Question

The Test Cover problem can be defined as follows:

Suppose we have a set of n diseases and a set of m tests we can perform to check for symptoms. We also are given the following:

• an nxn matrix A where A[i][j] is a binary value representing the result of running the jth test on a patient with the the ith disease (1 indicates a positive result, 0 indicates negative);
• the cost of running test j, c_j; and that
• any patient will have exactly one disease

The task is to find a set of tests that can uniquely identify each of the the n diseases at minimal cost.

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This problem can be formulated as an Integer Linear Program, where we want to minimize the objective function \sum_{j=1}^{m} c_j x_j, where x_j = 1 if we choose to include test j in our set, and 0 otherwise.

My question is:

What is the set of linear constraints for this problem?

Incidentally, I believe this problem is NP-hard (as is Integer Linear Programming in general).

Answer 1

Let T be the matrix that results from deleting the jth column of A for all j such that x_j = 0.

Then choosing a set of tests that can uniquely distinguish any two diseases is equivalent to ensuring that every row of T is unique.

Observe that two rows k and l are identical if and only if (T[k][j] XOR T[l][j]) = 0 for all j.

So, the constraints we want are

\sum_{j=1}^{m} x_j(A[k][j] XOR A[l][j]) >= 1

for all 1 <= k <= m and 1 <= l <= 1 such that k != l.

Note that the constraints above are linear, since we can just pre-compute the coefficient (A[k][j] XOR A[l][j]).