Assignment Algorithm

Assignment Algorithm-18
The main loop for Munkres as a step-wise algorithm is shown here implemented in set to some value outside the range 1..7 so that done will be set to true and the program will end.The following 6-step algorithm is a modified form of the original Munkres' Assignment Algorithm (sometimes referred to as the Hungarian Algorithm).

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The question is raised as to whether there is a better algorithm.

In fact there exists a polynomial runtime complexity algorithm for solving the assignment problem developed by James Munkre's in the late 1950's despite the fact that some references still describe this as a problem of exponential complexity.

Continue in this manner until there are no uncovered zeros left.

Save the smallest uncovered value and Go to Step 6. Continue until the series terminates at a primed zero that has no starred zero in its column.

Once minval has been found this value is subtracted from each element of that row in the second inner loop over j. In this step, we introduce the mask matrix M, which in the same dimensions as the cost matrix and is used to star and prime zeros of the cost matrix.

The value of step is set to 2 just before stepone ends. If there is no starred zero in its row or column, star Z. If M(i,j)=1 then C(i,j) is a starred zero, If M(i,j)=2 then C(i,j) is a primed zero. Before we go on to Step 3, we uncover all rows and columns so that we can use the cover vectors to help us count the number of starred zeros. If K columns are covered, the starred zeros describe a complete set of unique assignments. Once we have searched the entire cost matrix, we count the number of independent zeros found. If there is no starred zero in the row containing this primed zero, Go to Step 5.By applying Rule 4 to the step-algorithm we decide to make each step its own procedure.Now we can apply Rule 8 by using a case statement in a loop to control the ordering of step execution.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.The other possible way out of Step 4 is that there are no noncovered zeros at all, in which case the program goes to Step 6.At first it may seem that the erratic nature of this algorithm would make its implementation difficult.DONE: Assignment pairs are indicated by the positions of the starred zeros in the cost matrix.If C(i,j) is a starred zero, then the element associated with row i is assigned to the element associated with column j.Notice that there are two loops over the index j appearing inside an outer loop over the index i.The first inner loop over the index j searches a row for the minval.


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