Better Trees for Santa Claus

We revisit the problem max-min degree arborescence, which was introduced by Bateni et al. [STOC'09] as a central special case of the general Santa Claus problem, which constitutes a notorious open question in approximation algorithms. In the former problem we are given a directed graph with sources and sinks and our goal is to find vertex disjoint arborescences rooted in the sources such that at each non-sink vertex of an arborescence the out-degree is at least k, where k is to be maximized.

This problem is of particular interest, since it appears to capture much of the difficulty of the Santa Claus problem: (1) like in the Santa Claus problem the configuration LP has a large integrality gap in this case and (2) previous progress by Bateni et al. was quickly generalized to the Santa Claus problem (Chakrabarty et al. [FOCS'09]). These results remain the state-of-the-art both for the Santa Claus problem and for max-min degree arborescence and they yield a polylogarithmic approximation in quasi-polynomial time. We present an exponential improvement to this, a poly(log log n)-approximation in quasi-polynomial time for the max-min degree arborescence problem. To the best of our knowledge, this is the first example of breaking the logarithmic barrier for a special case of the Santa Claus problem, where the configuration LP cannot be utilized.

The main technical novelty of our result are locally good solutions: informally, we show that it suffices to find a poly(log n)-approximation that locally has stronger guarantees. We use a lift-and-project type of LP and randomized rounding, which were also used by Bateni et al., but unlike previous work we integrate careful pruning steps in the rounding. In the proof we extensively apply Lovász Local Lemma and a local search technique, both of which were previously used only in the context of the configuration LP.