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1. Shortest Paths in the Plane with Obstacle Violations

   https://doi.org/s00453-020-00673-y  

   Hershberger, J.; Kumar, N.; Suri, S. (January 2020, Algorithmica)

   We study the problem of finding shortest paths in the plane among h convex obstacles, where the path is allowed to pass through (violate) up to k obstacles, for 𝑘≤ℎ. Equivalently, the problem is to find shortest paths that become obstacle-free if k obstacles are removed from the input. Given a fixed source point s, we show how to construct a map, called a shortest k-path map, so that all destinations in the same region of the map have the same combinatorial shortest path passing through at most k obstacles. We prove a tight bound of 𝛩(𝑘𝑛) on the size of this map, and show that it can be computed in 𝑂(𝑘2𝑛log𝑛) time, where n is the total number of obstacle vertices. 

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2. A New Algorithm for Euclidean Shortest Paths in the Plane

   https://doi.org/10.1145/3580475  

   Wang, Haitao (April 2023, Journal of the ACM)

   Given a set of pairwise disjoint polygonal obstacles in the plane, finding an obstacle-avoiding Euclidean shortest path between two points is a classical problem in computational geometry and has been studied extensively. Previously, Hershberger and Suri (in SIAM Journal on Computing , 1999) gave an algorithm of O(n log n ) time and O(n log n ) space, where n is the total number of vertices of all obstacles. Recently, by modifying Hershberger and Suri’s algorithm, Wang (in SODA’21) reduced the space to O(n) while the runtime of the algorithm is still O(n log n ). In this article, we present a new algorithm of O(n+h log h ) time and O(n) space, provided that a triangulation of the free space is given, where h is the number of obstacles. The algorithm is better than the previous work when h is relatively small. Our algorithm builds a shortest path map for a source point s so that given any query point t , the shortest path length from s to t can be computed in O (log n ) time and a shortest s - t path can be produced in additional time linear in the number of edges of the path. 

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3. Multi-Objective Heuristics For Network Construction In An Obstacle-Dense Environment

   https://doi.org/10.1109/CASE59546.2024.10711579  

   Bernardini, Francesco; Becker, Aaron T (August 2024, IEEE)

   We present a heuristic method to construct an optimal communication network in an obstacle-dense environment. A set of immobile terminals must be connected by a network of straight-line edges by adding agents to serve as relays. Obstacles are represented by polygons, unaccessible by the agents of the network or by the edges. The problem with obstacles is reduced to a problem without obstacles by choosing the nodes of the optimal network among the obstacles’ vertices that are in mutual line of sight. A second heuristic method is developed to solve the bicriteria optimization problem with number of agents and length of the network as concurrent costs. 

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4. A* for Bounding Shortest Paths in the Graphs of Convex Sets

   https://doi.org/10.1609/icaps.v35i1.36127  

   Sundar, Kaarthik; Rathinam, Sivakumar (September 2025, Proceedings of the International Conference on Automated Planning and Scheduling)

   We present a novel algorithm that fuses the existing convex-programming based approach with heuristic information to find optimality guarantees and near-optimal paths for the Shortest Path Problem in the Graph of Convex Sets (SPP-GCS). Our method, inspired by A* initiates a best-first-like procedure from a designated subset of vertices and iteratively expands it until further growth is neither possible nor beneficial. Traditionally, obtaining solutions with bounds for an optimization problem involves solving a relaxation, modifying the relaxed solution to a feasible one, and then comparing the two solutions to establish bounds. However, for SPP-GCS, we demonstrate that reversing this process can be more advantageous, especially with Euclidean travel costs. In other words, we initially employ A* to find a feasible solution for SPP-GCS, then solve a convex relaxation restricted to the vertices explored by A* to obtain a relaxed solution, and finally, compare the solutions to derive bounds. We present numerical results to highlight the advantages of our algorithm over the existing approach in terms of the sizes of the convex programs solved and computation time. 

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5. Hardness of Motion Planning with Obstacle Uncertainty in Two Dimensions

   https://doi.org/10.1177/0278364921992787  

   Shimanuki, Luke; Axelrod, Brian (February 2021, The International Journal of Robotics Research)

   We consider the problem of motion planning in the presence of uncertain obstacles, modeled as polytopes with Gaussian-distributed faces (PGDFs). A number of practical algorithms exist for motion planning in the presence of known obstacles by constructing a graph in configuration space, then efficiently searching the graph to find a collision-free path. We show that such an exact algorithm is unlikely to be practical in the domain with uncertain obstacles. In particular, we show that safe 2D motion planning among PGDF obstacles is [Formula: see text]-hard with respect to the number of obstacles, and remains [Formula: see text]-hard after being restricted to a graph. Our reduction is based on a path encoding of MAXQHORNSAT and uses the risk of collision with an obstacle to encode variable assignments and literal satisfactions. This implies that, unlike in the known case, planning under uncertainty is hard, even when given a graph containing the solution. We further show by reduction from [Formula: see text]-SAT that both safe 3D motion planning among PGDF obstacles and the related minimum constraint removal problem remain [Formula: see text]-hard even when restricted to cases where each obstacle overlaps with at most a constant number of other obstacles. 

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