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Journal Articles RAIRO - Operations Research Year : 2019

On the star forest polytope for trees and cycles

Abstract

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 N P-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 SF P (G), which is the convex hull of the incidence vectors of the star forests of G. First, we prove several basic properties of SF P (G) and propose an integer programming formulation for MWSFP. Then, we give a class of facet-defining inequalities, called M-tree inequalities, for SF P (G). We show that for the case when G is a tree, the M-tree and the nonnegativity inequalities give a complete characterization of SF P (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 SF P (G) when G is a cycle.
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Dates and versions

hal-03071987 , version 1 (16-12-2020)

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Méziane Aïder, Lamia Aoudia, Mourad Baïou, A. Ridha Mahjoub, Viet Hung Nguyen. On the star forest polytope for trees and cycles. RAIRO - Operations Research, 2019, 53 (5), pp.1763-1773. ⟨10.1051/ro/2018076⟩. ⟨hal-03071987⟩
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