An Improved Approximation Guarantee for the Maximum Budgeted Allocation Problem

TL;DR

This paper presents an improved approximation algorithm for the Maximum Budgeted Allocation problem, surpassing the previous 3/4 ratio by leveraging the stronger Configuration-LP relaxation.

Contribution

It introduces a new algorithm that achieves a better approximation ratio for MBA by utilizing the Configuration-LP, demonstrating its superiority over the Assignment-LP.

Findings

Achieves a 3/4 + c approximation ratio for MBA

Shows Configuration-LP is strictly stronger than Assignment-LP

Provides a new rounding technique for LP solutions

Abstract

We study the Maximum Budgeted Allocation problem, which is the problem of assigning indivisible items to players with budget constraints. In its most general form, an instance of the MBA problem might include many different prices for the same item among different players, and different budget constraints for every player. So far, the best approximation algorithms we know for the MBA problem achieve a $3/4$-approximation ratio, and employ a natural LP relaxation, called the Assignment-LP. In this paper, we give an algorithm for MBA, and prove that it achieves a $3/4+c$-approximation ratio, for some constant $c>0$. This algorithm works by rounding solutions to an LP called the Configuration-LP, therefore also showing that the Configuration-LP is strictly stronger than the Assignment-LP (for which we know that the integrality gap is $3/4$) for the MBA problem.