Local search (optimization)
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Local search is a metaheuristic for solving computationally hard optimization problems. Local search can be used on problems that can be formulated as finding a solution maximizing 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) until a solution deemed optimal is found or a time bound is elapsed. Some problems where local search has been applied are:
- the vertex cover problem, in which a solution is a vertex cover of a graph, and the target is to find a solution with a minimal number of nodes;
- the travelling salesman problem, in which a solution is a cycle containing all nodes of the graph and the target is to minimize the total length of the cycle;
- the boolean satisfiability problem, in which a candidate solution is a truth assignment, and the target is to maximize the number of clauses satisfied by the assignment; in this case, the final solution is of use only if it satisfies all clauses.
Most problems can be formulated in terms of search space and target in several different manners. For example, for the travelling salesman problem a solution can be a cycle and the criterion to maximize is a combination of the number of nodes and the length of the cycle. But a solution can also be a path, and being a cycle is part of the target.
A local search algorithm starts from a candidate solution and then iteratively moves to a neighbor solution. This is only possible if a neighborhood relation is defined on the search space. As an example, the neighborhood of a vertex cover is another vertex cover only differing by one node. For boolean satisfiability, the neighbors of a truth assignment are usually the truth assignments only differing from it by the evaluation of a variable. The same problem may have multiple different neighborhoods defined on it; local optimization with neighborhoods that involve changing up to k components of the solution is often referred to as k-opt.
Typically, every candidate solution has more than one neighbor solution; the choice of which one to move to is taken using only information about the solutions in the neighborhood of the current one, hence the name local search. When the choice of the neighbor solution is done by taking the one locally maximizing the criterion, the metaheuristic takes the name hill climbing.
Termination of local search can be based on a time bound. Another common choice is to terminate when the best solution found by the algorithm has not been improved in a given number of steps. Local search algorithms are typically incomplete algorithms, as the search may stop even if the best solution found by the algorithm is not optimal. This can happen even if termination is due to the impossibility of improving the solution, as the optimal solution can lie far from the neighborhood of the solutions crossed by the algorithms.
Local search algorithms are widely applied to numerous hard computational problems, including problems from computer science (particularly artificial intelligence), mathematics, operations research, engineering, and bioinformatics. Examples of local search algorithm are WalkSAT and the 2-opt algorithm for the TSP.
