PROVE IT IS IN NP BY SELECTING ANOTHER NP PROBLEM, AND CALCULATE THE COMPLEXITY AND SHOW THAT IT RUNS POLYNOMIAL TIME. SOME NP PROBLEM LIKE 3D MATCHING, CLIQUE, PARTITION, VERTEX-COVER, HAMILTONIAN CIRCUIT, TRAVELING SALESMAN, 3SAT, SAT. JUST USE THESE NP PROBLEMS TO SOLVE THE 2ND PART. DO NOT SOLVE WITH OTHER NP PROBLEMS.

Suppose that you are given a set T of projects where each project tT is associated with - a positive integer et denoting the amount of effort (e.g., the amount of time or energy) required to complete the project t, and - a positive integer pt denoting the profit gained by completing the project t. Given a positive integer c denoting the maximum amount of effort that you can spend on these projects, the goal is to find the largest amount of profit that you can make by completing a subset of these projects such that the total amount of effort spent to complete the projects is at most c. 1. (50 points) State the decision version of this optimization problem: Input? Output? 2. (50 points) Prove that the decision version is in NP