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Smoothed Analysis of Algorithms and the Fujishige-Wolfe Algorithm to Minimize Submodular Functions

We study the method of smoothed analysis for understanding the complexity of algorithms. We first discuss the need for defining an alternative to worst case complexity and discuss some alternatives considered in analysis of algorithms. We then discuss smoothed analysis, introduced by Spielman and Teng [22] and describe how one could do a smoothed analysis of an algorithm. There are a few design questions here which we attempt to answer. As an instance we study the time complexity of the Simplex method with Two-Phase Shadow Vertex Pivot rule which was shown to have a polynomial smoothed complexity by Spielman and Teng [22]. We intend to apply smoothed analysis to analyze the time complexity of of a well known algorithm in combinatorial optimization for submodular function minimization, the Fujishige-Wolfe min-norm algorithm [9] . The time complexity has been unknown for a long time, although it has been shown to perform extremely well in practice. We study the Fujishige-Wolfe method to minimize a submodular function and prove properties of the algorithm that make the details clear. We then consider an approach to do a smoothed complexity analysis of this algorithm.