On the Shallow-Light Steiner Tree Problem

Let G = (V, E) be a given graph with nonnegative integral edge cost and delay, S ⊆ V be a terminal set and r ∈ S be the selected root. The shallow-light Steiner tree (SLST) problem is to compute a minimum cost tree spanning the terminals of S, such that the delay between r and every other terminal is bounded by a given delay constraint D ∈ ℤ₀⁺. It is known that the SLST problem is NP-hard and unless NP ⊆ DTIME(n^{log log n}) there exists no approximation algorithm with ratio (1, γ log2 n) for some fixed γ > 0 [12]. Nevertheless, under the same assumption it admits no approximation ratio better than (1, γ log² n) for some fixed γ > 0 even when D = 2 [2]. This paper first gives an exact algorithm with time complexity O(3^tnD + 2^tn²D² + n³D³), where n and t are the numbers of vertices and terminals of the given graph respectively. This is a pseudo polynomial time parameterized algorithm with respect to the parameterization “number of terminals”. Later, this algorithm is improved to a parameterized approximation algorithm with a time complexity O(3^t n²/∈ + 2^t n⁴/∈² + n⁶/∈³ ) and a bifactor approximation ratio (1 + ∈, 1). That is, for any small real number ∈ > 0, the algorithm computes a Steiner tree with delay and cost bounded by (1 + ∈)D and the optimum cost respectively.