Minimum Cut

Resources

The resources below include many clever applications of min cut, including the Closure Problem.

Minimum Node Covers

Solution - Coin Grid

This problem asks us to find a minimum node cover of a bipartite graph. Construct a flow graph with vertices labeled $0\ldots 2N+1$, source $0$, sink $2N+1$, and the following edges:

• Edges from $0\to i$ with capacity $1$ for each $1\le i\le N$. Cutting the $i$-th such edge corresponds to choosing the $i$-th row.
• Edges from $N+i\to 2N+1$ with capacity $1$ for each $1\le i\le N$. Cutting the $i$-th such edge corresponds to choosing the $i$-th column.
• If there exists a coin in $(r,c)$ add an edge from $r\to N+c$ with capacity $\infty$.

First we find a max flow, which tells us the number of edges with capacity 1 we need to cut. To find the min cut itself, BFS from the source once more time. Edges $(a,b)$ connecting vertices that are reachable from the source (lev[a] != -1) to vertices that aren't (lev[b] == -1) are part of the minimum cut. In this case, each of these edges must be of the form $(0,i)$ or $(i+N,2N+1)$ for $1\le i\le N$. Each cut edge corresponds to a row or column we remove coins from.

Note that edges of the form $r\to N+c$ can't be cut because they have capacity $\infty$.

Copy
struct Dinic {     // flow template
	using F = ll;  // flow type
	struct Edge {
		int to;
		F flo, cap;
	};
	int N;
	V<Edge> eds;
	V<vi> adj;
	void init(int _N) {

Minimum Path Covers

Solution - The Wrath of Kahn

Ignore all vertices of $G$ that can never be part of $S$. Then our goal is to find the size of a maximum antichain in the remaining graph, which as mentioned in CPH is just an instance of maximum path cover.

Copy
TopoSort<500> T;
int n, m;
bool link[500][500];
vi out[500];
Dinic<1005> D;

int main() {
	setIO();
	re(n, m);
	F0R(i, m) {

Problems