The goal is then to find for some instance an optimal solution, that is, a feasible solution with
For each optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure . For example, if there is a graph which contains vertices and , an optimization problem might be "find a path from to that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from to that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.
In the field of approximation algorithms, algorithms are designed to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is more naturally characterized as an optimization problem.
An NP-optimization problem (NPO) is an optimization problem with the following additional conditions.
This implies that the corresponding decision problem is in NP. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization (PO) problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are NP-hard. Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual Turing and Karp reductions. An example of such a reduction would be the L-reduction. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.
NPO is divided into the following subclasses according to their approximability:
Another class of interest is NPOPB, NPO with polynomially bounded cost functions. Problems with this condition has many desirable properties.
US Patent Issued to Indian Institute of Science on Jan. 29 for "Approach for Solving a Constrained Optimization Problem" (Indian Inventor)
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Dec 06, 2011; ALEXANDRIA, Va., Dec. 3 -- United States Patent no. 8,069,127, issued on Nov. 29, was assigned to 21 CT Inc. (Austin, Texas)....
US Patent Issued to International Business Machines on Jan. 11 for "Method for Machine Learning Using Online Convex Optimization Problem Solving with Minimum Regret" (California Inventors)
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