The Directed Planar Reachability Problem

The Planar k-Means Problem is NP-Hard

“…Let E be a planar combinatorial embedding of G; such an embedding can be obtained in polynomial time, and even in log space. (See for instance [1].) E corresponds to some plane drawing of G and specifies, for each vertex v, the cyclic ordering of the edges incident on v in this drawing.…”

“…Consider a plane drawing of H where vertices are embedded at points on an integer grid, and edges are embedded as rectilinear paths. Such a drawing can be obtained in polynomial time [15,21], and even in logarithmic space [1] For each vertex x ∈ X (and hence for each variable in F ), this process distorts the cycle C x in H into a circuit t. Let t i denote the circuit corresponding to variable x i . Since t i is a rectilinear circuit on a grid, it is of even length.…”

“…Given an instance I of planar 3-SAT, we construct a planar graph from I, embed it in an integer grid using the technique of [1], and construct the k-means instance from this grid embedding. (See Figure 4.) While clustering problems generally tend to be NP-hard even in the plane, there are surprising exceptions -the problem of covering a set of points by k balls so as to minimize the sum of the radii of the balls admits a polynomial time algorithm if we use L 1 balls, and a (1 + ε)-approximation algorithm that runs in time polynomial in the input size and log 1 ε for the usual Euclidean balls [10].…”

Space Complexity of the Directed Reachability Problem over Surface-Embedded Graphs

“…Subsequently in [BTV09], we show how to extend this weight function to general grid graphs (without restriction on the direction of edges). This implied that directed planar reachability is in UL since the directed planar reachability problem is known to be reducible to the grid graph reachability problem [ADR05]. In fact this even implied that the reachability problem for graphs embedded on constant genus surfaces and graphs that are K 3,3 -free and K 5 -free are in UL since the reachability problem for these classes of graphs reduces to the directed planar reachability problem [KV10,TW09] in log-space.…”

Connectivity check in 3-connected planar graphs with obstacles

“…The problem of reachability in directed planar graphs is not easy. It is known to be LOGSPACE-hard [1]. It can be solved efficiently with labelling schemes.…”