The Complexity of Separating Points in the Plane

We study the following separation problem: given $$n$$n connected curves and two points $$s$$s and $$t$$t in the plane, compute the minimum number of curves one needs to retain so that any path connecting $$s$$s to $$t$$t intersects some of the retained curves. We give the first polynomial $$(\fancyscript{O}(n^3))$$(O(n3)) time algorithm for the problem, assuming that the curves have reasonable computational properties. The algorithm is based on considering the intersection graph of the curves, defining an appropriate family of closed walks in the intersection graph that satisfies the 3-path-condition, and arguing that a shortest cycle in the family gives an optimal solution. The 3-path-condition has been used mainly in topological graph theory, and thus its use here makes the connection to topology clear. We also show that the generalized version, where several input points are to be separated, is NP-hard for natural families of curves, like segments in two directions or unit circles.