Fully Polynomial-Time Approximation Schemes for Fair Rent Division

Abstract

We study the problem of fair rent division that entails splitting the rent and allocating the rooms of an apartment among agents in a fair manner (i.e., under the imposed rents, no agent has a strictly stronger preference for any other agent’s room). The utility functions specify the cardinal preferences of the agents for the rooms for every possible room rent. Although envy-free solutions are guaranteed to exist under reasonably general utility functions, efficient algorithms for finding them were known only for quasilinear utilities. This work addresses this notable gap and develops a fully polynomial-time approximation scheme for fair rent division with minimal assumptions on the utility functions. Envy-free solutions correspond to equilibria of a two-sided matching market with monetary transfers; hence, this work also provides efficient algorithms for finding approximate equilibria in such markets. We complement the algorithmic results by proving that the fair rent division problem lies in the intersection of the complexity classes polynomial parity arguments on directed graphs and polynomial local search.

Maximin Fairness with Mixed Divisible and Indivisible Goods

“…A related line of research incorporates money into the fair division of indivisible goods, with the consideration of finding envy-free allocations (Alkan, Demange, and Gale 1991;Maskin 1987;Klijn 2000;Meertens, Potters, and Reijnierse 2002;Halpern and Shah 2019;Brustle et al 2020;Caragiannis and Ioannidis 2020;Aziz 2021). Another closely related problem is rent division (see (Su 1999;Haake, Raith, and Su 2002;Abdulkadiroglu, Sönmez, and Ünver 2004;Brams 2008;Gal et al 2017;Arunachaleswaran, Barman, and Rathi 2019)). Its cardinal utility version can be viewed as a special case of the mixed setting where one wants to allocate (indivisible) rooms and the (divisible) rent among agents.…”