Intractable problems are solvable in principle but require excessive time to compute, often growing exponentially with input size, making them impractical for large datasets. The Travelling Salesman Problem (TSP) exemplifies this, where finding the shortest route among cities becomes computationally untenable with increased cities due to factorial growth in possible routes. Understanding these limitations emphasizes the need for developing practical algorithms that can efficiently address complex real-world problems.

2. INTRODUCTION
 Definition of Intractable Problems: Intractable problems are those that, while
solvable in principle, cannot be solved in a feasible amount of time using any currently
known algorithm.
• Importance of Understanding the Limitations of Algorithmic Solutions:
• It's crucial to recognize that not all problems can be efficiently solved by algorithms.
Understanding the boundaries of what can be realistically computed helps in setting
appropriate expectations and exploring alternative approaches for solving complex
problems.

3. WHAT ARE INTRACTABLE
PROBLEMS?
 Intractable problems are those that can be solved in principle but not in a feasible
amount of time.
• These problems have algorithmic solutions but require excessive time to compute.
• In practical terms, the time required to solve these problems grows exponentially with
the size of the input, making them impractical to solve for large inputs.
• Example of Algorithmic Limitations: Problems like NP-hard problems fall into this
category.
• NP-hard problems are those for which no known polynomial-time algorithm exists.
• Examples include the Travelling Salesman Problem (TSP), the Knapsack Problem,
and the Hamiltonian Cycle Problem.

4. EXAMPLE - THE TRAVELLING
SALESMAN PROBLEM
• Description of the Problem:
• The travelling salesman problem (TSP) involves finding the shortest route that visits a set
of cities and returns to the starting point.
• The goal is to determine the most efficient path that minimizes the total travel distance or
cost.
• Complexity for a Small Number of Cities:
• For 10 cities, there are 9! (362,880) possible tours.
• This factorial growth means that the number of possible routes increases dramatically with
each additional city.
• Challenges:
• The primary challenge is the computational difficulty of examining all possible tours to find
the shortest one.
• As the number of cities increases, the number of potential routes grows factorially, making
it infeasible to compute all possible routes for even a modest number of cities.

5. COMPLEXITY WITH INCREASED
CITIES
• Increased Complexity:
• For 40 cities, the number of possible tours is 39! (approximately 10451045), which is an
astronomical number.
• This exponential growth demonstrates the rapid increase in complexity as the number of
cities increases.
• Computational Impracticality:
• Even with a computer that can examine 10151015 tours per second, it would take an
impractical amount of time.
• Specifically, evaluating all 10451045 possible routes would take 10301030 seconds.
• Illustration of Time Required:
• To put this into perspective, 10301030 seconds is equivalent to several billion lifetimes of
the universe.
• This illustrates why even theoretically solvable problems can become practically
impossible to solve due to the enormous time required.

6. CONCLUSION AND GOAL
• Summary:
• Solvable problems are not always practically feasible.
• The intractability of certain problems, like the travelling salesman problem, highlights the
limitations of current computational methods.
• Goal:
• The goal is to develop a formal mathematical theory to find practical feasible algorithms.
• This involves creating algorithms that can solve problems efficiently, within a reasonable
amount of time, for real-world applications.