The Graphical House Allocation 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 "aggregate local envy", i.e., 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, the latter of which has notable practical applications in organ exchanges. Recent work has studied the computational aspects of Graphical House Allocation 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 complete characterization of the approximability of the Graphical House Allocation problem. We present algorithms to approximate the optimal envy on general graphs, trees, planar graphs, bounded-degree graphs, bounded-degree planar graphs, and bounded-degree trees. 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 aforementioned tight upper bounds in general, and are significantly better approximations for many natural subclasses of graphs. Finally, we present constant factor approximation schemes for the special classes of complete binary trees and random graphs.

翻译：图形化房屋分配问题探讨：如何将$n$个（每个具有固定非负价值）房屋分配给无向图$G$的顶点，以最小化"聚合局部嫉妒"，即沿着$G$各边绝对差值之和？该问题泛化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——后者在器官交换等领域具有重要实际应用。近期研究关注了图形化房屋分配的计算复杂性，发现该问题即使在路径的不交并等简单图类上也属于NP难且不可逼近问题。然而，逼近性能与底层图结构特性的依赖关系此前尚未被研究。本文完整刻画了图形化房屋分配问题的可逼近性。我们提出了在一般图、树、平面图、有界度图、有界度平面图及有界度树上逼近最优嫉妒的算法。针对每个图类，我们随后证明了匹配的下界，表明除非P=NP，否则各情形均无法获得显著改进。我们还给出了作为底层图结构参数（如树宽）函数的通用逼近比，这些结果在一般情况下匹配上述紧上界，并对许多自然图子类实现显著更优的逼近。最后，我们针对完全二叉树和随机图等特殊图类提出了常数因子逼近方案。