How to Solve the Cake-Cutting Problem in Sublinear Time

The cake-cutting problem refers to the issue of dividing a cake into pieces and distributing them to players who have different value measures related to the cake, and who feel that their portions should be “fair.” The fairness criterion specifies that in situations where n is the number of players, each player should receive his/her portion with at least 1/n of the cake value in his/her measure. In this paper, we show algorithms for solving the cake-cutting problem in sublinear-time. More specifically, we preassign 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 Ω(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 “ǫnvictims,” which means that ǫn players (victims) may not get fair portions, where 0 < ǫ ≤ 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 ǫn 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/ǫ)-time algorithm for r ≤ ǫn/127 undesignated players with ǫn-victims, and an Õ(r/ǫ)-time algorithm for r ≤ ǫe √ lnn/7 designated players and ǫ ≤ 1/e with ǫn-victims are presented. ∗School of Informatics and Engineering, The University of Electro-Communications (UEC), Tokyo, Japan; and CREST, JST, Tokyo, Japan; [email protected] †Komatsu Ltd., Tokyo, Japan; [email protected]