Approximate Directed Minimum Degree Spanning Tree in Polynomial Time

Given a directed graph G and a sink vertex s, the directed minimum degree spanning tree problem requires computing an incoming spanning tree rooted at s whose maximum tree in-degree is the smallest among all such trees. The problem is known to be NP-hard, since it generalizes the Hamiltonian path problem. For the approximation version of this problem, a polynomial time algorithm with O(∆∗ logn) approximation guarantee and a quasi-polynomial time algorithm with O(∆∗ + logn) approximation guarantee are already known; here n = |V | and ∆∗ denotes the optimal tree in-degree. In this paper, we propose a simple polynomial time algorithm that also achieves an O(∆∗ +logn) approximation. Then we improve this algorithm to obtain a (1+ǫ)∆∗+O( logn log log n ) approximation for any constant 0 < ǫ < 1 in polynomial time. Institute for Interdisciplinary Information Sciences, Tsinghua University, [email protected] Institute for Interdisciplinary Information Sciences, Tsinghua University, [email protected]