An O*(2^n ) Algorithm for Graph Coloring and Other Partitioning Problems via Inclusion--Exclusion

We use the principle of inclusion and exclusion, combined with polynomial time segmentation and fast Mobius transform, to solve the generic problem of summing or optimizing over the partitions of n elements into a given number of weighted subsets. This problem subsumes various classical graph partitioning problems, such as graph coloring, domatic partitioning, and MAX k-CUT, as well as machine learning problems like decision graph learning and model-based data clustering. Our algorithm runs in O*(2^n ) time, thus substantially improving on the usual O*(3^n )-time dynamic programming algorithm; the notation O* suppresses factors polynomial in n. This result improves, e.g., Byskov's recent record for graph coloring from O*(2.4023^n ) to O*(2^n ). We note that twenty five years ago, R. M. Karp used inclusion--exclusion in a similar fashion to reduce the space requirement of the usual dynamic programming algorithms from exponential to polynomial.