Submodular Approximation: Sampling-based Algorithms and Lower Bounds

We introduce several generalizations of classical computer science problems obtained by replacing simpler objective functions with general submodular functions. The new problems include submodular load balancing, which generalizes load balancing or minimum-makespan scheduling, submodular sparsest cut and submodular balanced cut, which generalize their respective graph cut problems, as well as submodular function minimization with a cardinality lower bound. We establish upper and lower bounds for the approximability of these problems with a polynomial number of queries to a function-value oracle. The approximation guarantees for most of our algorithms are of the order of n/ln n. We show that this is the inherent difficulty of the problems by proving matching lower bounds. We also give an improved lower bound for the problem of approximately learning a monotone submodular function. In addition, we present an algorithm for approximately learning submodular functions with special structu...

• Type: Conference
• Year: 2008
• Where: FOCS
• Authors: Zoya Svitkina, Lisa Fleischer
• Keywords: FOCS 2008, General Submodular Functions, Lower Bounds, Submodular Functions, Theoretical Computer Science