A Constant Factor Approximation for Capacitated Min-Max Tree Cover

Given a graph G = (V,E) with non-negative real edge lengths and an integer parameter k, the (uncapacitated) Min-Max Tree Cover problem seeks to find a set of at most k trees which together span V and each tree is a subgraph of G. The objective is to minimize the maximum length among all the trees. In this paper, we consider a capacitated generalization of the above and give the first constant factor approximation algorithm. In the capacitated version, there is a hard uniform capacity (?) on the number of vertices a tree can cover. Our result extends to the rooted version of the problem, where we are given a set of k root vertices, R and each of the covering trees is required to include a distinct vertex in R as the root. Prior to our work, the only result known was a (2k-1)-approximation algorithm for the special case when the total number of vertices in the graph is k? [Guttmann-Beck and Hassin, J. of Algorithms, 1997]. Our technique circumvents the difficulty of using the minimum spanning tree of the graph as a lower bound, which is standard for the uncapacitated version of the problem [Even et al.,OR Letters 2004] [Khani et al.,Algorithmica 2010]. Instead, we use Steiner trees that cover ? vertices along with an iterative refinement procedure that ensures that the output trees have low cost and the vertices are well distributed among the trees

Approximation results for a min–max location-routing problem

AbstractThis paper studies a min–max location-routing problem, which aims to determine both the home depots and the tours for a set of vehicles to service all the customers in a given weighted graph, so that the maximum working time of the vehicles is minimized. The min–max objective is motivated by the needs of balancing or fairness in vehicle routing applications. We have proved that unless NP=P, it is impossible for the problem to have an approximation algorithm that achieves an approximation ratio of less than 4/3. Thus, we have developed the first constant ratio approximation algorithm for the problem. Moreover, we have developed new approximation algorithms for several variants, which improve the existing best approximation ratios in the previous literature