Add min cost flow

atcoder/mincostflow.py

@@ -0,0 +1,161 @@
+from __future__ import annotations
+
+from typing import NamedTuple, Optional, List
+from heapq import heappush, heappop
+
+
+class MCFGraph:
+    class Edge(NamedTuple):
+        src: int
+        dst: int
+        cap: int
+        flow: int
+        cost: int
+
+    class _Edge:
+        def __init__(self, dst: int, cap: int, cost: int) -> None:
+            self.dst = dst
+            self.cap = cap
+            self.cost = cost
+            self.rev: Optional[MCFGraph._Edge] = None
+
+    def __init__(self, n: int) -> None:
+        self._n = n
+        self._g: List[List[MCFGraph._Edge]] = [[] for _ in range(n)]
+        self._edges: List[MCFGraph._Edge] = []
+
+    def add_edge(self, src: int, dst: int, cap: int, cost: int) -> int:
+        assert 0 <= src < self._n
+        assert 0 <= dst < self._n
+        assert 0 <= cap
+        m = len(self._edges)
+        e = MCFGraph._Edge(dst, cap, cost)
+        re = MCFGraph._Edge(src, 0, -cost)
+        e.rev = re
+        re.rev = e
+        self._g[src].append(e)
+        self._g[dst].append(re)
+        self._edges.append(e)
+        return m
+
+    def get_edge(self, i: int) -> Edge:
+        assert 0 <= i < len(self._edges)
+        e = self._edges[i]
+        re = e.rev
+        return MCFGraph.Edge(
+            re.dst,
+            e.dst,
+            e.cap + re.cap,
+            re.cap,
+            e.cost
+        )
+
+    def edges(self) -> List[Edge]:
+        return [self.get_edge(i) for i in range(len(self._edges))]
+
+    def flow(self, s: int, t: int, flow_limit: Optional[int] = None) -> (int, int):
+        return self.slope(s, t, flow_limit)[-1]
+
+    def slope(self, s: int, t: int, flow_limit: Optional[int] = None) -> List[(int, int)]:
+        assert 0 <= s < self._n
+        assert 0 <= t < self._n
+        assert s != t
+        if flow_limit is None:
+            flow_limit = sum(e.cap for e in self._g[s])
+
+        dual = [0] * self._n
+        prev: List[Optional[(int, MCFGraph._Edge)]] = [None] * self._n
+
+        def refine_dual() -> bool:
+            pq = [(0, s)]
+            visited = [False] * self._n
+            dist: List[Optional[int]] = [None] * self._n
+            dist[s] = 0
+            while pq:
+                dist_v, v = heappop(pq)
+                if visited[v]:
+                    continue
+                visited[v] = True
+                if v == t:
+                    break
+                dual_v = dual[v]
+                for e in self._g[v]:
+                    w = e.dst
+                    if visited[w] or e.cap == 0:
+                        continue
+                    reduced_cost = e.cost - dual[w] + dual_v
+                    new_dist = dist_v + reduced_cost
+                    dist_w = dist[w]
+                    if dist_w is None or new_dist < dist_w:
+                        dist[w] = new_dist
+                        prev[w] = v, e
+                        heappush(pq, (new_dist, w))
+            else:
+                return False
+            dist_t = dist[t]
+            for v in range(self._n):
+                if visited[v]:
+                    dual[v] -= dist_t - dist[v]
+            return True
+
+        flow = 0
+        cost = 0
+        prev_cost_per_flow: Optional[int] = None
+        result = [(flow, cost)]
+        while flow < flow_limit:
+            if not refine_dual():
+                break
+            f = flow_limit - flow
+            v = t
+            while prev[v] is not None:
+                u, e = prev[v]
+                f = min(f, e.cap)
+                v = u
+            v = t
+            while prev[v] is not None:
+                u, e = prev[v]
+                e.cap -= f
+                e.rev.cap += f
+                v = u
+            c = -dual[s]
+            flow += f
+            cost += f * c
+            if c == prev_cost_per_flow:
+                result.pop()
+            result.append((flow, cost))
+            prev_cost_per_flow = c
+        return result
+
+
+# https://atcoder.jp/contests/practice2/tasks/practice2_e
+def main() -> None:
+    import sys
+    n, k = map(int, sys.stdin.readline().split())
+    s = n * 2
+    t = s + 1
+    g = MCFGraph(t + 1)
+    grid = [list(map(int, sys.stdin.readline().split())) for _ in range(n)]
+    big = 10 ** 9
+    edges = list([] for _ in range(n))
+    for i in range(n):
+        g.add_edge(s, i, k, 0)
+        g.add_edge(i + n, t, k, 0)
+        for j in range(n):
+            edges[i].append(g.add_edge(i, j + n, 1, big - grid[i][j]))
+    g.add_edge(s, t, n * k, big)
+
+    cost = g.flow(s, t, n * k)[1]
+    print(big * n * k - cost)
+
+    result = list(['.'] * n for _ in range(n))
+    for i in range(n):
+        for j in range(n):
+            if g.get_edge(edges[i][j]).flow > 0:
+                result[i][j] = 'X'
+
+    for s in result:
+        print("".join(s))
+
+
+if __name__ == '__main__':
+    main()