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Let G be a connected planar (but not yet embedded) graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of G + F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. An optimal solution to this problem approximates the crossing number of the graph G + F .Finding an exact solution to MEI is NP-hard for general F , but linear time solvable for the special case of |F | = 1 (SODA 01, Algorithmica) or when all of F are incident to a new vertex (SODA 09).The complexity for general F but with constant k = |F | was open, but algorithms both with relative and absolute approximation guarantees have been presented (SODA 11, ICALP 11). We show that the problem is fixed parameter tractable (FPT) in k forWe also mention that while the crossing number itself is in FPT w.r.t. the objective value [18,26], already a planar graph with one added edge may have unbounded crossing number.Both the aforementioned absolute MEI-approximation [10] and our new approach can use [11] to obtain the same relative ratio for approximating the crossing number. However, our new approach does so without any additional additive term:Corollary 2. Using the theorem relating an optimum MEI(G, F ) solution to the crossing number of the graph G + F [11], Theorem 1 gives a polynomial time k∆-approximation for the crossing number of G + F with constant k = |F |, where G is a planar graph and ∆ is its maximum degree.

An upward drawing of a DAG G is a drawing of G in which all arcs are drawn as curves increasing monotonically in the vertical direction. In this article, we present a new approach for upward crossing minimization, that is, finding an upward drawing of a DAG G with as few crossings as possible. Our algorithm is based on a two-stage upward planarization approach, which computes a feasible upward planar subgraph in the first step and reinserts the remaining arcs by computing constraint-feasible upward insertion paths. An experimental study shows that the new algorithm leads to much better results than existing algorithms for upward crossing minimization, including the classical Sugiyama approach.

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We consider the Steiner tree problem under a two-stage stochastic model with recourse and finitely many scenarios. In this problem, edges are purchased in the first stage when only probabilistic information on the set of terminals and the future edge costs is known. In the second stage, one of the given scenarios is realized and additional edges are puchased in order to interconnect the set of (now known) terminals. The goal is to decide on the set of edges to be purchased in the first stage while minimizing the overall expected cost of the solution. We provide a new semi-directed cut-set based integer programming formulation, which is stronger than the previously known undirected model. We suggest a two-stage branch-and-cut (B&C) approach in which L-shaped and integer-L-shaped cuts are generated. In our computational study we compare the performance of two variants of our algorithm with that of a B&C algorithm for the extensive form of the deterministic equivalent (EF). We show that, as the number of scenarios increases, the new approach significantly outperforms the (EF) approach.

Abstract. Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of G + F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. Finding an exact solution to MEI is NP-hard for general F . We present the first polynomial time algorithm for MEI that achieves an additive approximation guarantee -depending only on the size of F and the maximum degree of G, in the case of connected G. Our algorithm seems to be the first directly implementable one in that realm, too, next to the single edge insertion. It is also known that an (even approximate) solution to the MEI problem would approximate the crossing number of the F -almost-planar graph G+F , while computing the crossing number of G+F exactly is NP-hard already when |F | = 1. Hence our algorithm induces new, improved approximation bounds for the crossing number problem of F -almost-planar graphs, achieving constant-factor approximation for the large class of such graphs of bounded degrees and bounded size of F .