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Cygan [4] presented an algorithm that, for any ε>0, attains a (k+1+ε)/3 approximation. The run-time is polynomial in the number of sets and elements, but doubly-exponential in 1/ε. Furer and Yu [5] presented an algorithm that attains the same approximation, but with run-time singly-exponential in 1/ε.
The variant where variables are required to be 0 or 1, called zero-one linear programming, and several other variants are also NP-complete [2] [3]: MP1 Some problems related to Job-shop scheduling; Knapsack problem, quadratic knapsack problem, and several variants [2] [3]: MP9 Some problems related to Multiprocessor scheduling
The set of NP-complete problems is often denoted by NP-C or NPC. Although a solution to an NP-complete problem can be verified "quickly", there is no known way to find a solution quickly. That is, the time required to solve the problem using any currently known algorithm increases rapidly as the size of the problem grows.
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 n {\displaystyle n} items numbered from 1 up to n {\displaystyle n} , each with a weight w i {\displaystyle w_{i}} and a value v i {\displaystyle v_{i}} , along with a maximum weight capacity ...
The bin packing problem is strongly NP-complete.This can be proven by reducing the strongly NP-complete 3-partition problem to bin packing. [8]Furthermore, there can be no approximation algorithm with absolute approximation ratio smaller than unless =.
In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete.In his 1972 paper, "Reducibility Among Combinatorial Problems", [1] Richard Karp used Stephen Cook's 1971 theorem that the boolean satisfiability problem is NP-complete [2] (also called the Cook-Levin theorem) to show that there is a polynomial time many-one reduction ...
The hexagonal packing of circles on a 2-dimensional Euclidean plane. These problems are mathematically distinct from the ideas in the circle packing theorem.The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, for instance the plane or a sphere.
It is shown that finding an isomorphism for n-vertex graphs is equivalent to finding an n-clique in an M-graph of size n 2. This fact is interesting because the problem of finding a clique of order (1 − ε)n in a M-graph of size n 2 is NP-complete for arbitrarily small positive ε. [43] The problem of homeomorphism of 2-complexes. [44]