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Local search can be used on problems that can be formulated as finding a solution that maximizes a criterion among a number of candidate solutions. Local search algorithms move from solution to solution in the space of candidate solutions (the search space ) by applying local changes, until a solution deemed optimal is found or a time bound is ...
Constraint satisfaction problems (CSPs) are mathematical questions defined as a set of objects whose state must satisfy a number of constraints or limitations. CSPs represent the entities in a problem as a homogeneous collection of finite constraints over variables , which is solved by constraint satisfaction methods.
The solutions to the sub-problems are then combined to give a solution to the original problem. The divide-and-conquer technique is the basis of efficient algorithms for many problems, such as sorting (e.g., quicksort , merge sort ), multiplying large numbers (e.g., the Karatsuba algorithm ), finding the closest pair of points , syntactic ...
Finding the global minimum solution of a Hartree-Fock problem [37] Upward planarity testing [8] Hospitals-and-residents problem with couples; Knot genus [38] Latin square completion (the problem of determining if a partially filled square can be completed) Maximum 2-satisfiability [3]: LO5
A complete problem for a given complexity class C and reduction ≤ is a problem P that belongs to C, such that every problem A in C has a reduction A ≤ P. For instance, a problem is NP -complete if it belongs to NP and all problems in NP have polynomial-time many-one reductions to it.
More interesting is the Regular Post Embedding Problem, a further variant where one looks for solutions that belong to a given regular language (submitted, e.g., under the form of a regular expression on the set {, …,}). The Regular Post Embedding Problem is still decidable but, because of the added regular constraint, it has a very high ...
The NP-complete problems represent the hardest problems in NP. If some NP-complete problem has a polynomial time algorithm, all problems in NP do. 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.
For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure m 0. For example, if there is a graph G which contains vertices u and v , an optimization problem might be "find a path from u to v that uses the fewest edges".