Submodular goal value of Boolean functions

Submodular goal value of Boolean functions

Recently, Deshpande et al. introduced a new measure of the complexity of a Boolean function. We call this measure the “goal value” of the function. The goal value of f is defined in terms of a monotone, submodular utility function associated with f . As shown by Deshpande et al., proving that a Boolean function f has small goal value can lead to a good approximation algorithm for the Stochastic...

On submodular value functions of dynamic programming

We investigate in this paper submodular properties of the value function arrizing in complex Dynamic programming (DPs). We consider in particular DPs that include concatenation and linear combinations of standard DP operators, as well as combination of maximizations and minimizations. These DPs have many applications and interpretations, both in stochastic control (and stochastic zero-sum games...

Learning pseudo-Boolean k-DNF and submodular functions

We prove that any submodular function f : {0, 1}n → {0, 1, ..., k} can be represented as a pseudoBoolean 2k-DNF formula. Pseudo-Boolean DNFs are a natural generalization of DNF representation for functions with integer range. Each term in such a formula has an associated integral constant. We show that an analog of Håstad’s switching lemma holds for pseudo-Boolean k-DNFs if all constants associ...

On the Goal Value of a Boolean Function

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 log n.E + nlogn) where E is the time required to evaluate the function and n is the number of variables [32]. On the other hand, many computer vision and machine learning problems are defined...