Capacitated Domination Problem

Abstract

We consider a generalization of the well-known domination problem on graphs. The (soft) capacitated domination problem with demand constraints is to find a dominating set D of minimum cardinality satisfying both the capacity and demand constraints. The capacity constraint specifies that each vertex has a capacity that it can use to meet the demands of dominated vertices in its closed neighborhood, and the number of copies of each vertex allowed in D is unbounded. The demand constraint specifies the demand of each vertex in V to be met by the capacities of vertices in D dominating it. In this paper, we study the capacitated domination problem on trees from an algorithmic point of view. We present a linear time algorithm for the unsplittable demand model, and a pseudo-polynomial time algorithm for the splittable demand model. In addition, we show that the capacitated domination problem on trees with splittable demand constraints is NP-complete (even for its integer version) and provide a polynomial time approximation scheme (PTAS). We also give a primal-dual approximation algorithm for the weighted capacitated domination problem with splittable demand constraints on general graphs.

On combinatorial optimization for dominating sets (literature survey, new models)

“…(3) metaheuristics and hybrid methods (e.g., [9,16,112]). For some special cases of the problems polynomial approaches are suggested: (a) polynomial algorithms (e.g., [74,87,109,113,114]), and (b) polynomial time approximate schemes (PTAS) (e.g., [30,59,82,105,150]).…”

Capacitated Domination: Constant Factor Approximations for Planar Graphs

“…For this problem, Kao et al, [14], presented a (∆+1)-approximation for general graphs, where ∆ is the maximum vertex degree of the graph, and a polynomial time approximation scheme for trees, which they proved to be NP-hard. In a following work [13], they provided more approximation algorithms and complexity results for this problem.…”

“…Motivated by a general service-requirement assignment scenario, Kao et al, [13,14] considered a generalization of the dominating set problem called Capacitated Domination, which is defined as follows. Let G = (V, E) be a graph with three non-negative parameters defined on each vertex u ∈ V , referred to as the cost, the capacity, and the demand, further denoted by w(u), c(u), and d(u), respectively.…”

Optimal Algorithms and Approximation Algorithms for Replica Placement with Distance Constraints in Tree Networks

“…Many authors deal with variants of these replica placement problems in networks. Some variants of server location problems can be found in [18,21,4,12,14,5,15,19]. In most of these problems, a set of users in a network want to have access to a given service.…”