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 ∈ ℤ
₀
⁺
. 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 log
²
n) for some fixed γ > 0 even when D = 2 [2]. This paper first gives an exact algorithm with time complexity O(3
^t
nD + 2
^t
n
²
D
²
+ n
³
D
³
), 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
n
²
/∈ + 2
^t
n
⁴
/∈
²
+ n
⁶
/∈
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.