Language
Country/Area
No result found
How does this mathematical trick make NP-hard problems solvable?
In the field of mathematics, many problems are so computationally difficult that people cannot breathe. What can be done to break through these NP-hard barriers? Recently, mathematicians have conducted in-depth research on a key technology, which is "relaxation technology". The core of this technique is to relax the integer constraints and transform the problem into a linear programming problem that can be solved with a polynomial-time algorithm.
Relaxing the restrictions on integer problems greatly improves the solvability of the problem and opens up new ways to deal with various computing challenges.
For example, consider a "set coverage problem". In this problem, given a set of sets, we need to select a subset of them to cover all elements, and the number of selected sets should be as small as possible. This problem can be formalized as a 0-1 integer program, where each variable represents whether the set is selected. By relaxing the constraints and changing the choice of variables from 0 and 1 to real numbers between 0 and 1, we can solve the problem more easily.
Relaxation technology simplifies the original complex optimization problem, breaks the inherent computational difficulty, and allows the solution to emerge.
When we solve this kind of relaxed linear program, sometimes the solution we get happens to be an integer, which means we also solve the original integer problem. Although this situation is uncommon, it is still guaranteed that the relaxed solution is at least as good as the integer solution and can provide us with valuable information about the original problem.
In a specific example, suppose there are three sets F = {{a, b}, {b, c}, {a, c}}. The corresponding 0-1 integer program for the minimal set coverage designed for these sets would require minimizing the number of indicator variables. This example shows the importance of linear relaxation in the solution process, because through different solutions, we can not only find the lower bound of the integer solution, but also give a more accurate solution expectation.
Every time we perform a relaxation operation, we are laying the foundation for the next solution and gradually approaching the real optimal solution.
As to the quality of the solution, relaxation techniques provide valuable upper and lower bounds on solutions to integer programs. We typically examine the "integer gap," which is a measure of the gap between the original integer solution and its relaxation. If the gap is smaller, we are more confident that the solution to the original problem is accurately captured.
In addition to being the basis for approximation algorithms, this technique is also used in more complex branch-and-bound methods. When a non-integer solution is found, the algorithm breaks the problem into smaller sub-problems to search within a narrower scope.
Such a branch-and-bound method gives us hope to find integer solutions that are close to the optimal solution, and can still show its courage even in the face of NP-hard problems.
In addition, the "cutting plane method" is also a powerful technique. It helps us find more accurate integer solutions by finding cutting planes to exclude solutions outside the convex hull of the relaxed solution. This also shows that the use of these methods is not limited to specific problems, and the same ideas can be widely applied to a variety of computing challenges.
Combining these techniques, mathematicians are showing great promise in solving NP-hard problems. Through a combination of relaxation techniques, branching and bounding, and other methods, we are one step closer to solving problems that were once considered insurmountable. But do these methods often provide ideal solutions?
