LKH

LKH refers to the Lin-Kernighan-Helsgaun algorithm, a heuristic method used to find approximate solutions to the Traveling Salesperson Problem (TSP). The TSP is a classic optimization problem that seeks the shortest possible route that visits a set of cities and returns to the origin city. Exact solutions for the TSP become computationally intractable for large numbers of cities, making heuristic approaches like LKH essential.

The LKH algorithm is an iterative improvement heuristic. It starts with an initial tour and repeatedly applies a series of local search operations, known as 2-opt, 3-opt, and k-opt moves, to modify the tour in an attempt to shorten it. The "Lin-Kernighan" part of the name refers to the original adaptive k-opt procedure developed by Shen Lin and Benjamin Kernighan, which dynamically determines the value of k based on the improvement achieved. The "Helsgaun" addition signifies improvements and enhancements made by Donald Helsgaun, which further refined the algorithm's performance and efficiency.

The core idea behind the k-opt moves is to remove k edges from the current tour and reconnect the remaining segments in a different way to form a new, potentially shorter, tour. LKH is known for its ability to escape local optima by intelligently exploring the solution space and making significant improvements to the tour in each iteration. It is widely considered one of the most effective heuristics for the TSP, often producing solutions very close to the optimal tour, even for very large problem instances. Its application is crucial in fields where efficient routing is critical, such as logistics, vehicle scheduling, and circuit board drilling.