P versus NP, reductions, SAT, classic NP-complete problems, approximation algorithms and undecidability.
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- What is the P class in computational complexity?
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The set of decision problems solvable by a deterministic Turing machine in polynomial time.- What is the NP class in computational complexity?
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The set of decision problems verifiable by a deterministic Turing machine in polynomial time, given a certificate.- Is every problem in P also in NP?
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Yes. If a problem is solvable in polynomial time, it is also verifiable in polynomial time (ignore the certificate).- What is the P versus NP problem?
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The unsolved question of whether P equals NP (whether every polynomial-time verifiable problem is also polynomial-time solvable).- What does it mean for a problem to be NP-complete?
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A problem is NP-complete if it is in NP and every problem in NP can be reduced to it in polynomial time.- What is a polynomial-time reduction from problem A to problem B?
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An algorithm that transforms any instance of A into an instance of B in polynomial time, preserving the answer (yes-instance maps to yes-instance).- What is the Boolean satisfiability problem (SAT)?
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Given a Boolean formula in conjunctive normal form, determine if there exists an assignment of variables that makes it true.- Why is SAT the canonical NP-complete problem?
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The Cook-Levin theorem proves SAT is in NP and NP-hard, with every NP problem reducing to SAT in polynomial time.- What is 3-SAT?
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The satisfiability problem restricted to formulas in conjunctive normal form where each clause contains exactly 3 literals.- What is the clique problem?
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Given a graph and integer k, determine if the graph contains a complete subgraph (clique) of size at least k.- What is the vertex cover problem?
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Given a graph and integer k, determine if there exists a set of at most k vertices such that every edge has at least one endpoint in the set.- What is the independent set problem?
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Given a graph and integer k, determine if there exists a set of k vertices with no edges between them.- What is the Hamiltonian path problem?
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Given a graph, determine if there exists a path that visits every vertex exactly once.- What is the Traveling Salesman Problem (TSP)?
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Given n cities and distances, find the shortest tour that visits each city exactly once and returns to the start.- What is the subset sum problem?
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Given a set of integers and a target sum, determine if there is a subset that sums to the target.
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