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Distance Preserving Subtrees in Minimum Average Distance Spanning Trees
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Given an undirected graph with nonnegative costs on the edges, the routing cost of any of its spanning trees is the sum over all pairs of vertices of the cost of the path between the pair in the tree. Finding a spanning tree of minimum routing cost is NP-hard, even when the costs obey the triangle inequality. We show that the general case is in fact reducible to the metric case and present a polynomial-time approximation scheme valid for both versions of the problem. In particular, we show how to build a spanning tree of an n-vertex weighted graph with routing cost within (1+ε) from the minimum in time O(n O(1 ε) ). Besides the obvious connection to network design, trees with small routing cost also find application in the construction of good multiple sequence alignments in computational biology. The communication cost spanning tree problem is a generalization of the minimum routing cost tree problem where the routing costs of different pairs are weighted by different requirement amounts. We observe that a randomized O(log 2 n)-approximation for this problem follows directly from a recent result of Bartal, where n is the number of nodes in a metric graph. This also yields the same approximation for the generalized sum-of-pairs alignment problem in computational biology.
Given an undirected graph with nonnegative costs on the edges, the routing cost of any of its spanning trees is the sum over all pairs of vertices of the cost of the path between the pair in the tree. Finding a spanning tree of minimum routing cost is NP-hard, even when the costs obey the triangle inequality. We show that the general case is in fact reducible to the metric case and present a polynomial-time approximation scheme valid for both versions of the problem. In particulaf-, we show how to build a spanning tree of an n-vertex weighted graph with routing cost within (1 + E) from the minimum in time O(noct)). Besides the obvious connection to network design, trees with small routing cost also find application in the construction of good multiple sequence alignments in computational biology. The communication cost spanning tree problem is a Een- eralization of the minimum ro&ing c&t tree problem wi;ere the routinK costs of different pairs are weiahted bv differ- ent requirement amounts. W6 observe tha; a randomized O(log2 n)-approximation for this problem follows directly from a recent result of Bartal, where n is the number of nodes in a metric graph. This also yields the same approx- imation for the generalized sum-of-pairs alignment problem in computational biology.
The Optimal Network problem (as defined by Scott [16]) consists of selecting a subset of arcs that minimizes the sum of the shortest paths between all nodes subject to a budget constraint. This paper considers the worst-case behavior of heuristics for this prob'em. Let n be the number of nodes in the network and e be a constant between 0 and 1. For a general class of Optimal Network Problems, we show that the question of finding a solution which is always less than n times the optimal solution is NP-complete. This indicates that all polynomial-time heuristics for the problem most probably have poor worst-case performance. An upper bound for worst-case heuristic performance of 2n times the optimal solution is also derived. For a restricted version of the Optimal Network problem we describe a procedure whose maximum percentage of error is bounded by a constant. This research was supported, in part, by the U. S. Department of Transportation under Contract DOT-TSC-1058, Transportation Advanced Research Program (TARP).
Given an undirected and connected graph G, with a non-negative weight on each edge, the Minimum Average Distance (MAD) spanning tree problem is to find a spanning tree of G which minimizes the average distance between pairs of vertices. This network design problem is known to be NP-hard even when the edge-weights are equal. In this paper we make a step towards the proof of a conjecture stated by A.A. Dobrynin, R. Entringer and I. Gutman in 2001, and which says that the binomial tree B
n
is a MAD spanning tree of the hypercube H
n
. More precisely, we show that the binomial tree B
n
is a local optimum with respect to the 1-move heuristic which, starting from a spanning tree T of the hypercube H
n
, attempts to improve the average distance between pairs of vertices, by adding an edge e of H
n
-T and removing an edge e′ from the unique cycle created by e. We also present a greedy algorithm which produces good solutions for the MAD spanning tree problem on regular graphs such as the hypercube and the torus.
Given an undirected graph with a nonnegative weight on each edge, the shortest total path length spanning tree problem is to find a spanning tree of the graph such that the total path length summed over all pairs of the vertices is minimized. In this paper, we present several approximation algorithms for this problem. Our algorithms achieve approximation ratios of 2, 15/8, and 3/2 in time O(n2+f(G)),O(n3), and O(n4) respectively, in which f(G) is the time complexity for computing all-pairs shortest paths of the input graph G and n is the number of vertices of G. Furthermore, we show that the approximation ratio of (4/3+ε) can be achieved in polynomial time for any constant ε>0.