Provable Submodular Minimization via Wolfe’s Algorithm


March 10, 2015


Deeparnab Chakrabarty




Owing to several applications in large scale learning and vision problems, fast submodular function minimization (SFM) has become a critical problem. Theoretically, unconstrained SFM can be performed in polynomial time. However, these algorithms are typically not practical. In 1976, Wolfe proposed an algorithm to find the minimum Euclidean norm point in a polytope, and in 1980, Fujishige showed how Wolfe’s algorithm can be used for SFM. For general submodular functions, this Fujishige-Wolfe minimum norm algorithm seems to have the best empirical performance.

Despite its good practical performance, very little is known about Wolfe’s minimum norm algorithm theoretically. To our knowledge, the only result is an exponential time analysis due to Wolfe himself. In this paper we give a maiden convergence analysis of Wolfe’s algorithm. We prove that in t iterations, Wolfe’s algorithm returns an O(1/t)-approximate solution to the min-norm point on any polytope. We also prove a robust version of Fujishige’s theorem which shows that an O(1/n2)-approximate solution to the min-norm point on the base polytope implies exact submodular minimization. As a corollary, we get the first pseudo-polynomial time guarantee for the Fujishige-Wolfe minimum norm algorithm for unconstrained submodular function minimization.


Deeparnab Chakrabarty

I am currently a researcher in the Algorithms group at Microsoft Research, India. My main research interests are approximation algorithms, combinatorial optimization, and algorithmic economics.

Prior to moving to MSRI, I spent some time at the Dept. of CIS at UPenn and the Department of Combinatorics and Optimization at the University of Waterloo. Before that I was a graduate student at the College of Computing, Georgia Tech and an undergraduate one at the Indian Institute of Technology, Bombay in 2003.