Geospatial Optimization Problems

TL;DR

This paper introduces geospatial optimization problems (GOPs) relevant for resource allocation in geographic regions, proves their NP-hardness, and offers exact and approximate algorithms for solving them.

Contribution

It formally defines GOPs, analyzes their computational complexity, and proposes integer programming formulations and a polynomial-time approximation algorithm.

Findings

GOPs are NP-hard problems.

Integer programming models can solve GOPs exactly.

BMGOP-Compute provides efficient approximations.

Abstract

There are numerous applications which require the ability to take certain actions (e.g. distribute money, medicines, people etc.) over a geographic region. A disaster relief organization must allocate people and supplies to parts of a region after a disaster. A public health organization must allocate limited vaccine to people across a region. In both cases, the organization is trying to optimize something (e.g. minimize expected number of people with a disease). We introduce "geospatial optimization problems" (GOPs) where an organization has limited resources and budget to take actions in a geographic area. The actions result in one or more properties changing for one or more locations. There are also certain constraints on the combinations of actions that can be taken. We study two types of GOPs - goal-based and benefit-maximizing (GBGOP and BMGOP respectively). A GBGOP ensures that certain properties must be true at specified locations after the actions are taken while a BMGOP optimizes a linear benefit function. We show both problems to be NP-hard (with membership in NP for the associated decision problems). Additionally, we prove limits on approximation for both problems. We present integer programs for both GOPs that provide exact solutions. We also correctly reduce the number of variables in for the GBGOP integer constraints. For BMGOP, we present the BMGOP-Compute algorithm that runs in PTIME and provides a reasonable approximation guarantee in most cases.