How to solve the cake-cutting problem in sublinear time

TL;DR

This paper introduces sublinear-time algorithms for the cake-cutting problem, allowing fair division among many players with minimal time and some players possibly not receiving perfectly fair portions.

Contribution

It presents a novel framework and algorithms for approximate cake-cutting in sublinear time, including handling victims and designated players, which improves efficiency over existing methods.

Findings

Algorithms run in o(n) or poly(n) time for fair division

Framework accommodates eps n-victims, allowing some players to be unfairly treated

Provides specific algorithms with proven time complexities for different scenarios

Abstract

In this paper, we show algorithms for solving the cake-cutting problem in sublinear-time. More specifically, we preassign (simple) fair portions to o(n) players in o(n)-time, and minimize the damage to the rest of the players. All currently known algorithms require Omega(n)-time, even when assigning a portion to just one player, and it is nontrivial to revise these algorithms to run in $o(n)$-time since many of the remaining players, who have not been asked any queries, may not be satisfied with the remaining cake. To challenge this problem, we begin by providing a framework for solving the cake-cutting problem in sublinear-time. Generally speaking, solving a problem in sublinear-time requires the use of approximations. However, in our framework, we introduce the concept of "eps n-victims," which means that eps n players (victims) may not get fair portions, where 0< eps =< 1 is an arbitrary constant. In our framework, an algorithm consists of the following two parts: In the first (Preassigning) part, it distributes fair portions to r < n players in o(n)-time. In the second (Completion) part, it distributes fair portions to the remaining n-r players except for the eps n victims in poly}(n)-time. There are two variations on the r players in the first part. Specifically, whether they can or cannot be designated. We will then present algorithms in this framework. In particular, an O(r/eps)-time algorithm for r =< eps n/127 undesignated players with eps n-victims, and an O~(r^2/eps)-time algorithm for r =< eps e^{{sqrt{ln{n}}}/{7}} designated players and eps =< 1/e with eps n-victims are presented.