Provable Methods for Searching with an Imperfect Sensor

TL;DR

The paper addresses planning for a mobile robot with imperfect sensing under a time budget to maximize detection probability $Pr(oldsymbol{})$ when targeting a stationary location among a finite planar set. It decomposes the problem into subset selection, visitation order, and search-effort allocation, proving that optimal order can be achieved without revisits and that 1D search allocation can be solved by a pseudopolynomial DP (with NP-hardness in general). It further analyzes tractable cases under uniform priors and ordered point sets, develops approximation schemes (including a PTAS-based approach for $k$-TSP) and presents hardness results, complemented by empirical comparisons among an exact IP, DP-based methods, and a greedy heuristic. The work delivers fast, provably good algorithms suitable for SAR-like scenarios, demonstrated on map-based and synthetic benchmarks. Overall, it advances the algorithmic foundations for searching with imperfect sensors in geometric spaces and informs practical planning in real-world robotic search operations.

Abstract

Assume that a target is known to be present at an unknown point among a finite set of locations in the plane. We search for it using a mobile robot that has imperfect sensing capabilities. It takes time for the robot to move between locations and search a location; we have a total time budget within which to conduct the search. We study the problem of computing a search path/strategy for the robot that maximizes the probability of detection of the target. Considering non-uniform travel times between points (e.g., based on the distance between them) is crucial for search and rescue applications; such problems have been investigated to a limited extent due to their inherent complexity. In this paper, we describe fast algorithms with performance guarantees for this search problem and some variants, complement them with complexity results, and perform experiments to observe their performance.

Figures (6)

• Figure 1: A three-step approach to solve Imperfect Searcher. Say $n = 6$. First, we decide on which on subset to visit; next the order in which to visit them; and finally the number of times to search a point during each visit.
• Figure 2: Comparison of 3 algorithms on Stony Brook University buildings. We use Cartopy Cartopy and OpenStreetMap OpenStreetMap to obtain satellite data. Search costs are proportional to a building's footprint; time budget is a factor of the map's diameter. How do we find a target on one of these rooftops using an aerial vehicle with imperfect sensing capabilities?
• Figure 3: Lemma 2 fails if $V$ is not a metric space: schedule $\psi = (1, 1, 2, 1, 3)$ has less weight than $\psi' = (1, 1, 1, 2, 3)$.
• Figure 4: "Billionare's lane" in the Hamptons in New York, USA Brennan_2015 (obtained using Cartopy and OSM CartopyOpenStreetMap). Since there are no other roads, a SAR operation here reduces to 1D ImpSea.
• Figure 5: Gaps and runtime of all solvers and configurations on benchmarks. Instances of the same size are grouped together. Note that the $y$-axis in the bottom figure is a logarithmic scale.

Theorems & Definitions (20)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 3
• proof
• Lemma 4
• proof
• Theorem 5
• proof