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Hence, the Small Set Expansion Hypothesis, which postulates that SSE is hard to approximate, is a stronger (but closely related) assumption than the Unique Game Conjecture. [25] Some approximation problems are known to be SSE-hard [26] (i.e. at least as hard as approximating SSE).
The first definition is the basis for the abbreviation NP; "nondeterministic, polynomial time". These two definitions are equivalent because the algorithm based on the Turing machine consists of two phases, the first of which consists of a guess about the solution, which is generated in a nondeterministic way, while the second phase consists of ...
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. However, the opposite direction is not true: some problems are undecidable, and therefore even more difficult to solve than all problems in NP, but they are probably not NP-hard (unless ...
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 problem in NP can be transformed mechanically into a Boolean satisfiability problem in polynomial time.
Soft computing is an umbrella term used to describe types of algorithms that produce approximate solutions to unsolvable high-level problems in computer science. Typically, traditional hard-computing algorithms heavily rely on concrete data and mathematical models to produce solutions to problems.
The goal is then to find for some instance x an optimal solution, that is, a feasible solution y with (,) = {(, ′): ′ ()}. For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure m 0 .
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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]