📝 SummarXiv

Abstract

We investigate the Optimal Obstacle Placement (OOP) problem under uncertainty, framed as the dual of the Optimal Traversal Path problem in the Stochastic Obstacle Scene paradigm. We consider both continuous domains, discretized for analysis, and already discrete spatial grids that form weighted geospatial networks using 8-adjacency lattices. Our unified framework integrates OOP with stochastic geometry, modeling obstacle placement via Strauss (regular) and Mat\'ern (clustered) processes, and evaluates traversal using the Reset Disambiguation algorithm. Through extensive Monte Carlo experiments, we show that traversal cost increases by up to 40% under strongly regular placements, while clustered configurations can decrease traversal costs by as much as 25% by leaving navigable corridors compared to uniform random layouts. In mixed (with both true and false obstacles) scenarios, increasing the proportion of true obstacles from 30% to 70% nearly doubles the traversal cost. These findings are further supported by statistical analysis and stochastic ordering, providing rigorous insights into how spatial patterns and obstacle compositions influence navigation under uncertainty.