On the complexity of cake cutting

Abstract

In the cake cutting problem, n ≥ 2 players want to cut a cake into n pieces so that every player gets a 'fair' share of the cake by his/her own measure.We prove that in a certain, natural cake cutting model, every fair cake division protocol for n players must use Ω (n log n) cuts and evaluation queries in the worst case. Up to a small constant factor, this lower bound matches a corresponding upper bound in the same model obtained by Even and Paz, from 1984.

On Low-Envy Truthful Allocations

“…Since then, the problem of achieving a proportional allocation with the minimum number of operations has received much attention and is now well-understood [12,13,6,20,24]. The problem of achieving envyfreeness has been proven to be much more challenging [8,5,22]; in fact, under the most common computational model of cut and evaluation queries [20], no algorithm with bounded running time is known for more than 3 players.…”

The Efficiency of Fair Division

“…Furthermore, there are instances in which no optimal allocation is fair. Models similar to ours have been considered in the literature; the focus has been on the design of protocols for achieving proportionality [3,4,8], envyfreeness [3,6,7], and equitability [3] or on the design of approximation algorithms in settings where fulfilling the fairness objective exactly is impossible [2,5]. However, the related literature seems to have neglected the issue of efficiency.…”

Cake Cutting Algorithms

“…The result of Stromquist (2008) was generalized by Deng et al (2012). Other papers on the complexity of cake cutting include the ones by Woeginger and Sgall (2007), Magdon-Ismail et al (2003), and Balkanski et al (2014). A beautiful paper by Edmonds and Pruhs (2006b) circumvents Theorem 13.2 by designing a randomized, approximately proportional algorithm that requires only O(n) queries in the Robertson-Webb model.…”