Essentials of Tropical Combinatorics

The goal of this book is to explain, at the graduate student level, how tropical geometry can be accessed via geometric combinatorics. This way the book offers a viable path to a topic of very active research. At the same time the reader learns how a number of seemingly unrelated combinatorial results fall into place, once viewed through the "tropical lens". No attempt is made to cover the entire field of tropical geometry, which has been evolving too rapidly anyway to be covered by a book so small. The book's central concept is the "tropical convexity" introduced by Develin and Sturmfels, which is a version of "tropical linear algebra". This is used in the book as a general language to study classical subjects in combinatorial optimization including shortest paths, the assignment problem, the even dicycle problem, flow-type problems, and others. This book focuses on the polyhedral and combinatorial aspects of tropical geometry while requiring less prerequisites in algebraic geometry and commutative algebra, thus making the book more accessible to a wider audience. The main requirement beyond general mathematical maturity is a basic knowledge in polytope theory.