Pedigree Polytopes

📘 About This Book

This book defines and studies a combinatorial object called the pedigree and develops the theory for optimising a linear function over the convex hull of pedigrees (the Pedigree polytope). A strongly polynomial algorithm implementing the framework given in the book for checking membership in the pedigree polytope is a major contribution. This book challenges the popularly held belief in computer science that a problem included in the NP-complete class may not have a polynomial algorithm to solve. By showing STSP has a polynomial algorithm, this book settles the P vs NP question. This book has illustrative examples, figures, and easily accessible proofs for showing this unexpected result. This book introduces novel constructions and ideas previously not used in the literature. Another interesting feature of this book is it uses basic max-flow and linear multicommodity flow algorithms and concepts in these proofs establishing efficient membership checking for the pedigree polytope. Chapters 3-7 can be adopted to give a course on Efficient Combinatorial Optimization. This book is the culmination of the author's research that started in 1982 through a presentation on a new formulation of STSP at the XIth International Symposium on Mathematical Programming at Bonn.

📖 Summary

Pedigree Polytopes, authored by Tirukkattuppalli Subramanyam Arthanari and published in 2023, is a 235-page advanced computer science text that introduces and investigates a novel combinatorial object known as the pedigree. At its core, the book develops a comprehensive theoretical framework for optimizing a linear function over the convex hull of these pedigrees, which the author designates as the pedigree polytope. By diving deep into this mathematical structure, Arthanari establishes foundational concepts that challenge long-standing assumptions within the computational sciences. A major contribution of this work is the introduction of a strongly polynomial algorithm designed to implement the framework for checking membership within the pedigree polytope. This algorithmic breakthrough directly challenges the widely held computer science belief that problems classified as NP-complete cannot possess a polynomial-time solution. Most notably, the book attempts to settle the famous P versus NP question by demonstrating that the symmetric traveling salesperson problem, or STSP, actually has a polynomial algorithm. Throughout the text, the author supports these paradigm-shifting claims using illustrative examples, helpful figures, and proofs that are made accessible to readers willing to engage with the material. By introducing entirely novel constructions and fresh ideas that have never before appeared in academic literature, the book opens up radical new directions for research in combinatorial optimization and computational complexity theory. While the text deals with highly dense and abstract theoretical concepts, it maintains a structured approach to explaining its mechanics. The exploration of pedigree polytopes not only bridges gaps in current optimization theory but also invites researchers, students, and professionals to rethink the boundaries of algorithmic efficiency. Ultimately, this publication stands as a provocative and deeply technical contribution to the field of computer science, offering a bold perspective on one of the greatest unsolved problems in mathematics and computation.

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