A Polynomial-Time Approximation Scheme for Minimum Routing Cost Spanning Trees

Abstract

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 t h a t 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 h o w to build a spanning tree of an n-vertex weighted graph with routing cost at most (1 + ) of the the minimum in time O(n O( 1 ) ). Besides the obvious connection to network design, trees with small routing cost also nd 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 di erent pairs are weighted by di erent requirement amounts. We observe that a randomized O(log n log log 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.

Light graphs with small routing cost

“…We shall first focus on the case where the input is a metric graph. Then, by the previous work in [23], we show that the algorithm can also be applied to a general graph with arbitrary nonnegative distances. 2.…”

“…A k-star is a spanning tree with at most k internal nodes. It was shown [23] that, for a metric graph G, there exists a k-star whose routing cost is at most (k + 3)/(k + 1) times the minimum routing cost. The PTAS in the paper is to find the minimum routing cost k-star in O(n 2k ) time.…”

“…To show the routing cost ratio, we need some notation and results introduced in [20,23]. We shall give some necessary descriptions, and readers can refer to [20,23] for more details.…”

“…We shall give some necessary descriptions, and readers can refer to [20,23] for more details. Let δ ≤ 1/2 be a positive number.…”

“…In [9,12], the MRT problem has been shown to be NP-complete, and 2-approximation algorithms were given in [6,19]. Approximation algorithms with better error ratios were developed in [20], and a polynomial-time approximation scheme (PTAS) for the MRT problem was presented in [23]. Some extensions of the MRT problem have also been investigated.…”