网络流
最大流
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| struct Graph { struct Edge { int v; long w; Edge(int v = 0, long w = 0) : v(v), w(w) {} }; int S, T, cur[N], d[N]; vector<Edge> edges; vector<int> G[N]; void addedge(int u, int v, long w) { edges.emplace_back(v, w); G[u].emplace_back(edges.size() - 1); edges.emplace_back(u, 0); G[v].emplace_back(edges.size() - 1); } bool bfs(void) { memset(d, -1, sizeof d); queue<int> q; q.push(S); d[S] = 1; while (!q.empty()) { const int u = q.front(); q.pop(); for (const int id : G[u]) { Edge &e = edges[id]; if (e.w && d[e.v] == -1) { d[e.v] = d[u] + 1; q.push(e.v); if (e.v == T) return true; } } } return false; } long dinic(int x, long res) { if (x == T) return res; long flow = 0; for (int i = cur[x]; i < (int)G[x].size() && res; ++i) { cur[x] = i; Edge &e = edges[G[x][i]]; const long c = min(res, e.w); if (d[e.v] == d[x] + 1 && c) { long k = dinic(e.v, c); flow += k; res -= k; e.w -= k; edges[G[x][i] ^ 1].w += k; } } if (!flow) d[x] = -1; return flow; } long maxFlow(int s, int t) { S = s, T = t; long mxf = 0; while (bfs()) { memset(cur, 0, sizeof cur); while (long flow = dinic(S, 1e18)) mxf += flow; } return mxf; } } G;
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图匹配
完美匹配的存在性
Hall 定理:对于二分图 G=(X,Y,E),存在完美匹配的充要条件是对于任意 W⊆X,都有 ∣W∣≤∣N(W)∣。