The complexity theory companion

Invitation S e c r e t 1 Algorith~r~s are at the heart of comple~ty theory. T h a t is the dark secret of complexity theory. It is recognized by complexity theorists, but would be literally incredible to most others. In this book, we hope to make this secret credible. In fact, t h e real secret is even more dramatic. S e c r e t 2 ~imple algorithms are a~ the heart o/comple.citll theory. A corollary of Secret 2 is that every practitioner of computer science or student of computer science already possesses the ability required to understand, enjoy, and employ complexity theory. We realize that these secrets fly in the face of conventional wisdom. Most people view complexity theory as an arcane realm populated by pointy-hatted (ff not indeed pointy-headed) sorcerers stirring cauldrons of recursion theory with wands of combinatorics, while chanting incantations involving complexity classes whose very names contain hundreds of characters and sear the tongues of mere mortals. This stereotype has sprung up in part due to the small amount of esoteric research t h a t fits this bill, but the stereotype is more strongly attributable to the failuze of complexity theorists to commlmlcate in expository forums the central role t h a t algorithms play in complexity theory. T h r o u g h o u t this book-f~om the tree-pruning and interval-pruning algorithms that shape the first chapter to the query simulation procedures t h a t d o m i u a t e the last chapter-we will see t h a t proofs in complexity theory usually employ algorithms as their central tools. In fact, to more clearly highlight t h e role of algorithmic techniques in complexity theory, thgs book i8 organized by technique rather than by topic. T h a t is, in contrast to the organization of other books on complexity theory, each chapter of this book focuses on one te~hnlque what it is, and what results and applications it has yielded. The most thrilling times in complexity theory are when a new technique is introd~!ced and sweeps like fire over the field. In addition to highlighting the centrality of algorithm~ in t~e proof arsenal of complexity theory, we feel t h a t our technique-based approach more vividly conveys to the reader the flavor and excitement of such conflagrations. We invite t h e …

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