Question

Which of the following problems are in the class P? a. For a given set \(\mathrm{S}\) of \(\mathrm{n}\) integers, sort \(\mathrm{S}\). b. The traveling salesperson problem. c. The Hamilton Path problem. d. The Node Cover problem.

Short answer

Only problem a (sorting a set of integers) is in the class P.

Step-by-step solution

Understanding the Class P

The class P, in computational complexity theory, consists of decision problems that can be solved in polynomial time by a deterministic Turing machine. This means that there exists an algorithm that can solve the problem and its time complexity grows at a polynomial rate with the size of the input.

Evaluating Problem a

Problem a involves sorting a set of integers. Sorting algorithms such as QuickSort, MergeSort, and HeapSort can solve this problem in polynomial time (specifically O(n log n)), which falls under the class P.

Evaluating Problem b

Problem b is the Traveling Salesperson Problem (TSP). TSP is a well-known NP-complete problem. In the general case, there is no known polynomial-time solution for TSP, which means it is not in class P.

Evaluating Problem c

Problem c is the Hamilton Path problem, which asks if there exists a path in a graph that visits each vertex exactly once. This problem is NP-complete, meaning it is not in class P, as it cannot be solved in polynomial time by a deterministic Turing machine.

Evaluating Problem d

Problem d is the Node Cover problem, which is also a known NP-complete problem. Like other NP-complete problems, there is no known polynomial-time algorithm that solves it, so it is not in class P.

Key concepts

Polynomial Time

When we talk about polynomial time in computer science, we're discussing how long an algorithm takes to run, relative to the size of its input. The size of inputs can change with the given problem, like the number of items to sort, the number of nodes in a graph, etc. An algorithm is said to run in polynomial time if its runtime is bounded by a polynomial expression such as, for example, \(n^2\) or \(n^3\). Here, \(n\) represents the size of the input.

Polynomial time is significant because it is generally considered to be a measure of efficiency. Most computer scientists regard problems solvable in polynomial time as 'tractable' or 'feasible'. Algorithms that run in polynomial time are manageable for computers to solve, even as the input size becomes large.

• Examples include sorting a list, like using QuickSort or MergeSort, which run in \(O(n \log n)\) time.
• This concept is contrasted with exponential time algorithms, which become impractical as \(n\) grows.

NP-complete

NP-complete problems stand at the core of computational complexity theory. These problems are the toughest among those that sit in the NP category. NP stands for nondeterministic polynomial time. Essentially, this means that while these problems are verifiable in polynomial time given a solution, finding that solution might take much longer.

A problem is deemed NP-complete when two main criteria are met:

• It is in NP, meaning solutions can be verified quickly (in polynomial time) once found.
• It is NP-hard, indicating that every problem in NP can be transformed into this problem via a polynomial-time reduction.

These problems include classics like the Traveling Salesperson Problem and the Hamiltonian Path problem. If someone finds a polynomial-time solution for any NP-complete problem, it would mean all problems in NP can be solved in polynomial time, a question tightly linked to the unsolved "P vs. NP" problem.

Deterministic Turing Machine

The concept of a deterministic Turing machine (DTM) is foundational to understanding computational complexity. A deterministic Turing machine is a theoretical model of computation. It conceptualizes a machine that reads input from a tape, processes it, and writes output, all in a very precise, pre-defined order. At each step, a deterministic Turing machine decides the next action based strictly on its current state and the input it reads.

Here are a few key aspects:

• A DTM has a finite set of instructions. Each instruction tells it what move to make based on the current state and tape symbol.
• This contrasts with a nondeterministic Turing machine, which may 'guess' steps, with multiple possible outcomes from a single state.
• Problems solved by DTMs in polynomial time are considered part of the class P.

Understanding DTMs helps with grasping why certain problems are easily solvable - i.e., those in P - while others, like NP-complete problems, are much more complex to tackle.

Sorting Algorithms

Sorting algorithms play a vital role in computer science and many other fields. They organize data in a specific sequence or order, often numerical or lexicographical. This process is crucial because many algorithms and data structures require sorted data to function efficiently.

There are various popular sorting algorithms, with different time complexities and use cases. Some efficient algorithms you'll encounter include:

• QuickSort: A versatile algorithm with a best-case time complexity of \(O(n \log n)\), but can degrade to \(O(n^2)\) in the worst case.
• MergeSort: Always runs in \(O(n \log n)\) time, making it reliable even for large datasets and useful for linked lists.
• HeapSort: This also runs in \(O(n \log n)\) time, offering in-place sorting which can be beneficial when memory space is a concern.

Algorithms like these are optimized for speed and reliability, helping efficiently solve a variety of practical problems in software development and data processing. They're perfect examples of problems solvable in polynomial time, demonstrating their efficiency and importance in class P.