Bipartite Matching via Max-Flow

• If capacities are large (e.g. |f^*| = 2^{30}), this is exponential in the input size.
• Solution: choose augmenting paths smartly.

Key Idea

Always pick the shortest augmenting path (fewest edges) using BFS.

Result

• Shortest-path distances in G_f increase monotonically.
• At most O(VE) augmentations.
• Each BFS takes O(E).
• Total: O(VE^2) — independent of capacities!

Reduction to Max-Flow:

1. Add supersource s, connect to all u \in L with capacity 1.
2. Add supersink t, connect all v \in R to t with capacity 1.
3. Direct all original edges L \to R with capacity 1.
4. Find max flow. Max flow value = max matching size!

Each unit of flow = one matched pair

• Flow s \to u \to v \to t means u is matched to v.
• Unit capacity on s \to u ensures u is matched to at most one partner.
• Unit capacity on v \to t ensures v is matched to at most one partner.

The Integrality Theorem

If all capacities are integers, the max flow found by Ford-Fulkerson is integral (flow on every edge is an integer). * Since all capacities are 1, every edge has flow either 0 or 1. * The edges with flow 1 form a valid matching. ✓

• Max flow value \le |L| \le V.
• At most V BFS augmentations.
• Each BFS: O(E).
• Total: O(VE) — much better than general O(VE^2)!

Answer: The reduction still works! The unit capacities on s \to L and R \to t already enforce that each vertex is matched at most once. The internal edges don’t need the restriction.