New Exact Algorithms for the Two-Sided Bipartite Crossing Minimization Problem

The Two-sided Two-layer Crossing Minimization (TSCM) problem calls for minimizing a quadratic objective function subject to a set of ordering constraints, where all variables are binary. This NP-hard problem has important applications in graph drawing as a key step in drawing hierarchical graphs. In this paper, we propose two new exact algorithms for the TSCM problem. One is an LP-based branch-and-cut algorithm and the other is a hybrid algorithm integrating the strengths of integer and linear programming and constraint programming. We report exact solutions for various types of graphs including instances which are not solvable by the previous branch-and-cut algorithm. We show that our algorithms are capable of solving reasonable-size instances and very effective to compute optimal solutions for small sparse graphs.