Energy minimization via graph cuts: settling what is possible

The recent explosion of interest in graph cut methods in computer vision naturally spawns the question: what energy functions can be minimized via graph cuts? This question was first attacked by two papers of Kolmogorov and Zabih, in which they dealt with functions with pair-wise and triplewise pixel interactions. In this work, we extend their results in two directions. First, we examine the case of k-wise pixel interactions; the results are derived from a purely algebraic approach. Second, we discuss the applicability of provably approximate algorithms. Both of these developments should help researchers best understand what can and cannot be achieved when designing graph cut based algorithms.