Fast C++ implementation of an O(NM) algorithm for the assignment problem.

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# jamespayor/weighted-bipartite-perfect-matching

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## Fast C++ implementation of the Hungarian algorithm

This is an open-source implementation of the "O(N^3)" dynamic-programming version of the Hungarian algorithm, for weighted perfect bipartite matching. It's written with speed in mind, whilst trying to remain readable-ish.

It's also written to run in O(NM) time, where there are N nodes and M edges, meaning it will be substantially faster than a cost-matrix version on sparse graphs. (I also think the edge list representation turns out to speed things up even in the dense case. The inner loop exploits that edges only intermittently become tight/non-tight, and hence avoids a lot of work scanning all the edges every time.)

The best description of the algorithm I'm aware of is this TopCoder post, which I highly recommend should you attempt to understand what's going on.

If you're looking for competition code, I wouldn't recommend the approach taken here. You can write something vastly simpler if you assume a dense graph and only care about the asymptotic (worst-case) runtime (e.g. something on the order of 100 lines, rather than 400). For instance, this gist is an ancient O(N^4) version of max-weight matching from my past life.

Related: See https://github.com/weaversa/weighted-bipartite-perfect-matching for a C reimplementation.

### Usage

Choose your `n` (the number of nodes on each side of the bipartite graph), build a `vector` of `WeightedBipartiteEdge`s, then call `hungarianMinimumWeightPerfectMatching(n, edges)`.

You'll receive back either an empty `vector` (indicating that no perfect matching is possible), or a `vector` of size `n` giving the matching.

If `matching` is the result, then `matching[i]` gives the node on the right that the left node is matched to.

### Use cases

• Solving the assignment problem.
• In which we want to assign every node on the left to a node on the right, and minimize cost / maximize profit.
• General minimum-weight bipartite matching, where the right side has more nodes than the left.
• Solution sketch: add dummy nodes to the left that have high cost / low profit when matched.
• Maximum-weight bipartite matching.
• Just negate the costs. (The algorithm copes with costs as large and negative as you like, so long as no two of them add up to an integer overflow.)

### Sketch of what the algorithm does

Seriously consider reading the TopCoder post and the Wikipedia page as well. That said, here's my attempt at an explanation.

##### What are node "potentials" even?

The underlying theory is best understood viewing the node potentials in the algorithm as pricing functions, as seen in min-cost max-flow. I'll focus on what goes on in the Hungarian algorithm though.

First, note that adding X to the cost of all edges for a node on the left doesn't affect the final matching. It just increases the cost of every possible matching by X. Similarly, shifting the cost of all edges to a node on the right just changes the cost of every matching.

We've simply called "the amount we decrease the cost of edges connected to node i" the potential of node i. As we build a matching, we can freely move these potentials up and down, adjusting the edge costs, without changing the result.

##### Why is it useful to adjust costs in this way?

Let's call the costs adjusted by potential the reduced costs. The algorithm likes to proceed by maintaining a property:

• In the residual graph (in which edges in the matching go from right to left), there is no cycle with negative reduced cost.

We maintain this property from the beginning to the end. If at the end we have a perfect matching (i.e. a flow) with no negative residual cycles, then it's impossible to improve the matching (i.e. the flow is a min-cost flow).

(To see this, suppose we have a matching that costs more than optimal. Treat this as a flow, and subtract away any optimal flow. The difference must be a circulation, i.e. a set of cycles. One of these cycles must be negative-cost in our matching, otherwise it would have the same cost as the optimal one.)

##### What the algorithm does.

So, at the beginning, we have an empty matching, which has no residual cycles. We maintain always that edges have positive reduced cost. Then, we iterate:

• Adjust node potentials so that some of the edges have zero reduced cost.
• Attempt to augment our matching only using these zero-cost edges.

The algorithm can keep adding at least one new edge to the section of the graph it's exploring (with edges that have zero reduced cost), and so it must inexorably succeed.

##### Caveats

I've probably gotten something wrong in the above explanation, because linear programming isn't my thing, and I don't actually understand the algorithm well enough to have derived it. I hope this helps someone though!

Fast C++ implementation of an O(NM) algorithm for the assignment problem.

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