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.

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:

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