Abstract

A Steiner minimum tree (SMT) in the rectilinear plane is the shortest length tree interconnecting a set of points, called the regular points, possibly using additional vertices. Ak-size Steiner minimum tree (kSMT) is one that can be split into components where all regular points are leaves and all components have at mostkleaves. Thek-Steiner ratio in the rectilinear plane, ρ_k, is the infimum of the ratios SMT/kSMT over all finite sets of regular points. Thek-Steiner ratio is used to determine the performance ratio of several recent polynomial-time approximations for Steiner minimum trees. Previously it was known that in the rectilinear plane, ρ₂ = 2/3, ρ₃ = 4/5, and (2k − 2)/(2k − 1) ≤ ρ_k(L₁) ≤ (2k − 1)/(2k) fork ≥ 4. In 1991, P. Berman and V. Ramaiyer conjectured that in fact ρ_k = (2k − 1)/(2k) fork ≥ 4. In this paper we prove their conjecture.