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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.