The Graphical House Allocation (GHA) problem asks: how can $n$ houses (each with a fixed non-negative value) be assigned to the vertices of an undirected graph $G$, so as to minimize the sum of absolute differences along the edges of $G$? This problem generalizes the classical Minimum Linear Arrangement problem, as well as the well-known House Allocation Problem from Economics. Recent work has studied the computational aspects of GHA and observed that the problem is NP-hard and inapproximable even on particularly simple classes of graphs, such as vertex disjoint unions of paths. However, the dependence of any approximations on the structural properties of the underlying graph had not been studied. In this work, we give a nearly complete characterization of the approximability of GHA. We present algorithms to approximate the optimal envy on general graphs, trees, planar graphs, bounded-degree graphs, and bounded-degree planar graphs. For each of these graph classes, we then prove matching lower bounds, showing that in each case, no significant improvement can be attained unless P = NP. We also present general approximation ratios as a function of structural parameters of the underlying graph, such as treewidth; these match the tight upper bounds in general, and are significantly better approximations for many natural subclasses of graphs. Finally, we investigate the special case of bounded-degree trees in some detail. We first refute a conjecture by Hosseini et al. [2023] about the structural properties of exact optimal allocations on binary trees by means of a counterexample on a depth-$3$ complete binary tree. This refutation, together with our hardness results on trees, might suggest that approximating the optimal envy even on complete binary trees is infeasible. Nevertheless, we present a linear-time algorithm that attains a $3$-approximation on complete binary trees.

翻译：图式房屋分配（GHA）问题提出：如何将 n 栋房屋（每栋具有固定的非负价值）分配给无向图 G 的顶点，以最小化沿 G 边界的绝对差值之和？该问题推广了经典的最小线性排列问题，以及经济学中著名的房屋分配问题。近期工作研究了 GHA 的计算复杂性，发现该问题即使在特殊简单图类（如路径的顶点不交并）上也是 NP-难且不可近似的。然而，近似算法对底层图结构性质的依赖性尚未被研究。在本文中，我们给出了 GHA 可近似性的近乎完整刻画。我们提出了算法来近似一般图、树、平面图、有界度图以及有界度平面图上的最优嫉妒值。对于每一类图，我们随后证明了匹配的下界，表明在每种情况下，除非 P = NP，否则无法取得显著改进。我们还根据底层图的结构参数（如树宽）给出了通用近似比；这些比值在一般情况下匹配紧的上界，并对许多自然图子类提供了显著更好的近似。最后，我们详细研究了有界度树的特殊情况。我们首先通过一个深度为 3 的完全二叉树的反例，反驳了 Hosseini 等人 [2023] 关于二叉树上精确最优分配结构性质的猜想。这一反驳，结合我们在树上的难度结果，可能暗示即使在完全二叉树上近似最优嫉妒值也是不可行的。尽管如此，我们提出了一种线性时间算法，可在完全二叉树上实现 3-近似。