Partitioning planar graphs: a fast combinatorial approach for max-cut

Abstract

The max-cut problem asks for partitioning the nodes V of a graph G=(V,E) into two sets (one of which might be empty), such that the sum of weights of edges joining nodes in different partitions is maximum. Whereas for general instances the max-cut problem is NP-hard, it is polynomially solvable for certain classes of graphs. For planar graphs, there exist several polynomial-time methods determining maximum cuts for arbitrary choice of edge weights. Typically, the problem is solved by computing a minimum-weight perfect matching in some associated graph. The most efficient known algorithms are those of Shih et al. (IEEE Trans. Comput. 39(5):694–697, 1990) and that of Berman et al. (WADS, Lecture Notes in Computer Science, vol. 1663, pp. 25–36, Springer, Berlin, 1999). The running time of the former can be bounded by \(O(|V|^{\frac{3}{2}}\log|V|)\). The latter algorithm is more generally for determining T-joins in graphs. Although it has a slightly larger bound on the running time of \(O(|V|^{\frac{3}{2}}(\log|V|)^{\frac{3}{2}})\alpha(|V|)\), where α(|V|) is the inverse Ackermann function, it can solve large instances in practice.In this work, we present a new and simple algorithm for determining maximum cuts for arbitrary weighted planar graphs. Its running time is bounded by \(O(|V|^{\frac{3}{2}}\log|V|)\), the same bound achieved by Shih et al. It can easily determine maximum cuts in huge random as well as real-world graphs with up to 106 nodes. We present experimental results for our method using two different matching implementations. We furthermore compare our approach with those of Shih et al. and Berman et al. It turns out that our algorithm is considerably faster in practice than Shih et al. Moreover, it yields a much smaller associated graph. Its expanded graph size is comparable to that of Berman et al. However, whereas the procedure of generating the expanded graph in Berman et al. is very involved (thus needs a sophisticated implementation), implementing our approach is an easy and straightforward task.