### Near Optimal Bounds for Steiner Trees in the Hypercube

Tao Jiang, Zevi Miller, Dan Pritikin
2011 SIAM journal on computing (Print)
Given a set S of vertices in a connected graph G, the classic Steiner tree problem asks for the minimum number of edges of a connected subgraph of G that contains S. We study this problem in the hypercube. Given a set S of vertices in the n-dimensional hypercube Q n , the Steiner cost of S, denoted by cost(S), is the minimum number of edges among all connected subgraphs of Q n that contain S. We obtain the following results on cost(S). Let be any given small, positive constant, and set k = |S|.
more » ... t, and set k = |S|. (1) [upper bound] For every set S we have cost(S) < ( 1 3 k + 1 + 1 2 ln k)n. In particular, there is a constant c 1 depending only on such that if k > c 1 , then cost(S) < ( 1 3 + )kn. (2) We develop a randomized algorithm of running time O(kn) that produces a connected subgraph H of Q n containing S such that with probability approaching 1 as k, n → ∞ we have |E(H)| < ( 1 3 + )kn. (3) [lower bound] There are constants c 2 and b (with 1 < b < 2) depending only on such that if c 2 < k < b n , then as n → ∞ almost all sets S of size k in Q n satisfy cost(S) > ( 1 3 − )kn. Thus for k in this range with k → ∞, the upper bound (1) is asymptotically tight. We also show that for fixed k, as n → ∞, almost always a random family of k vertices in Q n satisfies k 3 + 2 9 (−1 + (− 1 2 ) k ) n − √ n ln n ≤ cost(S) ≤ k 3 + 2 9 (