Combinatorics of Polynomial Iterations

A Unified Approach for Design of Lp Polynomial Algorithms

By summarizing Khachiyan's algorithm and Karmarkar's algorithm forlinear program (LP) a unified methodology for the design of polynomial-time algorithms for LP is presented in this paper. A key concept is the so-called extended binary search (EBS) algorithm introduced by the author. It is used as a unified model to analyze the complexities of the existing modem LP algorithms and possibly, help ...

The minimal polynomial of 2cos(π/p) and its application to regular maps of large planar width

Closed-form expansions for the universal edge elimination polynomial

We establish closed-form expansions for the universal edge elimination polynomial of paths and cycles and their generating functions. This includes closed-form expansions for the bivariate matching polynomial, the bivariate chromatic polynomial, and the covered components polynomial.

A note on the cycle index polynomial of the symmetric group

An identity concerning the cycle index polynomial of the symmetric group is proved and a consequence of it presented.

An Exposition of Sanders' Quasi-Polynomial Freiman-Ruzsa Theorem

The polynomial Freiman-Ruzsa conjecture is one of the most important conjectures in additive combinatorics. It asserts that one can switch between combinatorial and algebraic notions of approximate subgroups with only a polynomial loss in the underlying parameters. This conjecture has also found several applications in theoretical computer science. Recently, Tom Sanders proved a weaker version ...