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POSecond best solution to an assignmentproblem using the Hungarian Algorithm
 

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    <p>For finding the best solution in the assignment problem it's easy to use the Hungarian Algorithm. For example:</p> <pre><code>A | 3 4 2 B | 8 9 1 C | 7 9 5 </code></pre> <p>When using the Hungarian Algorithm on this you become:</p> <pre><code>A | 0 0 1 B | 5 5 0 C | 0 1 0 </code></pre> <p>Which means A gets assigned to 'job' 2, B to job 3 and C to job 1. However, I want to find the second best solution, meaning I want the best solution with a cost strictly greater that the cost of the optimal solution. According to me I just need to find the assignment with the minimal sum in the last matrix without it being the same as the optimal. I could do this by just searching in a tree (with pruning) but I'm worried about the complexity (being O(n!)). Is there any efficient method for this I don't know about?</p> <p>I was thinking about a search in which I sort the rows first and then greedily choose the lowest cost first assuming most of the lowest costs will make up for the minimal sum + pruning. But assuming the Hungarian Algorithm can produce a matrix with a lot of zero's, the complexity is terrible again...</p>
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