最大的二分匹配问题可采用下面哪种算法求解：

The maximum bpartite matching problem can be solved using

Question

最大的二分匹配问题可采用下面哪种算法求解：

The maximum bpartite matching problem can be solved using

BlackTom AI · Accepted Answer

Construct a flow network from the bipartite graph and apply a max-flow algorithm. Specifically, create a source s connected to every vertex on the left side with capacity 1, connect each left vertex to its adjacent right vertices with capacity 1, and connect each right vertex to a sink t with capacity 1. This reduction is the core insight that makes maximum bipartite matching solvable by maximum flow methods.

1. Relevant concepts
- Bipartite matching: a set of vertex-disjoint edges between two partitions.
- Maximum flow (s–t flow): the largest amount of flow from source s to sink t respecting capacities.
- Ford–Fulkerson method: an algorithm that finds augmenting paths and increases flow until no augmenting path exists; with integral capacities it yields integral flows.

2. Step-by-step reasoning
- Build the network as described: capacities on all new edges = 1.
- Any feasible integral s–t flow corresponds to a matching: an edge (left, right) is in the matching if the flow on that edge is 1.
- Conversely, any matching corresponds to a flow of the same value (send 1 unit along s → left → right → t for each matched pair).
- Therefore the maximum s–t flow value equals the maximum matching size.

3. Intermediate steps / augmentation example
- Start with zero flow. Find an augmenting path s → u_i → v_j → t. Increase flow by 1 along that path. Each augmentation increases the matching cardinality by 1. Because capacities are 1 (integers), Ford–Fulkerson yields integer flows and thus an integral matching.

4. Why the correct answer is correct
- Ford–Fulkerson (or its implementation Edmonds–Karp) finds the maximum s–t flow in this constructed network; by the reduction above this gives the maximum bipartite matching.

5. Why other options are incorrect
- Prim’s algorithm finds a minimum spanning tree in a weighted undirected graph—unrelated to matching.
- Dijkstra’s algorithm computes shortest paths from a source—also unrelated.
- “None of these” is false because Ford–Fulkerson does solve the problem.

6. Conclusion
The maximum bipartite matching problem can be solved using the Ford–Fulkerson Algorithm for maximum flow (福特·富尔克森最大流量算法。The Ford-Fulkerson Algorithm for maximum flow.), matching the provided answer.

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