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© 2016

Combinatorics and Complexity of Partition Functions

Benefits

  • Contains an exposition of recent results

  • Demonstrates a unified approach to hard algorithmic problems

  • Provides an easy to read introduction to statistical physics phenomena

Book

Part of the Algorithms and Combinatorics book series (AC, volume 30)

Table of contents

  1. Front Matter
    Pages i-vi
  2. Alexander Barvinok
    Pages 1-7
  3. Alexander Barvinok
    Pages 9-45
  4. Alexander Barvinok
    Pages 47-92
  5. Alexander Barvinok
    Pages 93-143
  6. Alexander Barvinok
    Pages 145-179
  7. Alexander Barvinok
    Pages 181-227
  8. Alexander Barvinok
    Pages 229-267
  9. Alexander Barvinok
    Pages 269-292
  10. Back Matter
    Pages 293-303

About this book

Introduction

Partition functions arise in combinatorics and related problems of statistical physics as they encode in a succinct way the combinatorial  structure of complicated systems. The main focus of the book is on efficient ways to compute (approximate) various partition functions, such as permanents, hafnians and their higher-dimensional versions, graph and hypergraph matching polynomials, the independence polynomial of a graph and partition functions enumerating 0-1 and integer points in polyhedra, which allows one to make algorithmic advances in otherwise intractable problems. 

The book unifies various, often quite recent, results scattered in the literature, concentrating on the three main approaches: scaling, interpolation and correlation decay. The prerequisites include moderate amounts of real and complex analysis and linear algebra, making the book accessible to advanced math and physics undergraduates. 

Keywords

algorithms complexity partition function permanent mathing polynomial independence polynomial graph homomorphism integer flow stable polynomials correlation decay interpolation scaling

Authors and affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn Arbor, MIUSA

About the authors

Alexander Barvinok is a professor of mathematics at the University of Michigan in Ann Arbor, interested in computational complexity and algorithms in algebra, geometry and combinatorics. The reader might be familiar with his books “A Course in Convexity” (AMS, 2002) and “Integer Points in Polyhedra” (EMS, 2008)

Bibliographic information

Reviews

“The book is aimed at graduate students and researchers in theoretical computer science, combinatorics and statistical physics. … The author has the ability to make complicated proofs very accessible while not sacrificing any mathematical rigour, making it a pleasure to read. … The book also moves from the particular to the general … . An advantage of this is that it makes it easier to understand the key ideas.” (Guus Regts, Mathematical Reviews, August, 2018) ​