Bamas, E. (2024). Lift-and-Project Integrality Gaps for Santa Claus (arXiv:2406.18273v2). arXiv.
This paper investigates the effectiveness of lift-and-project methods, specifically the Sherali-Adams hierarchy, in providing polylogarithmic approximations for the MaxMinDegree Arborescence (MMDA) problem within polynomial time. The research aims to determine if a single round of the Sherali-Adams hierarchy is sufficient to achieve such an approximation.
The authors construct a specific instance of the MMDA problem on a layered graph of depth 3. They then analyze the integrality gap of the Sherali-Adams hierarchy after one round on this instance. To extend the findings to instances with greater depth, the authors introduce a "lifting" technique that generalizes the initial construction.
The research concludes that a single round of the Sherali-Adams hierarchy is insufficient to guarantee a polylogarithmic approximation for the MMDA problem in polynomial time. The "lifted" instances further suggest that achieving a polylogarithmic approximation using current lift-and-project techniques combined with existing reduction methods might necessitate a super-polynomial running time.
This work provides crucial insights into the limitations of widely used lift-and-project methods for approximation algorithms. The findings challenge the existing approaches to solving the MMDA and, consequently, the Santa Claus problem, encouraging the exploration of alternative techniques for achieving efficient approximations.
The authors propose a conjecture regarding the integrality gap of the Sherali-Adams hierarchy on the "lifted" instances for a higher number of rounds. Proving this conjecture remains an open problem. Further research could explore alternative lift-and-project hierarchies or develop novel algorithmic approaches to overcome the limitations identified in this study.
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