Efficient Extraction of Multiple Kuratowski Subdivisions

Abstract

A graph is planar if and only if it does not contain a Kuratowski subdivision. Hence such a subdivision can be used as a witness for non-planarity. Modern planarity testing algorithms allow to extract a single such witness in linear time. We present the first linear time algorithm which is able to extract multiple Kuratowski subdivisions at once. This is of particular interest for, e.g., Branch-and-Cut algorithms which require multiple such subdivisions to generate cut constraints. The algorithm is not only described theoretically, but we also present an experimental study of its implementation.

Exact Algorithms for the Maximum Planar Subgraph Problem

“…Although negative, this is an interesting observation. We should keep in mind that the thereby required efficient separation builds upon years of algorithmic development [3,12], and it is the only ILP where we currently know how to (heuristically) separate on fractional solutions. Equipped with similar tools, i.e., a sensible rounding scheme and a linear time separation routine (a modified left-right planarity test), the left-right edge coloring formulation might yield very competitive performance.…”

“…Instead, a sufficiently large but in many practical cases small subset of constraints is identified by a (heuristic) separation procedure. Over the years, the performance of this approach was improved by strong preprocessing [7], finding multiple violated constraints in linear time [12], and good heuristics [11].…”

Cycles to the Rescue! Novel Constraints to Compute Maximum Planar Subgraphs Fast

“…Still, since it has these guarantees on integral solutions, it suffices to obtain an exact algorithm. Over the years, the performance of this approach was improved by strong preprocessing [4], finding multiple Kuratowski subdivision in linear time [11], and strong primal heuristics [10]. We use all these identically in all considered algorithms.…”

“…Adding (D 2 + D)(n − 2) to both sides and afterwards dividing by D(D − 1) gives the right-hand sides of equation (11), divided by factor D − 1 (resp. D).…”

Computing Maximum C-Planar Subgraphs

“…Separating the connectivity constraints can be done in polynomial time by computing minimum cuts on the graph, using the fractional solution as edge capacities. On the other hand, there are no known polynomial algorithms for the Kuratowski constraint separation, and we have to resort to a heuristic routine, similar to the ones described in [12]: we round the fractional solution to an integer solution, which we can interpret as our support graph S, and search for Kuratowski subdivisions in S. For any such subdivision K we can test whether the current fractional solution violates the constraint induced by K. Traditional planarity test algorithms can extract a single Kuratowski subdivision per run; in our experiments we use the extended test algorithm presented in [1] which extracts multiple different subdivisions in linear time. Note that we separate all cut constraints before separating any Kuratowski constraints.…”