Efficient Minimization of Higher Order Submodular Functions using Monotonic Boolean Functions

Submodular function minimization is a key problem in a wide variety of applications in machine learning, economics, game theory, computer vision and many others. The general solver has a complexity of $O(n^6+n^5L)$ where $L$ is the time required to evaluate the function and $n$ is the number of variables \cite{orlin09}. On the other hand, many useful applications in computer vision and machine learning applications are defined over a special subclasses of submodular functions in which that can be written as the sum of many submodular cost functions defined over cliques containing few variables. In such functions, the pseudo-Boolean (or polynomial) representation \cite{BorosH02} of these subclasses are of degree (or order, or clique size) $k$ where $k<<n$. In this work, we develop efficient algorithms for the minimization of this useful subclass of submodular functions. To do this, we define novel mapping that transform submodular functions of order $k$ into quadratic ones, which can be efficiently minimized in $O(n^3)$ time using a max-flow algorithm. The underlying idea is to use auxiliary variables to model the higher order terms and the transformation is found using a carefully constructed linear program. In particular, we model the auxiliary variables as monotonic Boolean functions, allowing us to obtain a compact transformation using as few auxiliary variables as possible. Specifically, we show that our approach for fourth order function requires only 2 auxiliary variables in contrast to 30 or more variables used in existing approaches. In the general case, we give an upper bound for the number or auxiliary variables required to transform a function of order $k$ using Dedekind number, which is substantially lower than the existing bound of $2^{2^k}$.