The graph burning problem is an NP-hard combinatorial optimization problem that helps quantify the vulnerability of a graph to contagion. This paper introduces a simple farthest-first traversalbased approximation algorithm for this problem over general graphs. We refer to this proposal as the Burning Farthest-First (BFF) algorithm. BFF runs in O(n 3 ) steps and has a tight approximation factor of 3−2/b(G), where b(G) is the size of an optimal solution. The main attribute of BFF is that it has a better approximation factor than the state-of-the-art approximation algorithms for general graphs, which report an approximation factor of 3. Despite being simple, BFF proved practical when tested over some benchmark datasets.
The capacitated vertex k-center problem receives as input a complete weighted graph and a set of capacity constraints. Its goal is to find a set of k centers and an assignment of vertices that does not violate the capacity constraints. Furthermore, the distance from the farthest vertex to its assigned center has to be minimized. The capacitated vertex k-center problem models real situations where a maximum number of clients must be assigned to centers and the travel time or distance from the clients to their assigned center has to be minimized. These centers might be hospitals, schools, police stations, among many others. The goal of this paper is to explicitly state how the capacitated vertex k-center problem and the minimum capacitated dominating set problem are related. We present an exact algorithm that consists of solving a series of integer programming formulations equivalent to the minimum capacitated dominating set problem over the bottleneck input graph. Lastly, we present an empirical evaluation of the proposed algorithm using off-the-shelf optimization software.
The uniform capacitated vertex
k
-center problem is an
\(\mathcal {NP} \)
-hard combinatorial optimization problem that models real situations where
k
centers can only attend a maximum number of customers, and the travel time or distance from the customers to their assigned center has to be minimized. This paper introduces a polynomial-time constructive heuristic algorithm that exploits the relationship between this problem and the minimum capacitated dominating set problem. The proposed heuristic is based on the one-hop farthest-first heuristic that has proven effective for the uncapacitated version of the problem. We carried out different empirical evaluations of the proposed heuristics, including an analysis of the effect of a parallel implementation of the algorithm, which significantly improved the running time for relatively large instances.
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