关于最小割最大流定理的技术咨询：寻找边数最少的最小容量割

Hey there! Let's tackle your two questions about min-capacity cuts in flow networks—first the specific check for a cut with ≤100 edges, then the general method to find the min-edge min-capacity cut.

Checking for a Min-Capacity Cut with ≤100 Edges #

To answer this, we first need to find the edge-count minimal min-capacity cut (the smallest set of edges that forms a cut with the minimal possible total capacity). If this cut has ≤100 edges, then your answer is yes; if it has more, then no such cut exists.

Here's the step-by-step process:

1. Compute the maximum flow of the original network (by the max-flow min-cut theorem, this equals the capacity of any min-capacity cut).
2. Transform the network to prioritize edge count among all min-capacity cuts (details in the general approach below).
3. Find the min-cut in this transformed network—it will be the edge-count smallest min-capacity cut from the original network.
4. Count the number of edges in this cut. If it's ≤100, you have your answer.

General Approach: Finding the Min-Edge Min-Capacity Cut #

The core idea is to adjust edge weights so that when we compute the min-cut, it first optimizes for minimal capacity, then for minimal edge count. Here are two reliable methods (one avoids floating points, which is better for real-world implementation):

Method 1: Integer Weight Transformation (Recommended) #

This method avoids floating-point precision errors by scaling original capacities:

• Let total_edges be the total number of edges in your network.
• For each edge with original capacity c_e, set its new capacity to c_e * (total_edges + 1) + 1.
• Run your favorite max-flow algorithm (like Dinic's, which is efficient for large networks) on this transformed network to find the max-flow, then derive the corresponding min-cut.

Why this works:

• For any two cuts with different original capacities, the scaled capacity difference will be at least total_edges + 1—which is larger than the maximum possible difference in edge counts (which is total_edges). This ensures the scaled network still prioritizes the original minimal capacity first.
• For cuts with the same original capacity, the scaled capacity becomes original_capacity * (total_edges + 1) + edge_count. The cut with fewer edges will have a smaller scaled capacity, so it becomes the min-cut of the transformed network.

Method 2: Floating-Point Weight Transformation #

If you prefer a more intuitive approach (just watch out for precision issues):

• Choose a tiny value ε such that ε < 1 / total_edges (e.g., ε = 1 / (total_edges + 1) if all original capacities are integers).
• For each edge, set its new capacity to c_e + ε.
• Compute the max-flow and corresponding min-cut of this network.

The logic here is simple: the ε term adds a tiny penalty per edge. Among cuts with the same original capacity, the one with fewer edges will have a smaller total capacity in the transformed network, so it will be selected as the min-cut.

Extracting the Cut #

Once you have the max-flow of the transformed network:

• Perform a BFS or DFS on the residual network starting from the source node. All nodes reachable from the source belong to set S; the rest belong to set T.
• The edges that go from S to T in the original network form your edge-count minimal min-capacity cut.

内容的提问来源于stack exchange，提问作者toxin9