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This agenda outlines key concepts in the theory of computing, focusing on the class NP and its properties. We define NP languages and their polynomial-time verifiers, discuss the concept of NP-completeness, and address notable problems such as the Boolean satisfiability problem (SAT). The session will explore whether NP is closed under complementation and the implications of solving NP problems, including the open Millennium Problem of determining if P equals NP. Understanding these concepts is critical for advancing computer science and theoretical research.
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CSCI 2670Introduction to Theory of Computing December 1, 2004
Agenda • Yesterday • The Class NP • Value is exponential in length • Today • More on the class NP • Quiz December 1, 2004
The class NP Definition: A verifier for a language A is an algorithm V, where A={w|V accepts <w,c> for some string c} The string c is called a certificate of membership in A. Definition: NP is the class of languages that have polynomial-time verifiers. December 1, 2004
Is NP closed under complementation? • For example, can we verify in polynomial time that a graph cannot be 3-colored? • Not obviously • It seems we need to check many 3-colorings before we can conclude that none exist • The 3-coloring problem is in coNP December 1, 2004
Who wants $1,000,000? • In May, 2000, the Clay Mathematics Institute named seven open problems in mathematics the Millennium Problems • Anyone who solves any of these problems will receive $1,000,000 • Proving whether or not P equals NP is one of these problems December 1, 2004
NP P coNP What we know December 1, 2004
Are there any problems here? What we don’t know NP P coNP December 1, 2004
Solving NP problems • The best-known methods for solving problems in NP that are not known to be in P take exponential time • Brute force search • We don’t know if NP is actually in a smaller complexity class December 1, 2004
NP-completeness • A problem C is NP-complete if finding a polynomial-time solution for C would imply P=NP December 1, 2004
An NP-complete problem • A formula is Boolean if each of its variables can be assigned the values TRUE (1) or FALSE (0) • A Boolean formula is satisfiable if there is some assignment of values that results in the formula evaluating to TRUE SAT: Is a given Boolean formula satisfiable? • SAT is NP-complete December 1, 2004
Examples • (x y) ( x y) • Satisfiable – e.g., x = y = 1 • ((xy) (xz)) ((xy) (yz)) • Satisfiable – e.g., x = 0, y = z = 1 • ((xy) (xz)) ((xy) (yz)) • Unsatisfiable December 1, 2004
Proving a problem is NP-complete • A problem C is NP-complete if finding a polynomial-time solution for C would imply P=NP • If a polynomial-time solution is found for C, then that solution can be used to find a polynomial-time solution for any other problem in NP • What does this remind you of? • Reductions! December 1, 2004
Reductions and NP-completeness • If we can prove an NP-complete problem C can be polynomially reduced to a problem A, then we’ve shown A is NP-complete • A polynomial-time solution to A would provide a polynomial-time solution to C, which would imply P=NP December 1, 2004
Polynomial functions Definition: A function f:Σ*Σ* is a polynomial time computable function if some polynomial time Turing machine M exists that halts with just f(w) on its tape, when started on any input w. December 1, 2004
f f Polynomial reductions Definition: Language A is polynomial-time reducible to language B, written A ≤P B, if a polynomial time computable function f:Σ*Σ* exists, where for every w w A iff f(w) B December 1, 2004
