Polynomial time algorithm for a Bi-objective Spanning Star Forest Problem on Trees
Résumé
A star is a graph in which all edges share a common endpoint, called the center, and a star forest is a collection of disjoint stars. Given a connected graph with positive edge weights, a spanning star forest (SSF) is a star forest that contains all nodes of the graph. The total weight of a SSF is the sum of its edge weights in the weighted case, and it is the number of its edges in the unweighted case. Recently, the minimum weight spanning star forest problem with a constraint on the number of centers has received attention. In the context of the automobile industry, Agra et al. [1] studied the problem with at most k centers and proved it to be NP-hard. In related works, [5] investigated a variant with exactly k centers. However, both works require specifying the value of k in advance, and no criterion is provided for how k should be chosen. To avoid the need to predefine k, we study the Bi-objective Spanning Star Forest Problem (BSSFP), which aims to minimize the number of centers and the total edge weight simultaneously. Since the minimum version with bounded k centers is NP-hard, the BSSFP is also NP-hard on general graphs. To the best of our knowledge, no efficient results are known related to the enumeration of efficient solutions of constrained SSF problems involving both weight and center constraints, even for restricted graph classes. We focus on the case where the input graph is a tree and prove that, in this setting, all efficient solutions of the BSSFP can be enumerated through a two-phase dynamic programming algorithm. This result provides the first polynomial-time procedure for the BSSFP on trees.
| Origine | Fichiers produits par l'(les) auteur(s) |
|---|---|
| Licence |