Computing a multiway cut
With two labels:  classical min-cut problem
•Solvable by standard network flow algorithms
–polynomial time in theory, nearly linear in practice
More than 2 labels: NP-hard [Dahlhaus et al., STOC ‘92]
•But efficient approximation algorithms exist
–Within a factor of 2 of optimal
–Computes local minimum in a strong sense
»even very large moves will not improve the energy
–Yuri Boykov, Olga Veksler and Ramin Zabih, Fast Approximate Energy Minimization via Graph Cuts, International Conference on Computer Vision, September 1999.
•Basic idea
–reduce to a series of 2-way-cut sub-problems, using one of:
»swap move:  pixels with label l1 can change to l2, and vice-versa
»expansion move:  any pixel can change it’s label to l1