Yves Robert, École Normale Supérieure de Lyon, Institut Universitaire de France, and Université de Lyon, France

Anne Benoit and Frederic Vivien, École Normale Supérieure de Lyon, France

Presenting a complementary perspective to standard books on algorithms, A Guide to Algorithm Design: Paradigms, Methods, and Complexity Analysis provides a roadmap for readers to determine the difficulty of an algorithmic problem by finding an optimal solution or proving complexity results. It gives a practical treatment of algorithmic complexity and guides readers in solving algorithmic problems.

Divided into three parts, the book offers a comprehensive set of problems with solutions as well as in-depth case studies that demonstrate how to assess the complexity of a new problem.

• Part I helps readers understand the main design principles and design efficient algorithms.


• Part II covers polynomial reductions from NP-complete problems and approaches that go beyond NP-completeness.


• Part III supplies readers with tools and techniques to evaluate problem complexity, including how to determine which instances are polynomial and which are NP-hard.

Drawing on the authors' classroom-tested material, this text takes readers step by step through the concepts and methods for analyzing algorithmic complexity. Through many problems and detailed examples, readers can investigate polynomial-time algorithms and NP-completeness and beyond.

Polynomial-Time Algorithms: Exercises
Introduction to Complexity
On the complexity to compute xⁿ
Asymptotic notations: O, o, T, and ¿

Divide-and-Conquer
Strassen's algorithm
Master theorem
Solving recurrences

Greedy Algorithms
Motivating example: the sports hall
Designing greedy algorithms
Graph coloring
Theory of matroids

Dynamic Programming
The coin changing problem
The knapsack problem
Designing dynamic-programming algorithms

Amortized Analysis
Methods for amortized analysis

NP-Completeness and Beyond
NP-Completeness
A practical approach to complexity theory
Problem classes
NP-complete problems and reduction theory
Examples of NP-complete problems and reductions
Importance of problem definition
Strong NP-completeness
Why does it matter?

Exercises on NP-Completeness
Easy reductions
About graph coloring
Scheduling problems
More involved reductions
2-PARTITION is NP-complete

Beyond NP-Completeness
Approximation results
Polynomial problem instances
Linear programming
Randomized algorithms
Branch-and-bound and backtracking

Exercises Going beyond NP-Completeness
Approximation results
Dealing with NP-complete problems

Reasoning on Problem Complexity
Reasoning to Assess a Problem Complexity
Basic reasoning
Set of problems with polynomial-time algorithms
Set of NP-complete problems

Chains-on-Chains Partitioning
Optimal algorithms for homogeneous resources
Variants of the problem
Extension to a clique of heterogeneous resources
Conclusion

Replica Placement in Tree Networks
Access policies
Complexity results
Variants of the replica placement problem
Conclusion

Packet Routing
MEDP: Maximum edge-disjoint paths
PRVP: Packet routing with variable-paths
Conclusion

Matrix Product, or Tiling the Unit Square
Problem motivation
NP-completeness
A guaranteed heuristic
Related problems

Online Scheduling
Flow time optimization
Competitive analysis
Makespan optimization
Conclusion