Deterministic polylog approximation for minimum communication spanning trees

This paper considers the problem of selecting a minimum communication spanning tree (MCT) for a given weighted network, namely, a tree that minimizes the total cost of transmitting a given set of communication requirements between n sites over the tree edges [8]. A slightly stronger formulation of the problem [1] is based on the concept of a minimum average stretch spanning tree (MAST) for weighted connected multigraphs. In particular, a ρ-solution for the MAST problem (namely, an algorithm for constructing a spanning tree with average stretch p) in the special case of complete weighted graphs implies an approximation algorithm for the MCT problem with approximation ratio p. It is conjectured in [1] that for any given weighted multigraph there exists a spanning tree with average stretch O(log n) (which is the best possible, in view of the Ω(log n) lower bound given therein). However, the (deterministic) construction presented (which is the best construction to date) yields only a bound of exp(O(√log n log log n)) on the average stretch. For the restricted case of complete weighted graphs, there is a better, albeit randomized, construction yielding average stretch O(log² n) [2]. This implies a randomized approximation algorithm for MCT with the same ratio. This paper presents a deterministic algorithm that for every weighted complete multigraph constructs a spanning tree whose average stretch is bounded by O(log² n). This yields a deterministic polynomial-time approximation algorithm for MCT with ratio O(log² n). In addition, our solution approach confirms the conjecture of [1] in the special case of d-dimensional Euclidean complete multigraphs for fixed d, where our construction yields spanning trees with O(log n) average stretch.