Hamiltonian Cycles: The Problem That Could Win You $1 Million

In the world of computer science and mathematics, some problems are deceptively simple to describe but incredibly challenging to solve.

One such problem is the Hamiltonian Cycle Problem, a classic puzzle that has fascinated researchers for decades. Named after the Irish mathematician Sir William Rowan Hamilton, this problem lies at the intersection of graph theory, computational complexity, and real-world applications. Let’s dive into what makes this problem so intriguing and why it matters.

What if solving a single puzzle could make you a millionaire and revolutionize the world of computing?

Hamiltonian Cycle Problem, a deceptively simple question that has stumped mathematicians and computer scientists for decades. This problem is so challenging that anyone who cracks it could win the prestigious $1 million Millennium Prize.

But beware: it’s not just a puzzle; it’s one of the hardest problems in computer science, with implications that could change the way we solve real-world challenges.

Ready to dive into the mystery of the Hamiltonian Cycle and see why it’s worth a fortune?

What Is the Hamiltonian Cycle Problem?

The Hamiltonian Cycle Problem asks a simple question: 

Given a graph, does there exist a cycle that visits every vertex exactly once and returns to the starting point? Such a cycle is called a Hamiltonian cycle.

Example: Imagine a delivery driver who needs to visit a set of locations, each exactly once, and return to the starting point. Finding the most efficient route is essentially solving the Hamiltonian Cycle Problem.

While the problem is easy to state, solving it efficiently is anything but simple. In fact, the Hamiltonian Cycle Problem is one of the most famous NP-complete problems, a class of problems that are both notoriously difficult and deeply important in computer science.

Why Is It So Hard?

The Hamiltonian Cycle Problem is NP-complete(we will learn down in the thread), which means:

1. Verification is Easy: If someone gives you a potential cycle, you can quickly check whether it visits every vertex exactly once.
2. Solving is Hard: Finding such a cycle from scratch is computationally challenging, especially for large graphs.

No one has yet discovered an efficient algorithm to solve the Hamiltonian Cycle Problem for all cases. The best-known algorithms rely on brute-force search or backtracking, which become impractical as the size of the graph grows.

For example, a graph with just 20 vertices could require checking over 2.4 quintillion possible paths!

3. Applying This to a Graph with 20 Vertices

Let’s calculate the number of unique Hamiltonian cycles for n=20

Number of unique cycles=(20−1)! / 2=19! / 2

Number of unique cycles=

(20−1)!
------
​  2
​

Now, let’s compute 19!
19!=19×18×17×⋯×1=121,645,100,408,832,000


121,645,100,408,832,0002=60,822,550,204,416,000

Dividing by 2: ​
=60,822,550,204,416,000

So, there are approximately 60.8 quadrillion (6.08 × 10¹⁶) unique Hamiltonian cycles in a complete graph with 20 vertices.

Why Is This a Problem?

Exponential Growth: The number of possible paths grows factorially with the number of vertices. For example:

• 10!=3,628,800 (3.6 million)
• 20!=2.4×10^18 (2.4 quintillion)
• 30!=2.65×10^32 (265 nonillion)

Computational Feasibility: Even with modern computers, checking 2.4×10^18 paths is impractical. A computer checking 1 billion paths per second would take over 76 years to check all paths for a 20-vertex graph.

Let's first understand the complexity:

P (Polynomial Time):

Problems that can be solved quickly (in polynomial time) by a deterministic algorithm.

1. Sorting a list of numbers. Algorithms like QuickSort or MergeSort can sort a list in O(nlog⁡n).
2. O(n), O(n^2) time, which is polynomial time.
3. Key Idea: These are the easy problems because they can be solved efficiently.

Normal Time (Exponential Time):

An algorithm runs in exponential time if its running time grows exponentially with the input size.

• Example: If the input size is n
• n, then algorithms with running times like O(2^n) , O(n!) or O(3^n) are exponential time.

Why It Matters: Exponential time algorithms become impractical very quickly as the input size grows. For example, solving a problem with n=100

n=100 might take longer than the age of the universe!

NP (Nondeterministic Polynomial Time):

Problems where a solution can be verified quickly (in polynomial time), but finding the solution might be hard.

1. Example: The Traveling Salesman Problem (TSP). If someone gives you a route, you can quickly check if it visits all cities and is short enough. But finding the shortest route is hard.
2. Key Idea: NP includes all problems in P, plus problems where verifying a solution is easy, but solving it might not be.

NP-complete:

Problems that are:

1. In NP (solutions can be verified quickly).
2. NP-hard (at least as hard as the hardest problems in NP).

• Example: The Boolean Satisfiability Problem (SAT), Hamiltonian Cycle Problem, and Knapsack Problem.
• Key Idea: These are the hardest problems in NP. If you can solve one NP-complete problem quickly, you can solve all NP problems quickly.

NP-hard:

Problems that are at least as hard as the hardest problems in NP, but they don’t necessarily have to be in NP themselves.

• Example: The Halting Problem (determining whether a program will ever stop running) is NP-hard but not in NP.
• Key Idea: NP-hard problems are even harder than NP-complete problems. They might not even have solutions that can be verified quickly.

Theoretical Significance

The Hamiltonian Cycle Problem is more than just a puzzle, it’s a cornerstone of computational complexity theory. Here’s why it matters:

1. NP-completeness: The problem is one of the first to be proven NP-complete, meaning it is among the hardest problems in the class NP. If someone could solve it efficiently, they could solve all NP problems efficiently, which would imply P = NP, a question worth a $1 million Millennium Prize.
2. P vs NP: The problem is central to the P vs NP question, one of the most important unsolved problems in computer science. Solving the Hamiltonian Cycle Problem efficiently would revolutionize fields like cryptography, optimization, and artificial intelligence.

Real-World Applications

Despite its theoretical nature, the Hamiltonian Cycle Problem has practical applications in various fields:

1. Network Design and Routing:
2. Example: A delivery company wants to optimize routes so that drivers visit every customer exactly once and return to the depot. Finding a Hamiltonian cycle would solve this problem.
3. Circuit Design:
4. Example: In printed circuit boards (PCBs), engineers need to connect components without redundancy. A Hamiltonian cycle ensures every component is visited exactly once, minimizing waste.
5. Robotics and Path Planning:
6. Example: A cleaning robot needs to cover every room in a house without retracing its steps. A Hamiltonian cycle provides an optimal path.

Why Haven’t We Solved It Yet?

The Hamiltonian Cycle Problem remains unsolved because:

• Exponential Complexity: The number of possible paths grows exponentially with the size of the graph, making brute-force methods impractical.
• Lack of Insight: Despite decades of research, no one has found a clever way to avoid the exponential explosion in the general case.
• P vs NP: Solving the problem efficiently would imply P = NP, which is widely believed to be false.

What’s Next?

While the Hamiltonian Cycle Problem remains unsolved in the general case, researchers continue to explore:

• Heuristics and Approximation Algorithms: These methods provide practical solutions for specific types of graphs.
• Special Cases: Some graphs (e.g., complete graphs) always have Hamiltonian cycles, and algorithms exist to find them efficiently.
• Theoretical Advances: Progress in computational complexity theory could one day lead to a breakthrough.

Conclusion

The Hamiltonian Cycle Problem challenges our understanding of computation, drives the development of new algorithms, and has real-world applications in fields ranging from logistics to bioinformatics.

While it remains unsolved, its study continues to inspire researchers and push the boundaries of what we can achieve with computation.

So, the next time you’re stuck in traffic, solving a puzzle, or even planning the most efficient route for your errands, remember: you’re grappling with a problem that has stumped some of the brightest minds in computer science. And who knows?

Maybe the solution lies within you! 

Whether you’re a student, a researcher, or just someone who loves a good challenge, the Hamiltonian Cycle Problem is waiting for its next great solver.