Bumblebee visitation problem

Let G(V,E,c,w) be a weighted graph with vertex set V, edge set E, vertex-capacity function c:V→R₊, and edge-weight function w:E→R₊. In Bumblebee visitation problem, a mobile agent Bumblebee, denoted by B, begins by entering a vertex of the graph, and then moves along the edges of the graph. When B moves along an edge e=uv, both c(u) and c(v) are decreased by w(e). The Bumblebee visitation problem deals with placing and moving B in G such that the sum of the residual-capacities at the visited vertices is maximum. We consider four variants of this problem depending on edge-weights and constraints on the possible movements of B. The four variants are uniform-weight-constrained BUMBLEBEE VISITATION problem, variable-weight-constrained BUMBLEBEE VISITATION problem, uniform-weight-unconstrained BUMBLEBEE VISITATION problem, and variable-weight-unconstrained BUMBLEBEE VISITATION problem. We show that all four variants are NP-hard for general graphs, and the variable-weight constrained variant is NP-hard even for star graphs (K_1,n). On the positive side, for the uniform-weight constrained variant, we give a dynamic programming based linear-time algorithm for trees and a quadratic-time algorithm for cactus. We then extend these algorithms for the variable-weight unconstrained variant. We also give a 3-factor approximation algorithm for the uniform-weight unconstrained variant where each vertex-capacity is at least five.