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.

翻译：我们研究一种图结构下的房屋分配问题，其中智能体通过表示友谊关系的图相互连接。在此模型中，仅当两个智能体在图上是邻居（即朋友）时，他们才可能嫉妒对方。每个智能体拥有一组偏好（喜欢）的房屋，其余房屋则被视为不喜欢。若智能体$a$未分配到任何其喜欢的房屋，而其朋友$b$被分配到了$a$喜欢的房屋，则称$a$嫉妒$b$。该框架被称为图结构房屋分配。在此框架下，我们探究两个核心问题：第一个问题是计算使嫉妒智能体数量最小化的房屋分配方案。可能存在多个达到相同最小嫉妒水平的分配方案。在所有最小化嫉妒的分配方案中，第二个问题旨在找到使获得偏好房屋的智能体数量最大化的方案。我们对这些问题进行了细致的计算复杂性分析。特别地，我们证明当每个智能体最多偏好一套房屋时，两个问题均可在多项式时间内求解。然而，即使仅允许智能体最多偏好两套房屋，两个问题都会变为NP困难问题，从而清晰揭示了可处理性与难解性之间的严格边界。此外，我们针对智能体图的特定结构条件（如图稀疏、具有小规模平衡分隔符或允许小规模顶点覆盖）设计了两个问题的精确算法。这些算法相比朴素的暴力搜索方法具有显著的效率优势。