Advancements in Combinatorial Optimization and Spectral Graph Theory: Approaches to Permanents, Metric Embeddings, and Geometric Graphs

Abstract

In this thesis I will cover the five research papers that I co-authored during my Ph.D. studies. First, I will cover my results on planar and geometric graphs. Namely, in the papers ``On planar graphs of uniform polynomial growth'' and ``Non-existence of annular separators in geometric graphs'' we resolve several conjectures and open problems about spectral properties of a certain family of geometric graphs with uniform polynomial growth. Next, I will cover the paper ``Multiscale entropic regularization for MTS on general metric spaces'' which positively answers an open question in online algorithms by providing an $O(\log^n)$-competitive algorithm for the metrical task systems problem which is purely based on a mirror descent framework. Next, I will focus on our result on ``Counting and sampling perfect matchings in regular expanding non-bipartite graphs'' in which we provide algorithms to efficiently sample perfect matchings in those families of graphs. Finally, I will discuss our latest result, ``On approximability of the permanent of PSD matrices'', in which we improve both the lower-bound and upper-bounds for approximating the permanent of PSD matrices.