Presentation Script: Hamiltonian Path and Cycle Problems
Slide 1: Title Slide Good [morning/afternoon] everyone. We are Group 3, and today
we’ll be presenting on Hamiltonian Path and Cycle Problems, a key topic in graph
theory under Discrete Mathematics. We’ll cover what these paths are, explore their
complexity, and examine algorithmic solutions and real-world applications.
Slide 2: Team Members This presentation was prepared by Sheldon Shortridge,
Willard Welch, and Conroy Greaves.
Slide 3: Core Objectives Our core objectives today are:

To understand Hamiltonian Path and Cycle problems

Discuss the completeness of Hamiltonian problems

Explore two major algorithmic approaches: Backtracking and Dynamic
Programming

Conduct a comparative analysis

Explore real-world applications of Hamiltonian paths
Slide 4: Understanding the Problem Let’s begin with definitions: (Sheldon)

A Hamiltonian Path is a path in a graph that visits each vertex exactly once.

A Hamiltonian Cycle is a Hamiltonian Path that starts and ends at the same
vertex. This is different from Eulerian Paths, which visit every edge once.
Finding a Hamiltonian path is computationally hard. It’s classified as NP-Complete,
meaning there’s no known efficient algorithm to solve it for all cases. This makes it one
of the most challenging and fascinating problems in computer science.
Slide 5: Complexity and NP-Completeness We can verify a Hamiltonian path in
(Sheldon) polynomial time by checking the order of vertices. However, finding such a
path is NP-Hard—at least as hard as the hardest problems in NP. The proof involves
reducing a known NP-Complete problem, such as 3-SAT, to the Hamiltonian Path
problem. This indicates that solving this efficiently for all graphs would effectively solve
all NP problems, which is still an open question in computer science.
Slide 6: Backtracking Algorithm One approach to solve the problem is backtracking:
(Conroy)

We start with an empty path and recursively try to add vertices.

If a vertex is invalid (either already visited or not connected), we backtrack.
This algorithm has a time complexity of O(N!) in the worst case. Here’s a simplified
version of the pseudocode provided in the slide, and a visualization follows to illustrate
how the algorithm explores paths and backtracks when stuck.
Slide 7: Dynamic Programming Approach Another approach is Dynamic
Programming: (Conroy)

We define a state as (S, v) where S is the set of visited vertices and v is the last
vertex.

The recursive relation builds up possible paths by checking subsets.

Time complexity is O(n^2 * 2^n), which is significantly better than backtracking for
some instances. This approach uses memoization to avoid recalculating
overlapping subproblems, making it a more scalable solution.
Slide 8: Key Insights To summarize both algorithmic strategies: (Conroy)

Backtracking is simple but computationally expensive.

Dynamic Programming is more efficient but requires more memory.

Both methods are exponential in nature due to the complexity of the problem, but
they offer trade-offs between time and space.
Slide 9–11: Real-World Applications Let’s look at where Hamiltonian Paths are used
in the (Willard)real world.
Circuit Design: In integrated circuits and printed circuit boards (PCBs), Hamiltonian
paths help:

Minimize wiring complexity

Avoid crossover

Optimize layouts for manufacturability and cost
Genome Sequencing: In bioinformatics, Hamiltonian paths help reconstruct genomes
from DNA fragments:

Each DNA segment is a vertex

Overlaps between fragments are edges

A Hamiltonian Path reconstructs the sequence without repeats
This is essential in fields like disease research, vaccine development, and genetic
analysis.
Slide 12: Video [Optional: Play the short video that visually represents the real-world
applications. Narrate key moments or let the video speak for itself.] (Willard)
Slide 13: Conclusion and Reflection In conclusion, Hamiltonian Path and Cycle
Problems (Joevan)lie at the intersection of theory and real-world problem solving.
Although NP-Complete, various algorithms provide partial or practical solutions
depending on the case. Understanding these problems equips us with tools for tackling
complex computational challenges.
Slide 14: References We relied on a mix of scholarly and online resources such as:
(Joevan)

Cormen’s Introduction to Algorithms

Research by Karp, Held & Karp, and Bjorklund

GeeksforGeeks, InterviewBit, and Brilliant.org for practical explanations
Slide 15: Thank You Thank you for your attention. We are open to any questions you
may have.