A note on the integrality gap of the configuration LP for restricted Santa Claus

TL;DR

This paper improves the upper bound on the integrality gap of the configuration LP for the restricted Santa Claus problem from 4 to approximately 3.83, advancing understanding of the problem's approximation limits.

Contribution

The paper provides a tighter analysis of the integrality gap for the configuration LP in the restricted Santa Claus problem, reducing the known upper bound.

Findings

Integrality gap is at most 3.8333.

Previous bound was 4, with a known lower bound of 2.

The analysis refines the understanding of LP relaxation quality.

Abstract

In the restricted Santa Claus problem we are given resources $\mathcal R$ and players $\mathcal P$. Every resource $j\in\mathcal R$ has a value $v_j$ and every player $i$ desires a set $\mathcal R(i)$ of resources. We are interested in distributing the resources to players that desire them. The quality of a solution is measured by the least happy player, i.e., the lowest sum of resource values. This value should be maximized. The local search algorithm by Asadpour et al. and its connection to the configuration LP has proved itself to be a very influential technique for this and related problems. In the original proof, a local search was used to obtain a bound of $4$ for the ratio of the fractional to the integral optimum of the configuration LP (integrality gap). This bound is non-constructive since the local search has not been shown to terminate in polynomial time. On the negative side, the worst instance known has an integrality gap of $2$. Although much progress was made in this area, neither bound has been improved since. We present a better analysis that shows the integrality gap is not worse than $3 + 5/6 \approx 3.8333$.