Low-Degree Spanning Trees of Small Weight

The degree-d spanning tree problem asks for a minimum-weight spanning tree in which the degree of each vertex is at most d. When d=2 the problem is TSP, and in this case, the well-known Christofides algorithm provides a 1.5-approximation algorithm (assuming the edge weights satisfy the triangle inequality). In 1984, Christos Papadimitriou and Umesh Vazirani posed the challenge of finding an algorithm with performance guarantee less than 2 for Euclidean graphs (points in R^n) and d > 2. This paper gives the first answer to that challenge, presenting an algorithm to compute a degree-3 spanning tree of cost at most 5/3 times the MST. For points in the plane, the ratio improves to 3/2 and the algorithm can also find a degree-4 spanning tree of cost at most 5/4 times the MST.Comment: conference version in Symposium on Theory of Computing (1994

A Bound on the Sum of Weighted Pairwise Distances of Points Constrained to Balls

We consider the problem of choosing Euclidean points to maximize the sum of their weighted pairwise distances, when each point is constrained to a ball centered at the origin. We derive a dual minimization problem and show strong duality holds (i.e., the resulting upper bound is tight) when some locally optimal configuration of points is affinely independent. We sketch a polynomial time algorithm for finding a near-optimal set of points.Comment: Cornell ORIE Tech Repor

Bounded-Angle Spanning Tree: Modeling Networks with Angular Constraints

We introduce a new structure for a set of points in the plane and an angle $\alpha$, which is similar in flavor to a bounded-degree MST. We name this structure $\alpha$-MST. Let $P$ be a set of points in the plane and let $0 < \alpha \le 2\pi$ be an angle. An $\alpha$-ST of $P$ is a spanning tree of the complete Euclidean graph induced by $P$, with the additional property that for each point $p \in P$, the smallest angle around $p$ containing all the edges adjacent to $p$ is at most $\alpha$. An $\alpha$-MST of $P$ is then an $\alpha$-ST of $P$ of minimum weight. For $\alpha < \pi/3$, an $\alpha$-ST does not always exist, and, for $\alpha \ge \pi/3$, it always exists. In this paper, we study the problem of computing an $\alpha$-MST for several common values of $\alpha$. Motivated by wireless networks, we formulate the problem in terms of directional antennas. With each point $p \in P$, we associate a wedge $W_p$ of angle $\alpha$ and apex $p$. The goal is to assign an orientation and a radius $r_p$ to each wedge $W_p$, such that the resulting graph is connected and its MST is an $\alpha$-MST. (We draw an edge between $p$ and $q$ if $p \in W_q$, $q \in W_p$, and $|pq| \le r_p, r_q$.) Unsurprisingly, the problem of computing an $\alpha$-MST is NP-hard, at least for $\alpha=\pi$ and $\alpha=2\pi/3$. We present constant-factor approximation algorithms for $\alpha = \pi/2, 2\pi/3, \pi$. One of our major results is a surprising theorem for $\alpha = 2\pi/3$, which, besides being interesting from a geometric point of view, has important applications. For example, the theorem guarantees that given any set $P$ of $3n$ points in the plane and any partitioning of the points into $n$ triplets, one can orient the wedges of each triplet {\em independently}, such that the graph induced by $P$ is connected. We apply the theorem to the {\em antenna conversion} problem

A push–relabel approximation algorithm for approximating the minimum-degree MST problem and its generalization to matroids

AbstractIn the minimum-degree minimum spanning tree (MDMST) problem, we are given a graph G, and the goal is to find a minimum spanning tree (MST) T, such that the maximum degree of T is as small as possible. This problem is NP-hard and generalizes the Hamiltonian path problem. We give an algorithm that outputs an MST of degree at most 2Δopt (G)+o(Δopt (G)), where Δopt (G) denotes the degree of the optimal tree. This result improves on a previous result of Fischer [T. Fischer, Optimizing the degree of minimum weight spanning trees. Technical Report 14853, Dept. of Computer Science, Cornell University, Ithaca, NY, 1993] that finds an MST of degree at most bΔopt (G)+logbn, for any b>1.The MDMST problem is a special case of the following problem: given a k-ary hypergraph G=(V,E) and weighted matroid M with E as its ground set, find a minimum-cost basis (MCB) T of M such that the degree of T in G is as small as possible. Our algorithm immediately generalizes to this problem, finding an MCB of degree at most k2Δopt (G,M)+O(kkΔopt (G,M)).We use the push–relabel framework developed by Goldberg [A. V. Goldberg, A new max-flow algorithm, Technical Report MIT/LCS/TM-291, Massachusetts Institute of Technology, 1985 (Technical Report)] for the maximum-flow problem. To our knowledge, this is the first use of the push–relabel technique in an approximation algorithm for an NP-hard problem.The MDMST problem is closely connected to the bounded-degree minimum spanning tree (BDMST) problem. Given a graph G and degree bound B on its nodes, the BDMST problem is to find a minimum cost spanning tree among the spanning trees with maximum degree B. Previous algorithms for this problem by Könemann and Ravi [J. Könemann, R. Ravi, A matter of degree: Improved approximation algorithms for degree-bounded minimum spanning trees, SIAM Journal on Computing 31(6) (2002) 1783–1793; J. Könemann, R. Ravi, Primal-dual meets local search: Approximating MST’s with nonuniform degree bounds, in: Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing, 2003, pp. 389–395] and by Chaudhuri et al. [K. Chaudhuri, S. Rao, S. Riesenfeld, K. Talwar, What would Edmonds do? Augmenting paths and witnesses for bounded degree MSTs, in: Proceedings of APPROX/RANDOM, 2005, pp. 26–39] incur a near-logarithmic additive error in the degree. We give the first BDMST algorithm that approximates both the degree and the cost to within a constant factor of the optimum. These results generalize to the case of nonuniform degree bounds