Exact Exponential Algorithms

Today most computer scientists believe that NP-hard problems cannot be solved by polynomial-time algorithms. From the polynomial-time perspective, all NP-complete problems are equivalent but their exponential-time properties vary widely. Why do some NP-hard problems appear to be easier than others? Are there algorithmic techniques for solving hard problems that are significantly faster than the exhaustive, brute-force methods? The algorithms that address these questions are known as exact exponential algorithms.

The history of exact exponential algorithms for NP-hard problems dates back to the 1960s. The two classical examples are Bellman, Held and Karp's dynamic programming algorithm for the traveling salesman problem and Ryser's inclusion-exclusion formula for the permanent of a matrix. The design and analysis of exact algorithms leads to a better understanding of hard problems and initiates interesting new combinatorial and algorithmic challenges. The last decade has witnessed a rapid development of the area, with many new algorithmic techniques discovered. This has transformed exact algorithms into a very active research field. This book provides an introduction to the area and explains the most common algorithmic techniques, and the text is supported throughout with exercises and detailed notes for further reading.The book is intended for advanced students and researchers in computer science, operations research, optimization and combinatorics.

Auteur

The authors are highly regarded academics and educators in theoretical computer science, and in algorithmics in particular.

Texte du rabat

For a long time computer scientists have distinguished between fast and slow algo rithms. Fast (or good) algorithms are the algorithms that run in polynomial time, which means that the number of steps required for the algorithm to solve a problem is bounded by some polynomial in the length of the input. All other algorithms are slow (or bad). The running time of slow algorithms is usually exponential. This book is about bad algorithms. There are several reasons why we are interested in exponential time algorithms. Most of us believe that there are many natural problems which cannot be solved by polynomial time algorithms. The most famous and oldest family of hard problems is the family of NP complete problems. Most likely there are no polynomial time al gorithms solving these hard problems and in the worst case scenario the exponential running time is unavoidable. Every combinatorial problem is solvable in ?nite time by enumerating all possi ble solutions, i. e. by brute force search. But is brute force search always unavoid able? De?nitely not. Already in the nineteen sixties and seventies it was known that some NP complete problems can be solved signi?cantly faster than by brute force search. Three classic examples are the following algorithms for the TRAVELLING SALESMAN problem, MAXIMUM INDEPENDENT SET, and COLORING.

Contenu

Branching.- Dynamic Programming.- Inclusion-Exclusion.- Treewidth.- Measure & Conquer.- Subset Convolution.- Local Search and SAT.- Split and List.- Time Versus Space.- Miscellaneous.- Conclusions, Open Problems and Further Directions.