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Integer programming is NP-complete. In particular, the special case of 0–1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems. [1] If some decision variables are not discrete, the problem is known as a mixed-integer programming problem. [2]
Generalized assignment problem; Integer programming. 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
The problem is similar in nature to the year 2000 problem, the difference being the Year 2000 problem had to do with base 10 numbers, whereas the Year 2038 problem involves base 2 numbers. Analogous storage constraints will be reached in 2106 , where systems storing Unix time as an unsigned (rather than signed) 32-bit integer will overflow on 7 ...
In the guillotine cutting problem, both the items and the "bins" are two-dimensional rectangles rather than one-dimensional numbers, and the items have to be cut from the bin using end-to-end cuts. In the selfish bin packing problem, each item is a player who wants to minimize its cost. [53]
Two 0–1 integer programs that are equivalent, in that they have the same objective function and the same set of feasible solutions, may have quite different linear programming relaxations: a linear programming relaxation can be viewed geometrically, as a convex polytope that includes all feasible solutions and excludes all other 0–1 vectors ...
If all off-diagonal elements of are non-positive, the corresponding QUBO problem is solvable in polynomial time. [8] QUBO can be solved using integer linear programming solvers like CPLEX or Gurobi Optimizer. This is possible since QUBO can be reformulated as a linear constrained binary optimization problem.
The multifit algorithm uses binary search combined with an algorithm for bin packing. In the worst case, its approximation ratio is 8/7. The subset sum problem has an FPTAS which can be used for the partition problem as well, by setting the target sum to sum(S)/2.
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 ]