Obtaining optimal k -cardinality trees fast

Abstract

Given an undirected graph G = ( V , E ) with edge weights and a positive integer number k , the k -cardinality tree problem consists of finding a subtree T of G with exactly k edges and the minimum possible weight. Many algorithms have been proposed to solve this NP-hard problem, resulting in mainly heuristic and metaheuristic approaches. In this article, we present an exact ILP-based algorithm using directed cuts. We mathematically compare the strength of our formulation to the previously known ILP formulations of this problem, and show the advantages of our approach. Afterwards, we give an extensive study on the algorithm's practical performance compared to the state-of-the-art metaheuristics. In contrast to the widespread assumption that such a problem cannot be efficiently tackled by exact algorithms for medium and large graphs (between 200 and 5,000 nodes), our results show that our algorithm not only has the advantage of proving the optimality of the computed solution, but also often outperforms the metaheuristic approaches in terms of running time.