9

M-Coloring Problem

Difficulty: MEDIUM
Contributed By
Avg. time to solve
15 min
Success Rate
85%

Problem Statement

You are given an undirected graph in the form of an adjacency matrix along with an integer M. You need to tell if you can color the vertices of the graph using at most M colors such that no two adjacent vertices are of the same color.

For example:
If the given adjacency matrix is:
[0 1 0]
[1 0 1]
[0 1 0] and M = 3.
The given adjacency matrix tells us that 1 is connected to 2 and 2 is connected to 3. So if we color vertex 1 with 2, vertex 2 with 1, and vertex 3 with 2, it is possible to color the given graph with 2 colors: M.
Input Format:
The first line of input contains a single integer T, representing the number of test cases or queries to be run. 
Then the T test cases follow.

The first line of the test case contains two space-separated integers V and M, denoting the number of vertices in the undirected graph and the number of colors respectively.

Each of the next V lines contains V integers denoting the adjacency matrix of the undirected graph.
Output Format:
For each test case, you need to return “YES” if we can color the given graph with at most M colors. Otherwise, return “NO”. (without the inverted commas)
Note:
You are not required to print the expected output, it has already been taken care of. Just implement the function.
Constraints:
1 ≤ T ≤ 1000
1 ≤ V ≤ 20
1 ≤ M ≤ V

Time Limit : 1 sec 
Sample Input 1:
3
3 3
0 1 0
1 0 1
0 1 0
3 1
0 1 0
1 0 1
0 1 0
2 1
0 1
1 0
Sample Output 1:
YES
NO
NO
Explanation of Input 1:
The first test case has already been explained in the example.

In the second test case, the given adjacency matrix tells us that 1 is connected to 2 and 2 is connected to 3. We can see that minimum of 2 colors would be needed to color the graph. So it is not possible to color the graph in this case.

The third test case, the given adjacency matrix tells us that 1 is connected to 2. We can see that minimum 2 colors would be needed to color the graph. So it is not possible to color the graph in this case.
Sample Input 2:
3
3 3
0 0 0
0 0 1
0 1 0
4 2
0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 0
4 1
0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 0
Sample Output 2
YES
YES
NO
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