The Least-Dependency Constrained Spanning Tree Problem
Resumo
We introduce the Least-Dependency Constrained Spanning Tree Problem, which consists of finding a spanning tree where each edge has at least one edge of its dependency set, if non-empty, also in the tree. The dependencies on the input graph G are described by a digraph D where the vertices are the edges of G, and the in-neighbors of a vertex are its dependency set. We show that the optimization problem is NP-hard even if G is a chordal cactus with maximum degree 3 or diameter at most 2, and D is a disjoint union of arborescences of height 2. The same complexity is proved when G is planar bipartite, and each component of D is an oriented cycle or an anti-arborescence of height 1. We also report two polynomial cases.
Referências
Douglas, B. (2001). West. Graph Theory. Prentice Hall, Upper Saddle River, NJ.
Gary, M. R. and Johnson, D. S. (1979). Computers and Intractability: A Guide to the Theory of NP-completeness.
Samer, P. and Urrutia, S. (2015). A branch and cut algorithm for minimum spanning trees under conflict constraints. Optimization Letters, 9(1):41–55.
Viana, L. A. (2016). Árvore geradora com dependências mínima. Master’s thesis, Federal University of Ceará.
Zhang, R., Kabadi, S., and Punnen, A. (2011). The minimum spanning tree problem with conflict constraints and its variations. Discrete Optimization, 8(2):191 – 205.