Thick Non-crossing Paths

We consider the problem of finding shortest non-crossing thick paths in a polygonal domain, where a thick path is a Minkowski sum of a usual path and unit disk. We show that in a simple polygon, K shortest paths may be found in optimal O(K(n + K)) time and O(n + K) space. For polygons with holes we show that the problem becomes (weakly) NP-hard even for the case K = 2 and even when the paths are restricted to be monotone, if we wish to bound the length of the longest of the paths. We also observe that unless P=NP there exists no fully polynomial time approximation scheme for the problem. For the case K = 2 and L1 metric we suggest a pseudo-polynomial time algorithm. Next, we consider a special case of the minimum cost rectilinear flow problem. We observe that under several restrictions, the paths of minimum total length may be found in polynomial time by reducing the shortest paths problem to the flow problem in a path-preserving graph.