Optimized Longest Path is NP Complete

Optimized Longest Path Problem: The optimized longest path problem states that given a graph G, of a set of vertices V and edges E, the task is to prove that there is a path of length at least K between a set of nodes Vs and Ve.

Problem Statement: Given a graph G(V, E, K) and a set of nodes Vs and Ve with the sequence of nodes, of length ≥ K.

Explanation:
An instance of the problem is an input specified to the problem. An instance of the optimized-longest path problem is G(V, E, Vs, Ve, K). Since an NP-complete problem is a problem which is both in NP and NP-Hard, the proof for the statement that a problem is NP-Complete consists of two parts:

  1. The problem itself is in NP class.
  2. All other problems in NP class can be polynomial-time reducible to that.
    (B is polynomial-time reducible to C is denoted as B≤PC)

If the 2nd condition is only satisfied then the problem is called NP-Hard.

But it is not possible to reduce every NP problem into another NP problem to show its NP-Completeness all the time. That is why if we want to show a problem is NP-Complete we just show that the problem is in NP and any NP-Complete problem is reducible to that then we are done, i.e. if B is NP-Complete and B ≤ PC For C in NP, then C is NP-Complete. Thus, we can verify that the Optimized Longest Path Problem is NP-Complete using the following two propositions:



Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.


Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.


Article Tags :
Practice Tags :