Fair multi-cake cutting

In the classic problem of fair cake-cutting, a single interval (“cake”) has to be divided among n agents with different value measures, giving each agent a single sub-interval with a value of at least 1∕n of the total. This paper studies a generalization in which the cake is made of m disjoint intervals, and each agent should get at most k sub-intervals. The paper presents a polynomial-time algorithm that guarantees to each agent at least min(1∕n,k∕(m+n−1)) of the total value, and shows that this is the largest fraction that can be guaranteed. The algorithm simultaneously guarantees to each agent at least 1∕n of the value of his or her k most valuable islands. The main technical tool is envy-free matching in a bipartite graph. Some of the results remain valid even with additional fairness constraints such as envy-freeness. Besides the natural application of the algorithm to simultaneous division of multiple land-estates, the paper shows an application to a geometric problem — fair division of a two-dimensional land estate shaped as a rectilinear polygon, where each agent should receive a rectangular piece.