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.