Algorithmic Results for Clustering and Refined Physarum Analysis

In the first part of this thesis, we study the Binary -Rank- problem which given a binary matrix and a positive integer , seeks to find a rank- binary matrix minimizing the number of non-zero entries of . A central open question is whether this problem admits a polynomial time approximation scheme. We give an affirmative answer to this question by designing the first randomized almost-linear time approximation scheme for constant over the reals, , and the Boolean semiring. In addition, we give novel algorithms for important variants of -low rank approximation. The second part of this dissertation, studies a popular and successful heuristic, known as Approximate Spectral Clustering (ASC), for partitioning the nodes of a graph into clusters with small conductance. We give a comprehensive analysis, showing that ASC runs efficiently and yields a good approximation of an optimal -way node partition of . In the final part of this thesis, we present two results on slime mold computations: i) the continuous undirected Physarum dynamics converges for undirected linear programs with a non-negative cost vector; and ii) for the discrete directed Physarum dynamics, we give a refined analysis that yields strengthened and close to optimal convergence rate bounds, and shows that the model can be initialized with any strongly dominating point.