# Restricted Max-Min Allocation: Approximation and Integrality Gap

## TL;DR

This paper advances the approximation algorithms for the restricted max-min allocation problem, reducing the ratio to nearly 4 and tightening the integrality gap bounds, thus bringing us closer to optimal solutions.

## Contribution

It introduces a $(4+ ext{delta})$-approximation algorithm and improves the integrality gap upper bound for the configuration LP.

## Key findings

- Approximation ratio improved to $(4+ ext{delta})$
- Integrality gap upper bound refined to approximately 3.808
- Algorithm runs in polynomial time with respect to input size

## Abstract

Asadpour, Feige, and Saberi proved that the integrality gap of the configuration LP for the restricted max-min allocation problem is at most $4$. However, their proof does not give a polynomial-time approximation algorithm. A lot of efforts have been devoted to designing an efficient algorithm whose approximation ratio can match this upper bound for the integrality gap. In ICALP 2018, we present a $(6 + \delta)$-approximation algorithm where $\delta$ can be any positive constant, and there is still a gap of roughly $2$. In this paper, we narrow the gap significantly by proposing a $(4+\delta)$-approximation algorithm where $\delta$ can be any positive constant. The approximation ratio is with respect to the optimal value of the configuration LP, and the running time is $\mathit{poly}(m,n)\cdot n^{\mathit{poly}(\frac{1}{\delta})}$ where $n$ is the number of players and $m$ is the number of resources. We also improve the upper bound for the integrality gap of the configuration LP to $3 + \frac{21}{26} \approx 3.808$.