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Sam Kim Miller

Picture of Sam Kim Miller.

Thesis

Combinatorial Polynomial Hirsch Conjecture

Advisor
Mohamed Omar
Second Reader(s)
Nicholas J. Pippenger

Abstract

The Hirsch Conjecture states that for a $d$-dimensional polytope with $n$ facets, the diameter of the graph of the polytope is at most $n-d$. This conjecture was disproven in 2010 by Francisco Santos Leal. However, a polynomial bound in $n$ and $d$ on the diameter of a polytope may still exist. Finding a polynomial bound would provide a worst-case scenario runtime for the Simplex Method of Linear Programming. However working only with polytopes in higher dimensions can prove challenging, so other approaches are welcome. There are many equivalent formulations of the Hirsch Conjecture, one of which is the Combinatorial Polynomial Hirsch Conjecture, which turns the problem into a matter of counting sets.

This thesis explores the Combinatorial Polynomial Hirsch Conjecture.

Additional Materials

Poster