Minimum bottleneck spanning tree

Abstract: Given a graph G with edge lengths, the minimum bottleneck spanning tree (MBST) problem is to find a spanning tree where the length of the longest edge in tree is minimum. It is a well-known fact that every minimum spanning tree (MST) is a minimum bottleneck spanning tree. In this article, we introduce the δ-MBST problem, which is the problem of finding an MBST such that every vertex in the tree has degree at most δ. We show that optimal solutions to the similarly defined δ-MST problem are not necessarily optimal solutions to the δ-MBST, and we establish that the δ-MBST problem is NP-complete for any δ ≥ 2 . We show that when edge lengths of the graph are Euclidean distances between points in the plane, the problem is NP-hard for δ = 2 and 3, and tractable for δ ≥ 5 . We give a dual approximation scheme for the general graph version of the problem which is the best possible with respect to feasibility. For the Euclidean version, we give a 3 -factor approximation algorithm for the 4-MBST. We also give a 2-factor algorithm for the Euclidean 3-MBST and a 3-factor approximation algorithm for the general Euclidean δ-MBST, both of which can be generalized to metric spaces. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(4), 302–314 2016

Abstract: We consider general discrete Markov Random Fields(MRFs) with additional bottleneck potentials which penalize the maximum (instead of the sum) over local potential value taken by the MRF-assignment. Bottleneck potentials or analogous constructions have been considered in (i) combinatorial optimization (e.g. bottleneck shortest path problem, the minimum bottleneck spanning tree problem, bottleneck function minimization in greedoids), (ii) inverse problems with $L_{\infty}$-norm regularization and (iii) valued constraint satisfaction on the $(\min,\max)$-pre-semirings. Bottleneck potentials for general discrete MRFs are a natural generalization of the above direction of modeling work to Maximum-A-Posteriori (MAP) inference in MRFs. To this end we propose MRFs whose objective consists of two parts: terms that factorize according to (i) $(\min,+)$, i.e. potentials as in plain MRFs, and (ii) $(\min,\max)$, i.e. bottleneck potentials. To solve the ensuing inference problem, we propose high-quality relaxations and efficient algorithms for solving them. We empirically show efficacy of our approach on large scale seismic horizon tracking problems.