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NP-hard problems are those at least as hard as NP problems; i.e., all NP problems can be reduced (in polynomial time) to them. NP-hard problems need not be in NP; i.e., they need not have solutions verifiable in polynomial time. For instance, the Boolean satisfiability problem is NP-complete by the Cook–Levin theorem, so any instance of any ...
A polynomial-time problem can be very difficult to solve in practice if the polynomial's degree or constants are large enough. In addition, information-theoretic security provides cryptographic methods that cannot be broken even with unlimited computing power. "A large-scale quantum computer would be able to efficiently solve NP-complete problems."
That is, assuming a solution for H takes 1 unit time, H ' s solution can be used to solve L in polynomial time. [1] [2] As a consequence, finding a polynomial time algorithm to solve a single NP-hard problem would give polynomial time algorithms for all the problems in the complexity class NP.
The hardest problems in NP are called NP-complete problems. An algorithm solving such a problem in polynomial time is also able to solve any other NP problem in polynomial time. If P were in fact equal to NP, then a polynomial-time algorithm would exist for solving NP-complete, and by corollary, all NP problems. [4]
Although this problem seems easier, Valiant and Vazirani have shown [25] that if there is a practical (i.e. randomized polynomial-time) algorithm to solve it, then all problems in NP can be solved just as easily. MAX-SAT, the maximum satisfiability problem, is an FNP generalization of SAT. It asks for the maximum number of clauses which can be ...
As well as finding the first polynomial-time algorithm for 2-satisfiability, Krom (1967) also formulated the problem of evaluating fully quantified Boolean formulae in which the formula being quantified is a 2-CNF formula. The 2-satisfiability problem is the special case of this quantified 2-CNF problem, in which all quantifiers are existential ...
The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S {\displaystyle S} of integers and a target-sum T {\displaystyle T} , and the question is to decide whether any subset of the integers sum to precisely T {\displaystyle T} . [ 1 ]
This polynomial is further reduced to = + + which is shown in blue and yields a zero of −5. The final root of the original polynomial may be found by either using the final zero as an initial guess for Newton's method, or by reducing () and solving the linear equation. As can be seen, the expected roots of −8, −5, −3, 2, 3, and 7 were ...
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