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Chapter 9: Problem 1

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

Short Answer

Expert verified
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

01

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.
02

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.
03

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.

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Most popular questions from this chapter

Show that a graph problem using the number of vertices as the measure of the size of an instance is polynomially equivalent to one using the number of edges as the measure of the size of an instance.

When all instances of the CNF-Satisfiability Problem have exactly three literals per clause, it is called the 3 -Satisfiability Problem. Knowing that the 3-Satisfiability Problem is \(N P\) -complete, show that the Graph 3 -Coloring Problem is also \(N P\) -complete.

Show that a problem is \(N P\) -easy if and only if it reduces to an \(N P\) -complete problem.

Suppose that problem \(A\) and problem \(B\) are two different decision problems. Furthermore, assume that problem \(A\) is polynomial-time many-one reducible to problem \(B\), If problem \(A\) is \(N P\) -complete, is problem \(B\) NP-complete? Justify your answer.

Show that the reduction of the Traveling Salesperson (Undirected) Decision Problem to the Traveling Salesperson Decision Problem can be done in poly. nomial time.
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