On Solving Close Enough Orienteering Problems with Overlapped Neighborhoods

TL;DR

This paper addresses solving Close Enough Orienteering Problems by exploiting overlapped neighborhoods through Steiner Zones (SZ). It introduces Randomized Steiner Zone Discretization (RSZD) to efficiently discretize the problem and a hybrid CRaSZe-AntS solver that combines PSO for continuous waypoint placement with inherited Ant Colony System for discrete sequencing, tailored for both uniform CEOP and non-uniform CEOP-N costs. The approach yields competitive solution quality with substantial reductions in computation time, notably up to about 140% improvement in prize collection and over 55% faster runtimes on large instances, and demonstrates applicability to truck-and-drone delivery contexts. The methods advance practical UAV mission planning by enabling interior search within neighborhoods and scalable optimization under complex cost structures, with potential extensions to 3D generalized Steiner Zones and online, resource-constrained scenarios.

Abstract

Close Enough Traveling Salesman Problem (CETSP) is a well-known variant of TSP whereby the agent may complete its mission at any point within a target neighborhood. Heuristics based on overlapped neighborhoods, known as Steiner Zones (SZ), have gained attention in addressing CETSP. While SZs offer effective approximations to the original graph, their inherent overlap imposes constraints on search space, potentially conflicting with global optimization objectives. Here we show how such limitations can be converted into advantages in a Close Enough Orienteering Problem (CEOP) by aggregating prizes across overlapped neighborhoods. We further extend classic CEOP with Non-uniform Neighborhoods (CEOP-N) by introducing non-uniform costs for prize collection. To tackle CEOP and CEOP-N, we develop a new approach featuring a Randomized Steiner Zone Discretization (RSZD) scheme coupled with a hybrid algorithm based on Particle Swarm Optimization (PSO) and Ant Colony System (ACS), CRaSZe-AntS. The RSZD scheme identifies sub-regions for PSO exploration, and ACS determines the discrete visiting sequence. We evaluate the RSZD's discretization performance on CEOP instances derived from established CETSP instances and compare CRaSZe-AntS against the most relevant state-of-the-art heuristic focused on single-neighborhood optimization for CEOP instances. We also compare the performance of the interior search within SZs and the boundary search on individual neighborhoods in the context of CEOP-N. Our experimental results show that CRaSZe-AntS can yield comparable solution quality with significantly reduced computation time compared to the single neighborhood strategy, where we observe an average 140.44% increase in prize collection and a 55.18% reduction in algorithm execution time. CRaSZe-AntS is thus highly effective in solving emerging CEOP-N, examples of which include truck-and-drone delivery scenarios.

Figures (12)

• Figure 1: A Steiner zone of degree three, $|\Omega| = 3$, formed by three circular arcs $\widehat{v'_{1} v'_{2}}(c_3)$, $\widehat{v'_{1} v'_{3}}(c_2)$, and $\widehat{v'_{2} v'_{3}}(c_1)$. The red cross points refer to SZ vertices $v'_1, v'_2, v'_3$.
• Figure 2: Circle approximation wang2019steiner
• Figure 3: An example that the sweep line algorithm will find a wrong SZ of degree three. The interval $[l_2, l_5]$ is registered at $t_2$, overlapping with $[l_3, l_6], [l_1, l_4]$ and $[l_3, l_4]$. Three circles intersect pairwise. But in fact, only a SZ of degree two can be formed.
• Figure 5: An example of the first three steps of RSZD in concentricCircles1.
• Figure 6: Examples of the arc search algorithm for SZ of different degrees.