Steps To Solve Assignment Problem Critical Thinking Video Clips
For each vertex from left part (workers) find the minimal outgoing edge and subtract its weight from all weights connected with this vertex. Actually, this step is not necessary, but it decreases the number of main cycle iterations. Find the maximum matching using only 0-weight edges (for this purpose you can use max-flow algorithm, augmenting path algorithm, etc.). Step 2) Let and adjust the weights using the following rule: Step 3) Repeat Step 1 until solved.But there is a nuance here; finding the maximum matching in step 1 on each iteration will cause the algorithm to become O(n5).Let’s call this labeling feasible if it satisfies the following condition: .In other words, the sum of the labels of the vertices on both sides of a given edge are greater than or equal to the weight of that edge.The only thing in code that hasn't been explained yet is the procedure that goes after labels are updated.Say we've updated labels and now we need to complete our alternating tree; to do this and to keep algorithm in O(n3) time (it's only possible if we use each edge no more than one time per iteration) we need to know what edges should be added without iterating through all of them, and the answer for this question is to use BFS to add edges only from those vertices in Y, that are not in T and for which slack[y] = 0 (it's easy to prove that in such way we'll add all edges and keep algorithm to be O(n3)).(For example, (W4, J4, W3, J3, W2, J2) and (W4, J1, W1) are alternating paths)If the first and last vertices in alternating path are exposed, it is called (because we can increment the size of the matching by inverting edges along this path, therefore matching unmatched edges and vice versa). And now let’s illustrate these steps by considering an example and writing some code.
To clarify, let’s look at the step-by-step overview: Step 0)A. Apply the same procedure for the vertices in the right part (jobs). Otherwise find the minimum vertex cover (for the subgraph with 0-weight edges only), the best way to do this is to use Köning’s graph theorem.
We’ll handle the assignment problem with the Hungarian algorithm (or Kuhn-Munkres algorithm).
I’ll illustrate two different implementations of this algorithm, both graph theoretic, one easy and fast to implement with O(n4) complexity, and the other one with O(n3) complexity, but harder to implement.
See picture below for explanation: At last, here's the function that implements Hungarian algorithm: To see all this in practice let's complete the example started on step 0.| Build alternating tree V| Augmenting V path found| Build V alternating tree| Update V labels(Δ=1)| Build alternating V tree| Update V labels(Δ=1)→| Build V alternating tree| Augmenting V path found| Build V alternating tree| Update labels V (Δ=2)| Build V alternating tree| Update labels V (Δ=1)| Build V alternating tree| Augmenting V path found Optimalmatching found Finally, let's talk about the complexity of this algorithm.
On each iteration we increment matching so we have n iterations.