Search results
Results from the WOW.Com Content Network
As it is suspected, but unproven, that P≠NP, it is unlikely that any polynomial-time algorithms for NP-hard problems exist. [3] [4] A simple example of an NP-hard problem is the subset sum problem. Informally, if H is NP-hard, then it is at least as difficult to solve as the problems in NP.
Informally, an NP-complete problem is an NP problem that is at least as "tough" as any other problem in NP. 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.
Soon after it appeared, the book received positive reviews by reputed researchers in the area of theoretical computer science. In his review, Ronald V. Book recommends the book to "anyone who wishes to learn about the subject of NP-completeness", and he explicitly mentions the "extremely useful" appendix with over 300 NP-hard computational problems.
Euler diagram for P, NP, NP-complete, and NP-hard set of problems. Under the assumption that P ≠ NP, the existence of problems within NP but outside both P and NP-complete was established by Ladner. [1] In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems.
The hardest problems in PSPACE. PTAS: Polynomial-time approximation scheme (a subclass of APX). QIP: Solvable in polynomial time by a quantum interactive proof system. QMA: Quantum analog of NP. R: Solvable in a finite amount of time. RE: Problems to which we can answer "YES" in a finite amount of time, but a "NO" answer might never come. RL
The most common problem being solved is the 0-1 knapsack problem, which restricts the number of copies of each kind of item to zero or one. Given a set of items numbered from 1 up to , each with a weight and a value , along with a maximum weight capacity ,
As is common for complexity classes within the polynomial time hierarchy, a problem is called GI-hard if there is a polynomial-time Turing reduction from any problem in GI to that problem, i.e., a polynomial-time solution to a GI-hard problem would yield a polynomial-time solution to the graph isomorphism problem (and so all problems in GI).
Graph coloring is computationally hard. It is NP-complete to decide if a given graph admits a k-coloring for a given k except for the cases k ∈ {0,1,2}. In particular, it is NP-hard to compute the chromatic number. [31] The 3-coloring problem remains NP-complete even on 4-regular planar graphs. [32]