Parameterized Algorithms for Optimal Refugee Resettlement
Jiehua Chen, Ildikó Schlotter, Sofia Simola
TL;DR
The paper develops a comprehensive parameterized complexity analysis of Refugee Resettlement (RR) across three problem variants: Feasible-RR, MaxUtil-RR, and Pareto-RR. It introduces detailed mathematical modeling with multiple services, quotas, and utilities, and establishes both fixed-parameter tractable (FPT) algorithms and tight hardness results under natural parameters such as the number of places $m$, the number of families $n$, the number of services $t$, and the maximum service requirement $r_{\max}$. Key contributions include ILP/N-fold IP formulations, color-coding-based DP, and kernelization-style reductions that yield FPT results for several parameter regimes, as well as NP-hardness for important settings (e.g., with lower quotas, multiple services, or three or more places). The work bridges classical partition/knapsack problems with modern matching under quotas, providing both theoretical insights and algorithmic tools (including Lenstra-type ILPs and N-fold IPs) that could impact practical refugee settlement planning. Overall, the results map a rich tractability landscape and offer practical pathways for scalable, principled allocation under complex constraints.
Abstract
We study variants of the Optimal Refugee Resettlement problem where a set $F$ of refugee families need to be allocated to a set $L$ of possible places of resettlement in a feasible and optimal way. Feasibility issues emerge from the assumption that each family requires certain services (such as accommodation, school seats, or medical assistance), while there is an upper and, possibly, a lower quota on the number of service units provided at a given place. Besides studying the problem of finding a feasible assignment, we also investigate two natural optimization variants. In the first one, we allow families to express preferences over $P$, and we aim for a Pareto-optimal assignment. In a more general setting, families can attribute utilities to each place in $P$, and the task is to find a feasible assignment with maximum total utilities. We study the computational complexity of all three variants in a multivariate fashion using the framework of parameterized complexity. We provide fixed-parameter tractable algorithms for a handful of natural parameterizations, and complement these tractable cases with tight intractability results.