Hamiltonian cycle problem
Introduction
The Hamiltonian cycle problem is a mathematical problem that belongs to the field of graph theory. It is a classic computational problem that has been studied extensively in the realm of theoretical computer science and combinatorial optimization. The problem is named after the Irish mathematician Sir William Rowan Hamilton.
Definition
A Hamiltonian cycle, also known as a Hamiltonian circuit, is a path in an undirected or directed graph that visits each vertex exactly once and also returns to the original vertex. In other words, it is a closed loop on the graph where every node (vertex) is visited only once. A graph that contains a Hamiltonian cycle is known as a Hamiltonian graph.
The Hamiltonian cycle problem is the computational problem of finding a Hamiltonian cycle in a given graph (if one exists). It can be formally defined as follows:
Given a graph G = (V, E), does there exist a permutation of the vertices v1, v2, ..., vn such that (v1, v2), (v2, v3), ..., (vn-1, vn), (vn, v1) are all edges in E?
Complexity
The Hamiltonian cycle problem is known to be NP-complete, which means that it is computationally difficult to solve. In fact, it was one of Richard Karp's original 21 NP-complete problems proven to be NP-complete in 1972. This implies that unless P=NP, there is no efficient algorithm that solves all instances of the problem.
The Hamiltonian cycle problem remains NP-complete even for planar graphs, directed graphs, and graphs where each vertex has degree three. However, the problem can be solved in polynomial time for certain special classes of graphs, such as trees and graphs that are bipartite.
Algorithms
Several algorithms exist for solving the Hamiltonian cycle problem, or for finding Hamiltonian cycles in specific classes of graphs. These include:
- Brute force: This involves checking all possible permutations of the vertices to see if any form a Hamiltonian cycle. This approach has a time complexity of O(n!), which is infeasible for large graphs.
- Backtracking: This is a more efficient approach that involves exploring paths and backtracking whenever it is determined that a path cannot be extended to a Hamiltonian cycle.
- Dynamic programming: This approach can solve the problem in O(n^2 * 2^n) time, which is still exponential but significantly faster than brute force for small graphs.
- Heuristic algorithms: These are algorithms that can often find a solution quickly, but they do not guarantee to always find a solution even if one exists. Examples include the greedy algorithm and the simulated annealing algorithm.
Applications
The Hamiltonian cycle problem has several practical applications, particularly in the fields of operations research, computer science, and bioinformatics. For example, it is closely related to the famous traveling salesman problem, which is a central problem in operations research and theoretical computer science.
In bioinformatics, Hamiltonian paths are used in DNA sequencing. The problem of reconstructing a DNA sequence from its fragments can be reduced to finding a Hamiltonian path in a graph.
See Also
- Graph Theory - NP-Completeness - Traveling Salesman Problem - Operations Research - Bioinformatics