# Assignment Algorithm Rotate the matrix so that there are at least as many columns as rows and let k=min(n,m). If there is no starred zero in the row containing this primed zero, Go to Step 5.

Rotate the matrix so that there are at least as many columns as rows and let k=min(n,m). If there is no starred zero in the row containing this primed zero, Go to Step 5.

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Since each worker can perform only one job and each job can be assigned to only one worker the assignments constitute an independent set of the matrix C. A brute-force algorithm for solving the assignment problem involves generating all independent sets of the matrix C, computing the total costs of each assignment and a search of all assignment to find a minimal-sum independent set.

An arbitrary assignment is shown above in which worker a is assigned job q, worker b is assigned job s and so on. The complexity of this method is driven by the number of independent assignments possible in an nxn matrix.

We will assume that the cost matrix C(i,j) has already been loaded with the first index referring to the row number and the second index referring to the column number.

For each row of the matrix, find the smallest element and subtract it from every element in its row. We can define a local variable called minval that is used to hold the smallest value in a row.

The main loop for Munkres as a step-wise algorithm is shown here implemented in C#.

is set to some value outside the range 1..7 so that done will be set to true and the program will end.

Some of these descriptions require careful interpretation.

In Step 4, for example, the possible situations are, that there is a noncovered zero which get primed and if there is no starred zero in its row the program goes onto Step 5.

Unstar each starred zero of the series, star each primed zero of the series, erase all primes and uncover every line in the matrix. Step 6: Add the value found in Step 4 to every element of each covered row, and subtract it from every element of each uncovered column.

Return to Step 4 without altering any stars, primes, or covered lines.

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