Tropical Circuit Complexity

Tropical Circuit Complexity

Springer Nature, 2023, X+119 pages, 19 b/w illustrations

Book series: SpringerBriefs in Mathematics, ISBN 978-3-031-42353-6.

Frontmatter [PDF] List of Errata 
Supplementary material

About this book (prepared by Springer team):

This book presents an enticing introduction to tropical circuits and their use as a rigorous mathematical model for dynamic programming (DP), which is one of the most fundamental algorithmic paradigms for solving combinatorial, discrete optimization problems.

In DP, an optimization problem is broken up into smaller subproblems that are solved recursively(1). Many classical DP algorithms are pure in that they only use the basic (min,+) or (max,+) operations in their recursion equations. In tropical circuits, these operations are used as gates. Thanks to the rigorous combinatorial nature of tropical circuits, elements from the Boolean and arithmetic circuit complexity can be used to obtain lower bounds for tropical circuits, which play a crucial role in understanding the limitations and capabilities of these computational models. This book aims to offer a toolbox for proving lower bounds on the size of tropical circuits.

In this work, the reader will find lower-bound ideas and methods that have emerged in the last few years, with detailed proofs. Largely self-contained, this book is meant to be approachable by graduate students in mathematics and computer science with a special interest in circuit complexity.

(1) My addition: optimal solutions of smaller instances are found and retained for use in solving larger instances, that is, smaller instances are never solved again. This is the main difference between DP and more primitive divide-and-conquer like recursive algorithms. The difference is similar to that between Boolean (or arithmetic) circuits and formulas.   Jump back ☝

⇦   Back to my homepage