Strategyproof Matching of Roommates and Rooms

TL;DR

This work investigates matching 2n agents to n rooms with Leontief (complementary) utilities, focusing on social welfare and strategyproofness. It shows a sharp contrast to additive settings: while max-welfare is APX-hard in general, there exists a strategyproof mechanism that maximizes welfare under binary Leontief valuations, with both exponential-time and fixed-parameter tractable implementations. The paper further presents polynomial-time SP mechanisms providing constant-factor welfare guarantees (e.g., 1/3-SW for Triangle-then-L on Leontief, and 1/7–1/6-SW in additive/binary/symmetric cases) and derives tight impossibility results across valuation domains. Computational reductions to 3-Set Packing yield a 0.559-approximation for general utilities and establish NP-hardness/APX-hardness results, illustrating the nuanced landscape of efficiency and truthfulness. Overall, the results illuminate when strategyproofness can align with high welfare in complex roommate-room allocations and point to promising directions in restricted domains and algorithmic design.

Abstract

We initiate the study of matching roommates and rooms wherein the preferences of agents over other agents and rooms are complementary and represented by Leontief utilities. In this setting, 2n agents must be paired up and assigned to n rooms. Each agent has cardinal valuations over the rooms as well as compatibility values over all other agents. Under Leontief preferences, an agents utility for a matching is the minimum of the two values. We focus on the tradeoff between maximizing utilitarian social welfare and strategyproofness. Our main result shows that, in a stark contrast to the additive case, under binary Leontief utilities, there exist strategyproof mechanisms that maximize the social welfare. We further devise a strategyproof mechanism that implements such a welfare maximizing algorithm and is parameterized by the number of agents. Along the way, we highlight several possibility and impossibility results, and give upper bounds and lower bounds for welfare with or without strategyproofness.

Figures (9)

• Figure 1: An instance showing that a maximal matching cannot guarantee $\alpha$SW$\alpha$-SW for any $\alpha > 0$ under binary Leontief utilities. The optimal matching is shown in solid black edges and the wrong choice of triples is shown in dashed red edges.
• Figure 2: An instance showing that a maximal matching cannot guarantee $\alpha$SW$\alpha$-SW for any $\alpha > 1/4$ under binary additive utilities. The optimal matching is shown in solid black edges and the wrong choice of triple is shown in dashed red edges.
• Figure 3: Possible matching structures in a roommate matching problem. The first structure on the left is a triangle (T), the next one is L-shaped (L) and gives non-zero utility to agent $i$. The last three show the cases where both $i$ and $j$ get utility $0$ under Leontief.
• Figure 4: Tight Example of $1/6$SW$1/6$-SW for \ref{['thm:LT-maximal']}
• Figure 5: Agent compatibility values and misreports (in red) for the instance described in \ref{['thm:no-alpha-approx']}. The different colors on the edge correspond to the different types of maximum SW-SW matchings for this instance.

Theorems & Definitions (46)

• Definition 1: Roommate Matching
• Definition 2: Strategyproofness
• Proposition 3.0
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Definition 3
• Theorem 3.3