Search results
Results From The WOW.Com Content Network
In 1971, Stephen Cook published his paper "The complexity of theorem proving procedures" [2] in conference proceedings of the newly founded ACM Symposium on Theory of Computing. Richard Karp's subsequent paper, "Reducibility among combinatorial problems", [1] generated renewed interest in Cook's paper by providing a list of 21 NP-complete problems.
As noted above, this is the Cook–Levin theorem; its proof that satisfiability is NP-complete contains technical details about Turing machines as they relate to the definition of NP. However, after this problem was proved to be NP-complete, proof by reduction provided a simpler way to show that many other problems are also NP-complete ...
For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make (a AND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisfiable. SAT is the first problem that was proven to be NP-complete—this is the Cook–Levin theorem.
The concept of NP-completeness was introduced in 1971 (see Cook–Levin theorem), though the term NP-complete was introduced later. At the 1971 STOC conference, there was a fierce debate between the computer scientists about whether NP-complete problems could be solved in polynomial time on a deterministic Turing machine .
The circuit on the left is satisfiable but the circuit on the right is not. In theoretical computer science, the circuit satisfiability problem (also known as CIRCUIT-SAT, CircuitSAT, CSAT, etc.) is the decision problem of determining whether a given Boolean circuit has an assignment of its inputs that makes the output true. [1]
For example, just as counting cannot be done by an circuit family of subexponential size, many tautologies relating to the pigeonhole principle cannot have subexponential proofs in a proof system based on bounded-depth formulas (and in particular, not by resolution-based systems, since they rely solely on depth 1 formulas).
Sections 4.3 (The master method) and 4.4 (Proof of the master theorem), pp. 73–90. Michael T. Goodrich and Roberto Tamassia. Algorithm Design: Foundation, Analysis, and Internet Examples. Wiley, 2002. ISBN 0-471-38365-1. The master theorem (including the version of Case 2 included here, which is stronger than the one from CLRS) is on pp. 268 ...
Consider the case of two buyers, each with a value that is an independent draw from a distribution with support [0,1], cumulative distribution function F(v) and probability density function f(v). If buyers behave according to their dominant strategies, then a buyer with value v wins if his opponent's value x is lower. Thus his win probability is