Luzhiled's Library

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:heavy_check_mark: Knapsack Limitations(個数制限つきナップサック問題) $O(N^2 \max(v_i)^2)$
(dp/knapsack-limitations-2.hpp)

概要

個数制限つきナップサック問題を次に示す.

重さ $w_i$, 価値 $v_i$ であるような $N$ 種類の品物がある. $i$ 番目の品物は $m_i$ 個まで選ぶことができる. 重さの和が $W$ 以下となるように選ぶとき, 価値の最大値を求めよ.

まず「dp[価値の和]:= 重さの和の最小値」をある程度の大きさ($N \max(v_i)^2$)まで求める。残りの分は, 効率が良い順(価値/重さが大きい順)に貪欲にとってもいいことが示せて, なんかできる(c).

計算量

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Code

#include "knapsack-limitations.hpp"

template< typename T >
T knapsack_limitations(const vector< T > &w, const vector< T > &m, const vector< int > &v,
                       const T &W) {
  const int N = (int) w.size();
  auto v_max = *max_element(begin(v), end(v));
  if(v_max == 0) return 0;
  vector< int > ma(N);
  vector< T > mb(N);
  for(int i = 0; i < N; i++) {
    ma[i] = min< T >(m[i], v_max - 1);
    mb[i] = m[i] - ma[i];
  }
  T sum = 0;
  for(int i = 0; i < N; i++) sum += ma[i] * v[i];
  auto dp = knapsack_limitations(v, ma, w, sum, T(-1), less<>());
  vector< int > ord(N);
  iota(begin(ord), end(ord), 0);
  sort(begin(ord), end(ord), [&](int a, int b) {
    return v[a] * w[b] > v[b] * w[a];
  });
  T ret = T();
  for(int i = 0; i < dp.size(); i++) {
    if(dp[i] > W || dp[i] == -1) continue;
    T rest = W - dp[i], cost = i;
    for(auto &p : ord) {
      auto get = min(mb[p], rest / w[p]);
      if(get <= 0) continue;
      cost += get * v[p];
      rest -= get * w[p];
    }
    ret = max(ret, cost);
  }
  return ret;
}
#line 1 "dp/knapsack-limitations.hpp"
template< typename T, typename Compare = greater< T > >
vector< T > knapsack_limitations(const vector< int > &w, const vector< int > &m, const vector< T > &v,
                                 const int &W, const T &NG, const Compare &comp = Compare()) {
  const int N = (int) w.size();
  vector< T > dp(W + 1, NG), deqv(W + 1);
  dp[0] = T();
  vector< int > deq(W + 1);
  for(int i = 0; i < N; i++) {
    if(w[i] == 0) {
      for(int j = 0; j <= W; j++) {
        if(dp[j] != NG && comp(dp[j] + v[i] * m[i], dp[j])) {
          dp[j] = dp[j] + v[i] * m[i];
        }
      }
    } else {
      for(int a = 0; a < w[i]; a++) {
        int s = 0, t = 0;
        for(int j = 0; w[i] * j + a <= W; j++) {
          if(dp[w[i] * j + a] != NG) {
            auto val = dp[w[i] * j + a] - j * v[i];
            while(s < t && comp(val, deqv[t - 1])) --t;
            deq[t] = j;
            deqv[t++] = val;
          }
          if(s < t) {
            dp[j * w[i] + a] = deqv[s] + j * v[i];
            if(deq[s] == j - m[i]) ++s;
          }
        }
      }
    }
  }
  return dp;
}
#line 2 "dp/knapsack-limitations-2.hpp"

template< typename T >
T knapsack_limitations(const vector< T > &w, const vector< T > &m, const vector< int > &v,
                       const T &W) {
  const int N = (int) w.size();
  auto v_max = *max_element(begin(v), end(v));
  if(v_max == 0) return 0;
  vector< int > ma(N);
  vector< T > mb(N);
  for(int i = 0; i < N; i++) {
    ma[i] = min< T >(m[i], v_max - 1);
    mb[i] = m[i] - ma[i];
  }
  T sum = 0;
  for(int i = 0; i < N; i++) sum += ma[i] * v[i];
  auto dp = knapsack_limitations(v, ma, w, sum, T(-1), less<>());
  vector< int > ord(N);
  iota(begin(ord), end(ord), 0);
  sort(begin(ord), end(ord), [&](int a, int b) {
    return v[a] * w[b] > v[b] * w[a];
  });
  T ret = T();
  for(int i = 0; i < dp.size(); i++) {
    if(dp[i] > W || dp[i] == -1) continue;
    T rest = W - dp[i], cost = i;
    for(auto &p : ord) {
      auto get = min(mb[p], rest / w[p]);
      if(get <= 0) continue;
      cost += get * v[p];
      rest -= get * w[p];
    }
    ret = max(ret, cost);
  }
  return ret;
}
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