Submodular Functions and Optimization

Submodular Functions and Optimization covers mathematical theory and algorithmic aspects of submodular function minimization. The book focuses on both the theoretical foundations and practical applications in operations research, combinatorial optimization, and related fields. The text progresses from basic concepts to advanced techniques, including polynomial-time algorithms and extensions of classical results. Chapters address key topics such as discrete convex analysis, polymatroid intersection algorithms, and connections to network flows. Schrijver presents proofs with full mathematical rigor while maintaining accessibility through clear explanations and examples. The material builds systematically, with each section laying groundwork for subsequent developments. The work stands as a comprehensive treatment that bridges pure mathematics and computational applications, demonstrating the power of submodular optimization in solving real-world problems. The integration of theory and practice makes this text relevant for both researchers and practitioners.

There are not enough internet reviews to create a summary of this book. Instead, here is a summary of reviews of Alexander Schrijver's overall work: Readers value Schrijver's books as technical references but note they require advanced mathematical background. His "Theory of Linear and Integer Programming" receives attention for its thorough coverage and rigorous proofs. What readers liked: - Clear logical progression through topics - Comprehensive citation of historical developments - Detailed mathematical derivations - Quality of problem sets What readers disliked: - Dense notation makes books hard to read cover-to-cover - Limited worked examples - Few intuitive explanations for beginners - High price point of specialized volumes From Goodreads/Amazon: "Theory of Linear and Integer Programming" averages 4.4/5 stars (42 ratings) "Combinatorial Optimization" averages 4.7/5 stars (15 ratings) Notable reader comment: "Excellent reference but not suitable as first introduction to topic. Requires solid foundation in linear algebra and mathematical maturity." (Mathematics Stack Exchange review)

Convex Optimization by Stephen Boyd, Lieven Vandenberghe This text covers optimization theory with focus on convex functions, which connects to submodular optimization through their mathematical foundations and algorithmic approaches.

Combinatorial Optimization by Bernhard Korte, Jens Vygen The book explores discrete optimization problems and algorithms with sections on submodular functions as part of the broader mathematical framework.

Discrete Convex Analysis by Kazuo Murota This work bridges discrete and continuous optimization, presenting L-convex and M-convex functions which relate to submodular functions in optimization theory.

Theory of Linear and Integer Programming by Alexander Schrijver The text establishes fundamental concepts in optimization that complement submodular function theory through polyhedral theory and algorithmic methods.

Optimization Over Integers by Dimitris Bertsimas and Robert Weismantel The book presents integer optimization techniques with connections to discrete convex analysis and submodular optimization.

🔹 Alexander Schrijver is one of the most influential mathematicians in combinatorial optimization, having won the Fulkerson Prize three times for his groundbreaking work in the field. 🔹 Submodular functions have applications far beyond pure mathematics - they're used in machine learning, economics, and even social network analysis to model diminishing returns in complex systems. 🔹 The first edition of this book (1986) helped establish submodular function theory as a cornerstone of modern discrete optimization, leading to its widespread adoption in algorithm design. 🔹 The author developed several fundamental algorithms discussed in the book, including polynomial-time methods for submodular function minimization that revolutionized the field. 🔹 The book bridges pure and applied mathematics by connecting abstract convex analysis with practical optimization problems, making it valuable for both theoretical mathematicians and computer scientists.