Better Trees for Santa Claus

TL;DR

This paper improves the approximation ratio for the max-min degree arborescence problem, a key special case of the Santa Claus problem, achieving a polylogarithmic in log log n approximation in quasi-polynomial time, surpassing previous logarithmic bounds.

Contribution

It introduces the first polylogarithmic approximation for max-min degree arborescence, breaking the logarithmic barrier for this Santa Claus problem variant.

Findings

Achieves a poly(log log n) approximation ratio.

Breaks the logarithmic barrier for a Santa Claus problem case.

First to improve approximation bounds for this problem.

Abstract

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 $\mathrm{poly}(\log\log n)$-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.