the WTA problem asks whether at(G,w)≤k. The Unary (Binary resp.) WTA problem corresponds to the WTA problem when the weights are encoded in unary (binary resp.). We prove that Unary WTA is polynomial-time solvable in graphs of bounded treewidth and degree. When only the treewidth is bounded, this algorithm is quasi-polynomial, i.e., it runs in time WO(logW), where W is the sum of the weights of the vertices. Moreover, we show that Unary WTA is FPT in trees when parameterized by the maximum degree. On the negative side, we show that Unary WTA is NP-complete in trivially perfect graphs and split graphs, even when k=1 in the latter. We prove that the Binary WTA problem is NP-complete in trees of bounded degree, trees of bounded depth, and wheels, but that it is in XP for trees and wheels when parameterized by the solution size. Moreover, we show that Binary WTA is NP-complete in K3,n, planar graphs of pathwidth 2, and unit interval graphs even when k=1, and in trivially perfect graphs when k≥2 (but polynomial-time solvable when k=1).