House allocation is an extremely well-studied problem in the field of fair allocation, where the goal is to assign $n$ houses to $n$ agents while satisfying certain fairness criterion, e.g., envy-freeness. To model social interactions, the Graphical House Allocation framework introduces a social graph $G$, in which each vertex corresponds to an agent, and an edge $(u, v)$ corresponds to the potential of agent $u$ to envy the agent $v$, based on their allocations and valuations. In undirected social graphs, the potential for envy is in both the directions. In the Minimum Envy Graphical House Allocation (ME-GHA) problem, given a set of $n$ agents, $n$ houses, a social graph, and agent's valuation functions, the goal is to find an allocation that minimizes the total envy summed up over all the edges of $G$. Recent work, [Hosseini et al., AAMAS 2023, AAMAS 2024] studied ME-GHA in the regime of polynomial-time algorithms, and designed exact and approximation algorithms, for certain graph classes under identical agent valuations. We initiate the study of \gha with non-identical valuations, a setting that has so far remained unexplored. We investigate the multivariate (parameterized) complexity of \gha by identifying structural restrictions on the social graph and valuation functions that yield tractability. We also design moderately exponential-time algorithms for several graph classes, and a polynomial-time algorithm for {binary valuations that returns an allocation with envy at most one when the social graph has maximum degree at most one.

翻译：房屋分配是公平分配领域中一个被广泛研究的问题，其目标是在满足特定公平准则（如无嫉妒性）的条件下，将 $n$ 间房屋分配给 $n$ 个智能体。为建模社会互动，图房屋分配框架引入了一个社会图 $G$，其中每个顶点对应一个智能体，边 $(u, v)$ 则对应智能体 $u$ 基于分配结果和估值而可能嫉妒智能体 $v$ 的潜在可能性。在无向社会图中，嫉妒的可能性是双向的。在最小嫉妒图房屋分配问题中，给定一组 $n$ 个智能体、$n$ 间房屋、一个社会图以及智能体的估值函数，目标是找到一个分配方案，使得 $G$ 中所有边上的嫉妒总和最小。近期研究 [Hosseini et al., AAMAS 2023, AAMAS 2024] 在多项式时间算法框架下研究了 ME-GHA，并在智能体估值相同的假设下，为特定图类设计了精确算法和近似算法。我们首次开启了具有非相同估值的 \gha 研究，这一设定此前尚未被探索。我们通过识别社会图和估值函数上能带来可解性的结构约束，研究了 \gha 的多变量（参数化）复杂性。我们还为多个图类设计了中等指数时间算法，以及一个针对二元估值的多项式时间算法，该算法在社会图最大度不超过 1 时，能返回嫉妒值至多为 1 的分配方案。