## Description

(30 pts.)

1 Introduction

The Hamiltonian Path Problem is a classic computer science problem: Given a graph and two vertices i and j, determine whether there is a path from i to j in the graph that visits each vertex in the graph exactly once. Now, this problem is well-known to be NP-complete. I have written a solution to this problem (Hamiltonian_Path.cc) that uses the next_permutation function in C++ to generate all of the permutations (tours) of the vertices that start at vertex i and end in vertex j. This program will ﬁnd a Hamiltonian Path, if it exists. Otheriwse, it will say that no such path exists. However, this program is painfully slow. On a small graph (small_graph.dat) with 5 vertices, it ﬁnds the the tour 2 0 1 3 4 from 2 to 4 in much less than a second. But, on a bigger graph (big.dat) with 13 vertices, it takes over one minute to ﬁnd a solution. On bigger graphs (bigger.dat and biggest.dat), it takes much longer to solve. For example, on input

5 2 4 0 : 1 2 4 1 : 0 2 3 4 2 : 0 1 3 3 : 1 2 4 4 : 0 1 3

The output of the program should be “Tour = 2 0 1 3 4”. Your task is to make this code faster using parallel computing. Modify Hamiltonian_Path.cc using OpenMP so that

1. Your modiﬁed program still produces the correct results, and

2. It is at least 75% eﬃcient on bigger.dat on a machine with 4 cores/processors.