Hamiltonian Circuit

A Hamiltonian circuit is defined as a path within a graph that starts and ends at the same vertex, and passes through all other vertices in the graph exactly once. It is also described as a cycle that passes through every vertex of the graph precisely one time.

Formally, it can be viewed as a sequence of n + 1 adjacent vertices, where the initial and final vertices are identical, and all n - 1 intermediate vertices are distinct.

When searching for Hamiltonian circuits in a graph, Brute Force and Backtracking are two distinct algorithmic approaches:

Brute Force Method

The brute force method, also known as exhaustive search, is a straightforward approach that attempts to find a solution by generating every possible candidate solution and then checking each one to see if it satisfies all the problem's constraints.

• Approach: For a Hamiltonian circuit in a graph with n vertices, a brute-force algorithm would typically involve generating all permutations of the n vertices (or n-1 intermediate vertices if a starting point is fixed). Each generated permutation is then checked to determine if it forms a valid Hamiltonian circuit (i.e., if every successive pair of vertices in the sequence is connected by an edge, and the final vertex connects back to the starting vertex).
• Efficiency: This method is generally impractical for all but very small values of n. The number of permutations grows factorially (e.g., 1/2 * (n-1)! for undirected graphs, after fixing a starting vertex and another to define direction), making the exhaustive-search approach computationally infeasible as n increases. Its "principal weakness is the subpar efficiency".

Backtracking Method

Backtracking is considered a "more intelligent variation" of the brute-force approach. It systematically builds a solution one component at a time, and at each step, it evaluates whether the partially constructed solution has the potential to lead to a complete, valid solution.

• Approach:

  • The algorithm constructs the solution by making a series of choices for each component. For the Hamiltonian circuit problem, this means iteratively selecting the next vertex in the path.

  • It typically explores a state-space tree in a depth-first search (DFS) fashion. Each node in this tree represents a partially constructed path.

  • Constraint Checking and Pruning: If a partially constructed path is found to be "nonpromising" (e.g., it violates a constraint by adding a vertex already visited or leading to a dead end where no legitimate path can complete the circuit), the algorithm terminates that path and "backtracks" to the previous choice point to explore alternative options. This process is known as pruning, which significantly reduces the search space compared to brute force.

  • For example, when finding a Hamiltonian circuit, starting at a vertex (say, 'a'), the algorithm picks an adjacent vertex (e.g., 'b'), then from 'b' picks 'c', and so on. If it reaches a vertex (e.g., 'f') from which it cannot complete a circuit (e.g., it's a dead end), it undoes the last choice and tries another option for the previous vertex (e.g., from 'e' it backs up to 'd').

• Efficiency: By intelligently pruning branches that cannot lead to a solution, backtracking can solve many large instances of NP-hard problems in an acceptable amount of time. While still exponential in the worst case, it is generally much more efficient than pure brute force because it avoids redundant computations.

Comparison

Feature	Brute Force	Backtracking
Core Strategy	Generates and tests all possible complete solutions.	Builds solutions incrementally, extending partial solutions and abandoning paths early if they cannot lead to a valid solution.
"Intelligence"	Less intelligent; directly based on problem definition.	More intelligent; a "more intelligent variation".
State-Space Tree	Conceptually explores or generates the entire state-space tree to enumerate all candidates.	Explicitly constructs and traverses a state-space tree (usually DFS-based), but prunes non-promising branches to avoid unnecessary computation.
Efficiency	Highly inefficient due to factorial growth of candidates; impractical for most problem sizes.	More efficient than brute force due to early pruning; can solve larger problem instances in acceptable time, though still potentially exponential.
Constraint Check	Checks all constraints after a complete candidate solution is formed.	Checks constraints during the construction of the solution; prunes if any constraint is violated.