Tight Approximations for Graphical House Allocation

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

Parameterized complexity of envy-free resource allocation in social networks

We consider the classical problem of allocating resources among agents in an envy-free (and, where applicable, proportional) way. Recently, the basic model was enriched by introducing the concept of a social network which allows to capture situations where agents might not have full information about the allocation of all resources. We initiate the study of the parameterized complexity of these resource allocation problems by considering natural parameters which capture structural properties of the network and similarities between agents and items. In particular, we show that even very general fragments of the considered problems become tractable as long as the social network has bounded treewidth or bounded clique-width. We complement our results with matching lower bounds which show that our algorithms cannot be substantially improved

Parameterized Complexity of Envy-Free Resource Allocation in Social Networks

We consider the classical problem of allocating indivisible resources among agents in an envy-free (and, where applicable, proportional) way. Recently, the basic model was enriched by introducing the concept of a social network which allows to capture situations where agents might not have full information about the allocation of all resources. We initiate the study of the parameterized complexity of these resource allocation problems by considering natural parameters which capture structural properties of the network and similarities between agents and resources. In particular, we show that even very general fragments of the considered problems become tractable as long as the social network has constant treewidth or clique-width. We complement our results with matching lower bounds which show that our algorithms cannot be substantially improved