Minimizing Envy and Maximizing Happiness in Graphical House Allocation

We study the house allocation problem in a setting where agents are connected by a graph representing friendships. In this model, two agents can only envy each other if they are neighbors (i.e., friends) in the graph. Each agent has a set of preferred (liked) houses and dislikes the rest. An agent $a$ is said to envy a friend $b$ if $a$ is not assigned any house she likes, while $b$ is allocated a house that $a$ likes. This framework is known as graphical house allocation. Within this framework, we investigate two central problems. The first problem is to compute a house allocation that minimizes the number of envious agents. Multiple such allocations may exist that achieve the same minimum level of envy. Among all allocations that minimize envy, the second problem aims to find one that maximizes the number of agents who receive one of their preferred houses. We present a detailed complexity-theoretic analysis of these problems. In particular, we show that both problems can be solved in polynomial time when each agent prefers at most one house. However, both become NP-hard even when agents are allowed to prefer at most two houses, thereby highlighting the tight boundary between tractability and intractability. Additionally, we design exact algorithms for both problems under certain structural conditions on the agent graph, such as when the graph is sparse, has a small balanced separator, or admits a small vertex cover. These algorithms are significantly faster than the naive brute-force approach.

💡 Research Summary

The paper studies a novel variant of the classic house allocation problem in which agents are vertices of a social graph and envy can only occur between adjacent agents. An agent a envies a neighbor b if a does not receive any house from her preferred set P_a while b receives a house that a likes. Happiness is defined simply as receiving a preferred house. Two optimization goals are considered: (1) minimize the number of envious agents (the “minimum‑envy” problem) and (2) among all allocations achieving the minimum envy, maximize the number of happy agents (the “optimally‑happy” problem).

The authors first show that when every agent likes exactly one house (|P_a|=1 for all a), the problem becomes tractable. In this case each agent can be envious of at most one neighbor. They construct a complete bipartite graph between agents and houses, assigning to edge (a,h) a weight equal to the number of neighbors that would become envious if a receives house h. A minimum‑cost maximum matching in this weighted bipartite graph directly yields an allocation with the smallest possible number of envious agents; the matching cost equals that number. Since minimum‑cost matching can be solved in polynomial time, both the minimum‑envy and the optimally‑happy versions are polynomial‑time solvable under this restriction.

Next, the paper proves that allowing each agent to like up to two houses (|P_a|≤2) already makes both problems NP‑hard. The reduction works even when the agent graph is a complete bipartite graph or a 3‑regular graph, and it holds for the case where the number of agents equals the number of houses (|A|=|H|). This contrasts sharply with the classical setting, where the problem is polynomial when |A|=|H|. Consequently, the hardness of the minimum‑envy problem automatically transfers to the optimally‑happy problem.

Beyond these dichotomy results, the authors develop exact exponential‑time algorithms that exploit structural parameters of the agent graph. For a graph with m edges, they give an algorithm running in 2^{|A|+2|E|}·poly(|A|+|H|), which outperforms the naïve 2^{O(|A|log|H|)} brute force whenever |E|=o(|A|log|H|). They further design algorithms for graphs admitting small balanced separators (e.g., planar or bounded‑genus graphs) using a divide‑and‑conquer approach, achieving runtimes of the form 2^{O(√n)}·poly(n). When the graph has a vertex cover of size k, a dynamic‑programming scheme solves the problem in O(2^{k}·poly(n+m)); thus, if k=polylog(n) the algorithm is practically fast. Complementary hardness results show that even with vertex‑cover size n^{ε} (for any constant ε>0) the problem remains NP‑hard, ruling out quasi‑polynomial algorithms parameterized by such covers.

For the optimally‑happy variant, the authors observe that once a minimum‑envy allocation is obtained, maximizing happiness can be performed by a secondary optimization step (e.g., binary search on the number of happy agents combined with additional matching constraints). Hence all algorithmic results for the minimum‑envy problem extend naturally to the optimally‑happy problem.

The paper situates its contributions within a broad literature on fair division, top‑trading cycles, envy‑free allocations with subsidies, and prior work on graphical house allocation that minimized total envy across edges. By focusing on the count of envious agents and incorporating a secondary happiness objective, the work offers a fresh perspective on fairness when agents only compare outcomes with their immediate social contacts.

In summary, the study delivers: (i) a polynomial‑time algorithm for the case of single‑house preferences, (ii) NP‑hardness for the seemingly modest case of two‑house preferences, and (iii) a suite of exact algorithms that leverage sparsity, balanced separators, and small vertex covers to solve the problems faster than brute force on many realistic graph families. These results provide both theoretical insight into the complexity landscape of graphical house allocation and practical algorithmic tools for applications such as dormitory assignments, cloud resource distribution, or any setting where fairness must respect underlying social networks.