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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.
In theoretical computer science, the closest string is an NP-hard computational problem, [1] which tries to find the geometrical center of a set of input strings. To understand the word "center", it is necessary to define a distance between two strings. Usually, this problem is studied with the Hamming distance in mind.
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.
This variant is known as VM packing [7] since when virtual machines (VMs) are packed in a server, their total memory requirement could decrease due to pages shared by the VMs that need only be stored once. If items can share space in arbitrary ways, the bin packing problem is hard to even approximate.
NP is the set of decision problems for which the problem instances, where the answer is "yes", have proofs verifiable in polynomial time by a deterministic Turing machine, or alternatively the set of problems that can be solved in polynomial time by a nondeterministic Turing machine. [2] [Note 1]
Goldbach’s Conjecture. One of the greatest unsolved mysteries in math is also very easy to write. Goldbach’s Conjecture is, “Every even number (greater than two) is the sum of two primes ...
A problem is hard for a class of problems C if every problem in C can be polynomial-time reduced to . Thus no problem in C is harder than , since an algorithm for allows us to solve any problem in C with at most polynomial slowdown. Of particular importance, the set of problems that are hard for NP is called the set of NP-hard problems.
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 ]