It has widely been recognized that submodular functions play essential roles in efficiently solvable combinatorial optimization problems. Since the publication of the 1st edition of this book fifteen years ago, submodular functions have been showing further increasing importance in optimization, combinatorics, discrete mathematics, algorithmic computer science, and algorithmic economics, and there have been made remarkable developments of theory and algorithms in submodular functions. The 2nd edition of the book supplements the 1st edition with a lot of remarks and with new two chapters: "Submodular Function Minimization" and "Discrete Convex Analysis." The present 2nd edition is still a unique book on submodular functions, which is essential to students and researchers interested in combinatorial optimization, discrete mathematics, and discrete algorithms in the fields of mathematics, operations research, computer science, and economics.

Key features:

- Self-contained exposition of the theory of submodular functions.
- Selected up-to-date materials substantial to future developments.
- Polyhedral description of Discrete Convex Analysis.
- Full description of submodular function minimization algorithms.
- Effective insertion of figures.
- Useful in applied mathematics, operations research, computer science, and economics.
- Self-contained exposition of the theory of submodular functions.
- Selected up-to-date materials substantial to future developments.
- Polyhedral description of Discrete Convex Analysis.
- Full description of submodular function minimization algorithms.
- Effective insertion of figures.
- Useful in applied mathematics, operations research, computer science, and economics.

This book constitutes the thoroughly refereed post-conference proceedings of the 4th International Symposium on Combinatorial Optimization, ISCO 2016, held in Vietri sul Mare, Italy, in May 2016. The 38 revised full papers presented in this book were carefully reviewed and selected from 98 submissions. They present original research on all aspects of combinatorial optimization, such as algorithms and complexity; mathematical programming; operations research; stochastic optimization; and graphs and combinatorics.