Lift-and-Project Integrality Gaps for Santa Claus

TL;DR

This paper investigates the integrality gaps of lift-and-project hierarchies for the Santa Claus problem, demonstrating limitations of current techniques and establishing bounds on the Sherali-Adams hierarchy's effectiveness.

Contribution

It constructs specific instances showing that one round of Sherali-Adams cannot close the integrality gap for certain layered graph problems related to Santa Claus.

Findings

An integrality gap of n^{Ω(1)} survives 1 round of Sherali-Adams.

After 2 rounds, the gap reduces to polylogarithmic for depth-3 graphs.

The constructed instances can be extended to any depth within a certain range, challenging existing techniques.

Abstract

This paper is devoted to the study of the MaxMinDegree Arborescence (MMDA) problem in layered directed graphs of depth $\ell\le O(\log n/\log \log n)$, which is an important special case of the Santa Claus problem. Obtaining a polylogarithmic approximation for MMDA in polynomial time is of high interest as it is a necessary condition to improve upon the well-known 2-approximation for makespan scheduling on unrelated machines by Lenstra, Shmoys, and Tardos [FOCS'87]. The only way we have to solve the MMDA problem within a polylogarithmic factor is via an elegant recursive rounding of the $(\ell-1)^{th}$ level of the Sherali-Adams hierarchy, which needs time $n^{O(\ell)}$ to solve. However, it remains plausible that one could obtain a polylogarithmic approximation in polynomial time by using the same rounding with only $1$ round of the Sherali-Adams hierarchy. As a main result, we rule out this possibility by constructing an MMDA instance of depth $3$ for which an integrality gap of $n^{\Omega(1)}$ survives $1$ round of the Sherali-Adams hierarchy. This result is tight since it is known that after only $2$ rounds the gap is at most polylogarithmic on depth-3 graphs. Second, we show that our instance can be ``lifted'' via a simple trick to MMDA instances of any depth $\ell\in \Omega(1)\cap o(\log n/\log \log n)$ (the whole range of interest), for which we conjecture that an integrality gap of $n^{\Omega(1/\ell)}$ survives $\Omega(\ell)$ rounds of Sherali-Adams. We show a number of intermediate results towards this conjecture, which also suggest that our construction is a significant challenge to the techniques used so far for Santa Claus.