Bang Ye Wu

A polynomial time approximation scheme for minimum routing cost spanning trees

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 at most

Approximation algorithms for some optimum communication spanning tree problems

(1+\epsilon)

Exact algorithms for the minimum latency problem

of the minimum in time

Constructing the Maximum Consensus Tree from Rooted Triples

O(n^{O({\frac{1}{\epsilon}}% )})

Approximation algorithms for the shortest total path length spanning tree problem

. 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 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.

A polynomial time approximation scheme for the two-source minimum routing cost spanning trees

Abstract Let G =( V , E , w ) be an undirected graph with nonnegative edge length function w and nonnegative vertex weight function r . The optimal product-requirement communication spanning tree (PROCT) problem is to find a spanning tree T minimizing ∑ u , v ∈ V r ( u ) r ( v ) d T ( u , v ), where d T ( u , v ) is the length of the path between u and v on T . The optimal sum-requirement communication spanning tree (SROCT) problem is to find a spanning tree T such that ∑ u , v ∈ V ( r ( u )+ r ( v )) d T ( u , v ) is minimized. Both problems are special cases of the optimum communication spanning tree problem, and are reduced to the minimum routing cost spanning tree (MRCT) problem when all the vertex weights are equal to each other. In this paper, we present an O( n 5 )-time 1.577-approximation algorithm for the PROCT problem, and an O( n 3 ) time 2-approximation algorithm for the SROCT problem, where n is the number of vertices. We also show that a 1.577-approximation solution for the MRCT problem can be obtained in O( n 3 )-time, which improves the time complexity of the previous result.

An efficient algorithm for the length-constrained heaviest path problem on a tree

Let G = (V, E,w) be an undirected graph with positive weight w(e) on each edge e ∈ E. Given a starting vertex s ∈ V and a subset U ⊂ V as the demand vertex set, the minimum latency problem (MLP) asks for a tour P starting at s and visiting each demand vertex at least once such that the total latency of all demand vertices is minimized, in which the latency of a vertex is the length of the path from s to the first visit of the vertex. The MLP is an important problem in computer science and operations research, and is also known as the delivery man problem or the traveling repairman problem. Similar to the well-known traveling salesperson problem (TSP), in the MLP we are asked to find an “optimal” way for routing a server passing through the demand vertices. The difference is the objective functions. The latency of a vertex can be thought of as the delay of the service. In the MLP we care about the total delay (service quality), while the total length (service cost) is concerned in the TSP. The MLP on a metric space is NP-hard and also MAX-SNP-hard [4]. Polynomial time algorithms are only known for very special graphs, such as paths [1, 6], edge-unweighted trees [9], trees of diameter 3 [4], trees of constant number of leaves [8], or graphs with similar structure [12]. Even for caterpillars (paths with edges sticking out), no polynomial time algorithm has been reported. In a recent work, it is shown that the MLP on edge-weighted trees is NP-hard [11]. Due to the NP-hardness, many works ∗corresponding author (bangye@mail.stu.edu.tw)

Polynomial time algorithms for some minimum latency problems

We investigated the problem of constructing the maximum consensus tree from rooted triples. We showed the NP-hardness of the problem and developed exact and heuristic algorithms. The exact algorithm is based on the dynamic programming strategy and runs in O((m + n2)3n) time and O(2n) space. The heuristic algorithms run in polynomial time and their performances are tested and shown by comparing with the optimal solutions. In the tests, the worst and average relative error ratios are 1.200 and 1.072 respectively. We also implemented the two heuristic algorithms proposed by Gasieniec et al. The experimental result shows that our heuristic algorithm is better than theirs in most of the tests.

A Polynomial Time Approximation Scheme for Optimal Product-Requirement Communication Spanning Trees

Given an undirected graph with a nonnegative weight on each edge, the shortest total path length spanning tree problem is to nd 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(n 2 +f(G)); O(n 3 ), and O(n 4 ) 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+) can be achieved in polynomial time for any constant >0. ? 2000 Elsevier Science B.V. All rights reserved.

Light graphs with small routing cost

Let G be an undirected graph with nonnegative edge lengths. Given two vertices as sources and all vertices as destinations, we investigated the problem how to construct a spanning tree of G such that the sum of distances from sources to destinations is minimum. In the paper, we show the NP-hardness of the problem and present a polynomial time approximation scheme. For any e > 0, the approximation scheme finds a (1 + e)- approximation solution in O(n⌈1/e+1⌈) time. We also generalize the approximation algorithm to the weighted case for distances that form a metric space.