Question

List three problems that have polynomial-time algorithms. Justify your answer.

Short answer

Sorting, Shortest Path, and Maximum Flow are problems that have polynomial-time algorithms since their most efficient solving algorithms show polynomial time complexities. For instance, Quick Sort, Merge Sort, and Heap Sort for Sorting, Dijkstra's and Bellman-Ford for Shortest Path, and the Edmonds-Karp implementation of the Ford-Fulkerson method for Maximum Flow all have polynomial time complexities, classifying them as P-Time algorithms.

Step-by-step solution

Problem 1 - Sorting

Sorting is the problem of arranging items in ascending or descending order. There are many polynomial-time algorithms available to solve this problem such as Quick Sort, Merge Sort, Heap Sort, etc. These algorithms have worst-case time complexities of \( O(n^2) \), \( O(n \log n) \), and \( O(n \log n) \) respectively, which classifies them as P-Time algorithms.

Problem 2 - Shortest Path

The shortest path problem is about finding the shortest paths between vertices in a graph. Algorithms like Dijkstra's and Bellman Ford can solve this problem in polynomial time. Dijkstra's algorithm runs in \( O(|E|+|V|\log |V|) \), and Bellman Ford runs in \( O(|V||E|) \), both of which are polynomial.

Problem 3 - Maximum Flow

The maximum flow problem involves finding a feasible flow through a single-source, single-sink flow network that is maximum. The Ford-Fulkerson method can solve this problem in polynomial time. Although its worst-case time complexity is infinite, the Edmonds-Karp implementation provides a strong polynomial-time bound of \( O(|V|^2|E|^2) \) which makes it a P-time algorithm.