Computing Socially-Efficient Cake Divisions

TL;DR

This paper studies the problem of dividing a divisible good among multiple players to maximize social welfare, proving NP-hardness results and providing approximation algorithms, including FPTAS for small numbers of players.

Contribution

It introduces the first complexity and approximation results for contiguous cake divisions optimizing social welfare functions.

Findings

NP-hard to find optimal divisions for utilitarian and egalitarian welfare.

Provides a constant factor approximation algorithm for utilitarian welfare.

Proves no FPTAS exists for utilitarian welfare unless P=NP.

Abstract

We consider a setting in which a single divisible good ("cake") needs to be divided between n players, each with a possibly different valuation function over pieces of the cake. For this setting, we address the problem of finding divisions that maximize the social welfare, focusing on divisions where each player needs to get one contiguous piece of the cake. We show that for both the utilitarian and the egalitarian social welfare functions it is NP-hard to find the optimal division. For the utilitarian welfare, we provide a constant factor approximation algorithm, and prove that no FPTAS is possible unless P=NP. For egalitarian welfare, we prove that it is NP-hard to approximate the optimum to any factor smaller than 2. For the case where the number of players is small, we provide an FPT (fixed parameter tractable) FPTAS for both the utilitarian and the egalitarian welfare objectives.