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#include <algorithm>
#include <limits>
#include <iostream>
#include <vector>
#include <cstdio>
using namespace std;
// #define DEBUG
typedef vector<vector<double>> matrix;
void print(matrix A, vector<double> b)
{
#ifdef DEBUG
for (int i = 0; i < A.size(); i++) {
for (int j = 0; j < A[0].size(); j++) {
cerr << A[i][j] << " ";
}
cerr << ": " << b[i] << endl;
}
cerr << endl;
#endif
}
#define DELTA 0.000001
static inline bool same(double a, double b)
{
double d = (a - b);
return (d < 0 ? -d : d) < DELTA;
}
void gaussian_elimination(matrix& m)
{
int rows = m.size();
int cols = m[0].size();
int np_row = 0;
// each column
for (int col = 0; col < (cols - 1); col++) {
// each non-pivot row
for (int row = np_row; row < rows; row++) {
// leftmost non-zero
if (m[row][col] != 0) {
// swap row with np_row
if (row != np_row)
std::swap(m[np_row], m[row]);
vector<double>& pivot_row = m[np_row];
// scale pivot row
if (pivot_row[col] != 1.0) {
double f = 1.0 / pivot_row[col];
for (int i = 0; i < cols; i++)
pivot_row[i] *= f;
}
// scale then add to non-zero other rows
for (int i = 0; i < rows; i++) {
if (i != np_row && m[i][col] != 0) {
vector<double>& r = m[i];
double f = r[col] / pivot_row[col];
for (int j = 0; j < cols; j++)
r[j] -= pivot_row[j] * f;
}
}
np_row++;
break;
}
}
}
}
#define BIG_NUM 1000000000
#define BEST -std::numeric_limits<double>::max()
pair<int, vector<double>> solve(
matrix A,
vector<double> b,
vector<double> c)
{
int cstrs = A.size(); // n
int vars = A[0].size(); // m
matrix M(vars, vector<double>(vars + 1));
double best = BEST;
vector<double> sol(vars);
// build permutations of A
int nbits = 0;
int v = 0;
int max = (1 << cstrs) - 1;
vector<bool> bits(cstrs, false);
bool augmented = false; // will be set to true when the last constraint is used
// for each permutation of size vars
for (int v = 0; v < max; v++) {
// build next value
for (int i = 0; i < bits.size(); i++) {
if (!bits[i]) {
bits[i] = true;
nbits++;
break;
}
bits[i] = !bits[i];
nbits--;
}
#ifdef DEBUG
cerr << "####" << v + 1 << " : ";
for (int i = 0; i < bits.size(); i++)
cerr << bits[i] << " ";
cerr << endl;
#endif
if (nbits != vars)
continue;
if (!augmented && bits[bits.size() - 1]) {
// no solution found so far
if (same(best, BEST))
return {-1, sol}; // no solution
augmented = true;
}
// build matrix to solve
int x = 0;
for (int i = 0; i < bits.size(); i++) {
if (bits[i]) {
for (int j = 0; j < vars; j++)
M[x][j] = A[i][j];
M[x][vars] = b[i];
x++;
}
}
print(M, b);
gaussian_elimination(M);
print(M, b);
bool ok = true;
// check that solution matrix pivots are 1's
for (int i = 0; i < vars; i++) {
if (!same(M[i][i], 1.0)) {
ok = false;
break;
}
}
if (!ok)
continue;
// check that other inequalities holds
for (int i = 0; i < bits.size(); i++) {
if (!bits[i]) {
double r = 0;
for (int j = 0; j < vars; j++)
r += (M[j][vars] * A[i][j]);
if (r > (b[i] + DELTA)) {
ok = false;
break;
}
}
}
if (!ok)
continue;
// compute fitness
double r = 0;
for (int j = 0; j < vars; j++)
r += (M[j][vars] * c[j]);
if (r > best) {
if (augmented)
return { 1, sol}; // infinity, see last paragraph in What To Do
best = r;
for (int j = 0; j < vars; j++)
sol[j] = M[j][vars];
}
}
if (same(best, BEST))
return {-1, sol}; // no solution
return {0, sol};
}
pair<int, vector<double>> solve_diet_problem(
int n,
int m,
matrix A,
vector<double> b,
vector<double> c)
{
// add m constraints : -var <= 0
for (int i = 0; i < m; i++) {
vector<double> v(m, 0);
v[i] = -1;
A.push_back(v);
b.push_back(0);
}
// add augmented constraint : sum vars < BIG_NUM
A.push_back(vector<double>(m, 1));
b.push_back(BIG_NUM);
return solve(A, b, c);
}
int main(){
int n, m;
cin >> n >> m;
matrix A(n, vector<double>(m));
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
cin >> A[i][j];
}
}
vector<double> b(n);
for (int i = 0; i < n; i++) {
cin >> b[i];
}
vector<double> c(m);
for (int i = 0; i < m; i++) {
cin >> c[i];
}
pair<int, vector<double>> ans = solve_diet_problem(n, m, A, b, c);
switch (ans.first) {
case -1:
printf("No solution\n");
break;
case 0:
printf("Bounded solution\n");
for (int i = 0; i < m; i++) {
printf("%.14f%c", ans.second[i], " \n"[i + 1 == m]);
}
break;
case 1:
printf("Infinity\n");
break;
}
return 0;
}
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