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Minimum Cut

Author: Benjamin Qi

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Resources

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

Resources
CPCSlides from "Algorithm Design." Min-Cut Max-Flow Theorem, applications of flow / min cut.

Minimum Node Covers

Focus Problem – read through this problem before continuing!

Resources
CPHbrief mentions of Hall's Theorem, Konig's Theorem

Solution - Coin Grid

This problem asks us to find a minimum node cover of a bipartite graph. Construct a flow graph with vertices labeled 02N+10\ldots 2N+1, source 00, sink 2N+12N+1, and the following edges:

  • Edges from 0i0\to i with capacity 11 for each 1iN1\le i\le N. Cutting the ii-th such edge corresponds to choosing the ii-th row.
  • Edges from N+i2N+1N+i\to 2N+1 with capacity 11 for each 1iN1\le i\le N. Cutting the ii-th such edge corresponds to choosing the ii-th column.
  • If there exists a coin in (r,c)(r,c) add an edge from rN+cr\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)(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)(0,i) or (i+N,2N+1)(i+N,2N+1) for 1iN1\le i\le N. Each cut edge corresponds to a row or column we remove coins from.

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

1struct Dinic { // flow template
2 using F = ll; // flow type
3 struct Edge { int to; F flo, cap; };
4 int N; V<Edge> eds; V<vi> adj;
5 void init(int _N) { N = _N; adj.rsz(N), cur.rsz(N); }
6 /// void reset() { trav(e,eds) e.flo = 0; }
7 void ae(int u, int v, F cap, F rcap = 0) { assert(min(cap,rcap) >= 0);
8 adj[u].pb(sz(eds)); eds.pb({v,0,cap});
9 adj[v].pb(sz(eds)); eds.pb({u,0,rcap});
10 }

Minimum Path Covers

Focus Problem – read through this problem before continuing!

Resources
CPHbrief mentions of node-disjoint and general path covers, Dilworth's theorem
Wikipediaproof via Konig's theorem

Solution - The Wrath of Kahn

Ignore all vertices of GG that can never be part of SS. 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.

1TopoSort<500> T;
2int n,m;
3bool link[500][500];
4vi out[500];
5Dinic<1005> D;
6
7int main() {
8 setIO(); re(n,m);
9 F0R(i,m) {
10 int x,y; re(x,y);

Problems

StatusSourceProblem NameDifficultyTagsSolution
CSESEasyView Solution
Old GoldEasy
Show Tags

Max Flow

External Sol
CSANormalCheck CSA
CFNormalCheck CF
CFNormalCheck CF
CFHardCheck CF
ACHardCheck AC
FHCHardCheck FHC

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