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Abstract
This paper presents a new formulation and solution of a mixed-integer program for the hierarchical orthogonal hypergraph drawing problem, and the number of hyperedge crossings is minimized. The novel feature of the model is in combining several stages of the Sugiyama framework for graph drawing: vertex ordering, the assignment of vertices’ x-coordinates, and orthogonal hyperedge routing. The hyperedges of a hypergraph are assumed to be multi-source and multi-target, and vertices are depicted as rectangles with ports on their top and bottom sides. Such hypergraphs are used in data-flow diagrams and in a scheme of cooperation. The numerical results demonstrate the correctness and effectiveness of the proposed approach compared to mathematical heuristics. For instance, the proposed exact approach yields a 67.3% reduction of the number of crossings compared to that obtained by using a mathematical heuristic for a dataset of non-planar graphs.
Highlights
Mathematics 2022, 10, 689. https://A new optimization model for the layout problem of a hierarchical orthogonal hypergraph is discussed in this paper
This paper presents a new formulation and solution of a mixed-integer program for the hierarchical orthogonal hypergraph drawing problem, and the number of hyperedge crossings is minimized
Three approaches were applied to obtain the numerical results, namely, the mathematical heuristic, mixed-integer program (5)–(21) with a higher priority of the objective function (5), and the same mixed-integer program enhanced by an initial feasible solution derived with the mathematical heuristic approach
Summary
A new optimization model for the layout problem of a hierarchical orthogonal hypergraph is discussed in this paper. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations To make this algorithm applicable to the hypergraph layout problem, Sander [6] first proposed reordering the nodes on each layer to minimize the number of edge crossings, and calculated the preliminary coordinates for the nodes and segments to avoid hyperedge crossings, obtaining the coordinates that rendered a balanced drawing. An additional optimization problem arises if one takes ports for the source and target nodes into account because each vertical segment of a hyperedge must be assigned to a unique port This problem should be considered as a part of a hierarchical hypergraph drawing problem, together with the reordering of nodes and horizontal segments. There are some novel approaches to nanostructures and material science with application of graph theory [13,14], where there is a large variety of applications connected with hierarchical hypergraphs with orthogonal hyperedges
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