On the Complexity of Chore Division

TL;DR

This paper establishes an $oldsymbol{ ext{Omega}(n ext{ log } n)}$ lower bound for the complexity of proportional chore division, demonstrating its close relation to the well-studied cake cutting problem.

Contribution

It proves that chore division requires at least $oldsymbol{ ext{Omega}(n ext{ log } n)}$ queries, matching the known bounds for cake cutting, thus resolving an open problem.

Findings

Chore division and cake cutting are closely related problems.

A lower bound of $oldsymbol{ ext{Omega}(n ext{ log } n)}$ queries is established for chore division.

The result aligns chore division complexity with that of cake cutting.

Abstract

We study the proportional chore division problem where a protocol wants to divide an undesirable object, called chore, among $n$ different players. The goal is to find an allocation such that the cost of the chore assigned to each player be at most $1/n$ of the total cost. This problem is the dual variant of the cake cutting problem in which we want to allocate a desirable object. Edmonds and Pruhs showed that any protocol for the proportional cake cutting must use at least $\Omega(n \log n)$ queries in the worst case, however, finding a lower bound for the proportional chore division remained an interesting open problem. We show that chore division and cake cutting problems are closely related to each other and provide an $\Omega(n \log n)$ lower bound for chore division.