Rent Division with Picky Roommates

TL;DR

This work tackles rent division where m people share n rooms with preferences over both rooms and roommates, establishing NP-hardness and the nonexistence of guaranteed envy-free welfare allocations in general. It develops a tripartite methodology: a greedy algorithm, a bipartite-matching refinement, and a maximum-weight independent set (MWIS) formulation that yields a (4/3+ε)-approximation to maximum social welfare, complemented by an integer-programming approach to achieve room envy-freeness. Empirical results show MWIS attains the exact welfare optimum on small instances and often yields fairer outcomes than greedy approaches, while theoretical results connect REF feasibility to price vectors via the Second Welfare Theorem. The paper also proposes a conjecture that MWIS can be solved in polynomial time for the relevant graph class, signaling a potential exact algorithm for this challenging problem and offering practical tools for real-world roommate and housing allocations.

Abstract

How can one assign roommates and rooms when tenants have preferences for both where and with whom they live? In this setting, the usual notions of envy-freeness and maximizing social welfare may not hold; the roommate rent-division problem is assumed to be NP-hard, and even when welfare is maximized, an envy-free price vector may not exist. We first construct a novel greedy algorithm with bipartite matching before exploiting the connection between social welfare maximization and the maximum weighted independent set (MWIS) problem to construct a polynomial-time algorithm that gives a $\frac{3}{4}+\varepsilon$-approximation of maximum social welfare. Further, we present an integer program to find a room envy-free price vector that minimizes envy between any two tenants. We show empirically that a MWIS algorithm returns the optimal allocation in polynomial time and conjecture that this problem, at the forefront of computer science research, may have an exact polynomial algorithm solution. This study not only advances the theoretical framework for roommate rent division but also offers practical algorithmic solutions that can be implemented in real-world applications.

Figures (11)

• Figure 1: Illustrated Example of the Roommates Problem.
• Figure 2: Maximum Weight Independent Set with Four Tenants and Two Rooms.
• Figure 3: Mean percent of max social welfare generated by the greedy and greedy with bipartite matching algorithms for $10$ trials each of $200$ simulations.
• Figure 4: Mean percent of max social welfare generated for two greedy algorithms plotted against single preference for $50$ trials each of $200$ simulations.
• Figure 5: A comparison of greedy, greedy with bipartite matching, and MWIS algorithms. MWIS consistently outperforms both alternatives and returns a welfare-maximizing assignment when compared against brute force.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Definition 3
• Claim 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Corollary 3