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 Min-Max k-Tree Cover problem seeks to find a set of at most $k$ subtrees of $G$, such that the union of the trees is the vertex set $V$. The objective is to minimize the maximum length among all the trees. We give the first constant factor approximation for the hard uniform capacitated version of this problem, where, an input parameter $\lambda$ upper bounds the number of vertices that can be covered by any of the trees. 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 $k\lambda$ [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 $\lambda$ 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.