Purely Combinatorial Algorithms for Approximate Directed Minimum Degree Spanning Trees

Given a directed graph $G$ on $n$ vertices with a special vertex $s$, the directed minimum degree spanning tree problem requires computing a 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. The best LP-based polynomial time algorithm can achieve an approximation of $Δ^*+2$ [Bansal et al, 2009], where $Δ^*$ denotes the optimal maximum tree in-degree. As for purely combinatorial algorithms (algorithms that do not use LP), the best approximation is $O(Δ^*+\log n)$ [Krishnan and Raghavachari, 2001] but the running time is quasi-polynomial. In this paper, we focus on purely combinatorial algorithms and try to bridge the gap between LP-based approaches and purely combinatorial approaches. As a result, we propose a purely combinatorial polynomial time algorithm that also achieves an $O(Δ^* + \log n)$ approximation. Then we improve this algorithm to obtain a $(1+ε)Δ^* + O(\frac{\log n}{\log\log n})$ for any constant $0<ε<1$ approximation in polynomial time.