Volume 53, Number 5, November-December 2019
|Page(s)||1763 - 1773|
|Published online||09 October 2019|
On the star forest polytope for trees and cycles
Laromad, Faculty of Mathematics, University of Sciences and Technology Houari Boumediéne, Algeirs, Algeria
2 LIMOS CNRS UMR 6158, Clermont-Ferrand, France
3 Université Paris-Dauphine, PSL, CNRS UMR 7243 LAMSADE, Place du Maréchal De Lattre de Tassigny, 75775, Paris Cedex 16, France
4 Sorbonne Université, CNRS UMR 7606, LIP6 – 4 place Jussieu, Paris, France
* Corresponding author: Hung.Nguyen@lip6.fr
Accepted: 4 September 2018
Let G = (V, E) be an undirected graph where the edges in E have non-negative weights. A star in G is either a single node of G or a subgraph of G where all the edges share one common end-node. A star forest is a collection of vertex-disjoint stars in G. The weight of a star forest is the sum of the weights of its edges. This paper deals with the problem of finding a Maximum Weight Spanning Star Forest (MWSFP) in G. This problem is NP-hard but can be solved in polynomial time when G is a cactus [Nguyen, Discrete Math. Algorithms App. 7 (2015) 1550018]. In this paper, we present a polyhedral investigation of the MWSFP. More precisely, we study the facial structure of the star forest polytope, denoted by SFP(G), which is the convex hull of the incidence vectors of the star forests of G. First, we prove several basic properties of SFP(G) and propose an integer programming formulation for MWSFP. Then, we give a class of facet-defining inequalities, called M-tree inequalities, for SFP(G). We show that for the case when G is a tree, the M-tree and the nonnegativity inequalities give a complete characterization of SFP(G). Finally, based on the description of the dominating set polytope on cycles given by Bouchakour et al. [Eur. J. Combin. 29 (2008) 652–661], we give a complete linear description of SFP(G) when G is a cycle.
Mathematics Subject Classification: 90C27 / 90C10 / 05Cxx
Key words: Combinatorial optimization / polyhedral combinatorics / star forest / facility location / dominating set
© The authors. Published by EDP Sciences, SMAI 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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