We prove that it is NP-hard to decide whether two points in a polygonal domain with holes can be connected by a wire. This implies that finding any approximation to the shortest path for a long snake amidst polygonal obstacles is NP-hard. On the positive side, we show that snake's problem is "length-tractable": if the snake is "fat", i.e., its length width ratio is small, the shortest path can be computed in polynomial time.
In robotics thick paths were studied as routes for a circular robot. In this context, path self-overlap poses no problem as even a self-overlapping path may be traversed by the robot; that is, in contrast to VLSI, robotics research should not care about finding non-selfoverlapping paths. In @cite_2 , Chew gave an efficient algorithm for finding a shortest thick path in a polygonal domain. In a sense, our algorithm for shortest path for a short snake () may be viewed as an extension of Chew's.
@cite_2
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Many algorithms have been proposed that create a path for a robot in an environment with obstacles. Most methods are aimed at finding a solution. However, for many applications, the path must be of a good quality as well. That is, a path should be short and should keep some amount of minimum clearance to the obstacles. Traveling along such a path reduces the chances of collisions due to the difficulty of measuring and controlling the precise position of the robot. This paper reports a new technique, called Partial shortcut, which decreases the path length. While current methods have difficulties in removing all redundant motions, the technique efficiently removes these motions by interpolating one degree of freedom at a time. Two algorithms are also studied that increase the clearance along paths. The first one is fast but can only deal with rigid, translating bodies. The second algorithm is slower but can handle a broader range of robots, including three-dimensional free-flying and articulated robots, which may reside in arbitrary high-dimensional configuration spaces. A big advantage of these algorithms is that clearance along paths can now be increased efficiently without using complex data structures and algorithms. Finally, we combine the two criteria and show that high-quality paths can be obtained for a broad range of robots. Consider a setting where robots must visit sites represented as nodes in a graph, but each robot may fail when traversing an edge. The goal is to find a set of paths for a team of robots which maximizes the expected number of nodes collectively visited, while guaranteeing that the paths satisfy a notion of “independence” formalized by a matroid (e.g. limits on team size, number of visits to regions), and that the probabilities that each robot survives to its destination are above a given threshold. We call this problem the Matroid Team Surviving Orienteers (MTSO) problem, which has broad applications such as environmental monitoring in risky regions and search and rescue in dangerous conditions. We present the MTSO formally and detail numerous examples of matroids in a path planning context. We then propose an approximate greedy algorithm for selecting a feasible set of paths and prove that the value of the output is within a factor p s p s + λ of the optimum, where p s is the per-robot survival probability threshold and 1 λ ≤ 1 is the approximation factor of an oracle routine for the well known orienteering problem. We demonstrate the efficiency of our approach by applying it to a scenario where a team of robots must gather information while avoiding pirates in the Coral Triangle. In this paper, we study the structure and computational complexity of optimal multi-robot path planning problems on graphs. Our results encompass three formulations of the discrete multi-robot path planning problem, including a variant that allows synchronous rotations of robots along fully occupied, disjoint cycles on the graph. Allowing rotation of robots provides a more natural model for multi-robot path planning because robots can communicate. Our optimality objectives are to minimize the total arrival time, the makespan (last arrival time), and the total distance. On the structure side, we show that, in general, these objectives demonstrate a pairwise Pareto optimal structure and cannot be simultaneously optimized. On the computational complexity side, we extend previous work and show that, regardless of the underlying multi-robot path planning problem, these objectives are all intractable to compute. In particular, our NP-hardness proof for the time optimal versions, based on a minimal and direct reduction from the 3-satisfiability problem, shows that these problems remain NP-hard even when there are only two groups of robots (i.e. robots within each group are interchangeable). We present new techniques for establishing lower bounds in robot motion planning problems. Our scheme is based on path encoding and uses homotopy equivalence classes of paths to encode state. We first apply the method to the shortest path problem in 3 dimensions. The problem is to find the shortest path under an Lp metric (e.g. a euclidean metric) between two points amid polyhedral obstacles. Although this problem has been extensively studied, there were no previously known lower bounds. We show that there may be exponentially many shortest path classes in single-source multiple-destination problems, and that the single-source single-destination problem is NP-hard. We use a similar proof technique to show that two dimensional dynamic motion planning with bounded velocity is NP-hard. Finally we extend the technique to compliant motion planning with uncertainty in control. Specifically, we consider a point in 3 dimensions which is commanded to move in a straight line, but whose actual motion may differ from the commanded motion, possibly involving sliding against obstacles. Given that the point initially lies in some start region, the problem of finding a sequence of commanded velocities which is guaranteed to move the point to the goal is shown to be non-deterministic exponential time hard, making it the first provably intractable problem in robotics.