#line 1 "verify/optimization/project_selection.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#include <cassert>
#line 1 "utilities/fast_io.hpp"
#include <algorithm>
#include <array>
#include <cerrno>
#include <charconv>
#include <cstddef>
#include <cstdio>
#include <cstdlib>
#include <cstdint>
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#include <type_traits>
#include <utility>
#include <unistd.h>
#include <vector>
namespace m1une {
namespace utilities {
struct FastOutput;
namespace internal {
// Shared with the convenience helpers in template.hpp.
inline FastOutput* standard_output_instance = nullptr;
// Detect std::begin(x), std::end(x).
template <class T, class = void>
struct is_range : std::false_type {};
template <class T>
struct is_range<T, std::void_t<
decltype(std::begin(std::declval<T&>())),
decltype(std::end(std::declval<T&>()))
>> : std::true_type {};
template <class T>
inline constexpr bool is_range_v = is_range<T>::value;
template <class T>
using range_reference_t = decltype(*std::begin(std::declval<T&>()));
template <class T>
using range_value_t = std::remove_cv_t<std::remove_reference_t<range_reference_t<T>>>;
template <class T, class = void>
struct range_stored_value {
using type = range_value_t<T>;
};
template <class T>
struct range_stored_value<T, std::void_t<typename std::remove_cv_t<std::remove_reference_t<T>>::value_type>> {
using type = typename std::remove_cv_t<std::remove_reference_t<T>>::value_type;
};
template <class T>
using range_stored_value_t = typename range_stored_value<T>::type;
// Treat strings and C strings as scalar output objects, not as ranges.
template <class T>
struct is_char_array : std::false_type {};
template <class T, std::size_t N>
struct is_char_array<T[N]>
: std::bool_constant<std::is_same_v<std::remove_cv_t<T>, char>> {};
template <class T>
struct is_string_like
: std::bool_constant<
std::is_same_v<std::decay_t<T>, std::string>
|| std::is_same_v<std::decay_t<T>, const char*>
|| std::is_same_v<std::decay_t<T>, char*>
|| is_char_array<std::remove_reference_t<T>>::value
> {};
template <class T>
inline constexpr bool is_string_like_v = is_string_like<T>::value;
// ModInt-like type: x.val() is printable, and x can be assigned from long long.
template <class T, class = void>
struct has_val_method : std::false_type {};
template <class T>
struct has_val_method<T, std::void_t<decltype(std::declval<const T&>().val())>>
: std::true_type {};
template <class T>
inline constexpr bool has_val_method_v = has_val_method<T>::value;
template <class T, class = void>
struct has_static_mod_raw : std::false_type {};
template <class T>
struct has_static_mod_raw<
T, std::void_t<decltype(T::mod()), decltype(T::raw(std::declval<uint32_t>()))>>
: std::true_type {};
template <class T>
inline constexpr bool has_static_mod_raw_v = has_static_mod_raw<T>::value;
// libstdc++ before GCC 16 does not classify __int128 as an integral type in
// strict ISO modes such as -std=c++23. Keep the fast-I/O interface independent
// of that implementation detail.
template <class T>
inline constexpr bool is_integral_v =
std::is_integral_v<T>
|| std::is_same_v<std::remove_cv_t<T>, __int128_t>
|| std::is_same_v<std::remove_cv_t<T>, __uint128_t>;
template <class T>
inline constexpr bool is_signed_v =
std::is_signed_v<T>
|| std::is_same_v<std::remove_cv_t<T>, __int128_t>;
template <class T>
struct make_unsigned {
using type = std::make_unsigned_t<T>;
};
template <>
struct make_unsigned<__int128_t> {
using type = __uint128_t;
};
template <>
struct make_unsigned<__uint128_t> {
using type = __uint128_t;
};
template <class T>
using make_unsigned_t = typename make_unsigned<std::remove_cv_t<T>>::type;
} // namespace internal
struct FastInput {
static constexpr int buffer_size = 1 << 20;
private:
std::FILE* _stream;
char _buffer[buffer_size];
int _position;
int _length;
int _file_descriptor;
bool _streaming;
bool refill() {
_position = 0;
if (_streaming) {
ssize_t length;
do {
length = ::read(_file_descriptor, _buffer, buffer_size);
} while (length < 0 && errno == EINTR);
if (length <= 0) {
_length = 0;
return false;
}
_length = int(length);
} else {
_length = int(std::fread(_buffer, 1, buffer_size, _stream));
}
return _length != 0;
}
template <class T>
bool read_integer_from_stream(T& value) {
if (!skip_spaces()) return false;
int c = read_char_raw();
bool negative = false;
if (c == '-') {
negative = true;
c = read_char_raw();
}
if constexpr (internal::is_signed_v<T>) {
T result = 0;
while ('0' <= c && c <= '9') {
result = negative ? result * 10 - (c - '0')
: result * 10 + (c - '0');
c = read_char_raw();
}
value = result;
} else {
T result = 0;
while ('0' <= c && c <= '9') {
result = result * 10 + T(c - '0');
c = read_char_raw();
}
value = negative ? T(0) - result : result;
}
return true;
}
bool prepare_number() {
if (_length - _position >= 64) return true;
const int remaining = _length - _position;
if (remaining > 0) std::memmove(_buffer, _buffer + _position, remaining);
const int added = int(std::fread(_buffer + remaining, 1, buffer_size - remaining, _stream));
_position = 0;
_length = remaining + added;
if (_length < buffer_size) _buffer[_length] = '\0';
return _length != 0;
}
public:
explicit FastInput(std::FILE* stream = stdin)
: _stream(stream),
_position(0),
_length(0),
_file_descriptor(::fileno(stream)),
_streaming([&] {
struct stat status;
return _file_descriptor >= 0
&& ::fstat(_file_descriptor, &status) == 0
&& !S_ISREG(status.st_mode);
}()) {}
FastInput(const FastInput&) = delete;
FastInput& operator=(const FastInput&) = delete;
int read_char_raw() {
if (_position == _length && !refill()) return EOF;
return _buffer[_position++];
}
bool skip_spaces() {
int c = read_char_raw();
while (c != EOF && c <= ' ') c = read_char_raw();
if (c == EOF) return false;
--_position;
return true;
}
bool read(char& value) {
if (!skip_spaces()) return false;
value = char(read_char_raw());
return true;
}
bool read(std::string& value) {
if (!skip_spaces()) return false;
value.clear();
while (true) {
const int begin = _position;
while (_position < _length &&
static_cast<unsigned char>(_buffer[_position]) > ' ') {
++_position;
}
value.append(_buffer + begin, _position - begin);
if (_position < _length) {
++_position;
return true;
}
if (!refill()) return true;
}
}
bool read(bool& value) {
int x;
if (!read(x)) return false;
value = x != 0;
return true;
}
template <class T>
std::enable_if_t<
internal::is_integral_v<T>
&& !std::is_same_v<std::remove_cv_t<T>, bool>
&& !std::is_same_v<std::remove_cv_t<T>, char>,
bool
>
read(T& value) {
if (_streaming) return read_integer_from_stream(value);
if (!prepare_number()) return false;
int c = static_cast<unsigned char>(_buffer[_position++]);
while (c <= ' ') c = static_cast<unsigned char>(_buffer[_position++]);
bool negative = false;
if (c == '-') {
negative = true;
c = static_cast<unsigned char>(_buffer[_position++]);
}
if constexpr (internal::is_signed_v<T>) {
T result = 0;
while ('0' <= c && c <= '9') {
const int first = c - '0';
const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
if (0 <= second && second <= 9) {
result = negative ? result * 100 - (first * 10 + second)
: result * 100 + (first * 10 + second);
++_position;
} else {
result = negative ? result * 10 - first : result * 10 + first;
}
c = static_cast<unsigned char>(_buffer[_position++]);
}
value = result;
} else {
T result = 0;
while ('0' <= c && c <= '9') {
const unsigned first = unsigned(c - '0');
const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
if (0 <= second && second <= 9) {
result = result * 100 + T(first * 10 + unsigned(second));
++_position;
} else {
result = result * 10 + T(first);
}
c = static_cast<unsigned char>(_buffer[_position++]);
}
value = negative ? T(0) - result : result;
}
if (_position > _length) _position = _length;
return true;
}
template <class T>
std::enable_if_t<std::is_floating_point_v<T>, bool>
read(T& value) {
if (!skip_spaces()) return false;
int c = read_char_raw();
bool negative = false;
if (c == '-' || c == '+') {
negative = c == '-';
c = read_char_raw();
}
long double result = 0;
while ('0' <= c && c <= '9') {
result = result * 10 + (c - '0');
c = read_char_raw();
}
if (c == '.') {
long double place = 0.1L;
c = read_char_raw();
while ('0' <= c && c <= '9') {
result += (c - '0') * place;
place *= 0.1L;
c = read_char_raw();
}
}
if (c == 'e' || c == 'E') {
c = read_char_raw();
bool exponent_negative = false;
if (c == '-' || c == '+') {
exponent_negative = c == '-';
c = read_char_raw();
}
int exponent = 0;
while ('0' <= c && c <= '9') {
exponent = exponent * 10 + (c - '0');
c = read_char_raw();
}
long double scale = 1;
long double power = 10;
while (exponent > 0) {
if (exponent & 1) scale *= power;
power *= power;
exponent >>= 1;
}
result = exponent_negative ? result / scale : result * scale;
}
value = static_cast<T>(negative ? -result : result);
return true;
}
template <class T>
std::enable_if_t<
internal::has_val_method_v<T>
&& !internal::is_integral_v<T>
&& !internal::is_range_v<T>,
bool
>
read(T& value) {
long long x;
if (!read(x)) return false;
if constexpr (internal::has_static_mod_raw_v<T>) {
if (x >= 0 && uint64_t(x) < uint64_t(T::mod())) {
value = T::raw(uint32_t(x));
} else {
value = T(x);
}
} else {
value = T(x);
}
return true;
}
template <class First, class Second>
bool read(std::pair<First, Second>& value) {
if (!read(value.first)) return false;
return read(value.second);
}
template <class Range>
std::enable_if_t<
internal::is_range_v<Range>
&& !internal::is_string_like_v<Range>,
bool
>
read(Range& range) {
using StoredValue = internal::range_stored_value_t<Range>;
constexpr bool nested = internal::is_range_v<StoredValue>
&& !internal::is_string_like_v<StoredValue>;
for (auto&& value : range) {
if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
bool x;
if (!read(x)) return false;
value = x;
} else {
if (!read(value)) return false;
}
}
return true;
}
template <class First, class Second, class... Rest>
bool read(First& first, Second& second, Rest&... rest) {
if (!read(first)) return false;
return read(second, rest...);
}
template <class T>
FastInput& operator>>(T& value) {
if (!read(value)) std::abort();
return *this;
}
};
struct FastOutput {
static constexpr int buffer_size = 1 << 20;
private:
inline static const auto digit_quads = [] {
std::array<char, 40000> result{};
for (int i = 0; i < 10000; i++) {
int value = i;
for (int j = 3; j >= 0; j--) {
result[4 * i + j] = char('0' + value % 10);
value /= 10;
}
}
return result;
}();
std::FILE* _stream;
char _buffer[buffer_size];
int _position;
int _precision;
std::chars_format _float_format;
char _range_separator;
std::string* _capture = nullptr;
template <class T>
std::string format_cell(const T& value) {
std::string result;
struct CaptureGuard {
std::string*& target;
std::string* previous;
~CaptureGuard() { target = previous; }
} guard{_capture, _capture};
_capture = &result;
write(value);
return result;
}
template <class Matrix>
void write_aligned_matrix(const Matrix& matrix) {
std::vector<std::vector<std::string>> rows;
std::vector<std::size_t> widths;
for (const auto& row : matrix) {
auto& cells = rows.emplace_back();
std::size_t column = 0;
for (const auto& value : row) {
cells.push_back(format_cell(value));
if (column == widths.size()) widths.push_back(0);
widths[column] = std::max(widths[column], cells.back().size());
++column;
}
}
bool first = true;
for (const auto& row : rows) {
if (!first) write_char('\n');
first = false;
for (std::size_t column = 0; column < row.size(); ++column) {
if (column != 0) write_char(_range_separator);
for (std::size_t padding = row[column].size();
padding < widths[column]; ++padding) {
write_char(' ');
}
write(row[column]);
}
}
}
public:
explicit FastOutput(std::FILE* stream = stdout)
: _stream(stream),
_position(0),
_precision(6),
_float_format(std::chars_format::general),
_range_separator(' ') {
if (_stream == stdout
&& internal::standard_output_instance == nullptr) {
internal::standard_output_instance = this;
}
}
FastOutput(const FastOutput&) = delete;
FastOutput& operator=(const FastOutput&) = delete;
~FastOutput() {
flush();
if (internal::standard_output_instance == this) {
internal::standard_output_instance = nullptr;
}
}
void flush() {
if (_position != 0) {
std::fwrite(_buffer, 1, _position, _stream);
_position = 0;
}
std::fflush(_stream);
}
void write_char(char c) {
if (_capture != nullptr) {
_capture->push_back(c);
return;
}
if (_position == buffer_size) flush();
_buffer[_position++] = c;
}
void write(const char* s) {
while (*s != '\0') write_char(*s++);
}
void write(const std::string& s) {
if (_capture != nullptr) {
_capture->append(s);
return;
}
std::size_t position = 0;
while (position < s.size()) {
if (_position == buffer_size) flush();
const std::size_t copied =
std::min<std::size_t>(buffer_size - _position, s.size() - position);
std::memcpy(_buffer + _position, s.data() + position, copied);
_position += int(copied);
position += copied;
}
}
void write(char c) {
write_char(c);
}
void write(bool value) {
write_char(value ? '1' : '0');
}
template <class T>
std::enable_if_t<std::is_floating_point_v<T>>
write(T value) {
char digits[128];
auto [end, error] = std::to_chars(
digits,
digits + sizeof(digits),
value,
_float_format,
_precision
);
if (error != std::errc()) std::abort();
for (const char* pointer = digits; pointer != end; pointer++) {
write_char(*pointer);
}
}
template <class T>
std::enable_if_t<
internal::is_integral_v<T>
&& !std::is_same_v<std::remove_cv_t<T>, bool>
&& !std::is_same_v<std::remove_cv_t<T>, char>
>
write(T value) {
using Raw = std::remove_cv_t<T>;
using Unsigned = internal::make_unsigned_t<Raw>;
Unsigned magnitude;
if constexpr (internal::is_signed_v<Raw>) {
if (value < 0) {
write_char('-');
magnitude = Unsigned(0) - Unsigned(value);
} else {
magnitude = Unsigned(value);
}
} else {
magnitude = value;
}
if (magnitude == 0) {
write_char('0');
return;
}
unsigned chunks[16];
int count = 0;
while (magnitude >= 10000) {
const Unsigned quotient = magnitude / 10000;
chunks[count++] = unsigned(magnitude - quotient * 10000);
magnitude = quotient;
}
if (_capture == nullptr && _position > buffer_size - 64) flush();
char captured[64];
char* const begin = _capture != nullptr ? captured : _buffer + _position;
char* destination = begin;
const unsigned leading = unsigned(magnitude);
const char* first = digit_quads.data() + 4 * leading;
int skip = leading < 10 ? 3 : leading < 100 ? 2 : leading < 1000 ? 1 : 0;
for (; skip < 4; skip++) *destination++ = first[skip];
while (count--) {
const char* digits = digit_quads.data() + 4 * chunks[count];
std::memcpy(destination, digits, 4);
destination += 4;
}
if (_capture != nullptr) {
_capture->append(begin, destination - begin);
} else {
_position += int(destination - begin);
}
}
template <class T>
std::enable_if_t<
internal::has_val_method_v<T>
&& !internal::is_integral_v<T>
&& !internal::is_range_v<T>
>
write(const T& value) {
write(value.val());
}
template <class First, class Second>
void write(const std::pair<First, Second>& value) {
write(value.first);
write_char(' ');
write(value.second);
}
template <class Range>
std::enable_if_t<
internal::is_range_v<Range>
&& !internal::is_string_like_v<Range>
>
write(const Range& range) {
using StoredValue = internal::range_stored_value_t<const Range>;
constexpr bool nested = internal::is_range_v<StoredValue>
&& !internal::is_string_like_v<StoredValue>;
bool first = true;
for (const auto& value : range) {
if (!first) write_char(nested ? '\n' : _range_separator);
first = false;
if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
write(static_cast<bool>(value));
} else {
write(value);
}
}
}
template <class First, class... Rest>
void print(const First& first, const Rest&... rest) {
write(first);
((write_char(' '), write(rest)), ...);
}
void println() {
write_char('\n');
}
void set_precision(int precision) {
_precision = precision;
}
void set_fixed(int precision = 6) {
_float_format = std::chars_format::fixed;
_precision = precision;
}
void set_general(int precision = 6) {
_float_format = std::chars_format::general;
_precision = precision;
}
void set_range_separator(char separator) {
_range_separator = separator;
}
template <class Matrix>
void write_aligned(const Matrix& matrix) {
using Row = internal::range_stored_value_t<const Matrix>;
using Cell = internal::range_stored_value_t<const Row>;
static_assert(internal::is_range_v<Row> && !internal::is_string_like_v<Row>,
"write_aligned requires a two-dimensional range");
static_assert(!internal::is_range_v<Cell> || internal::is_string_like_v<Cell>,
"write_aligned requires scalar cells");
write_aligned_matrix(matrix);
}
template <class Matrix>
void println_aligned(const Matrix& matrix) {
write_aligned(matrix);
write_char('\n');
}
template <class... Args>
void println(const Args&... args) {
print(args...);
write_char('\n');
}
template <class T>
FastOutput& operator<<(const T& value) {
write(value);
return *this;
}
};
} // namespace utilities
} // namespace m1une
#line 5 "verify/optimization/project_selection.test.cpp"
#include <limits>
#line 7 "verify/optimization/project_selection.test.cpp"
#line 1 "optimization/all.hpp"
#line 1 "optimization/hungarian.hpp"
#line 9 "optimization/hungarian.hpp"
namespace m1une {
namespace opt {
template <class T>
struct HungarianResult {
T cost;
std::vector<int> row_to_col;
std::vector<int> col_to_row;
int matching_size() const {
int result = 0;
for (int col : row_to_col) {
if (col != -1) result++;
}
return result;
}
std::vector<std::pair<int, int>> matching() const {
std::vector<std::pair<int, int>> result;
for (int row = 0; row < int(row_to_col.size()); row++) {
if (row_to_col[row] != -1) result.push_back({row, row_to_col[row]});
}
return result;
}
};
namespace detail {
template <class T>
T assignment_cost(const std::vector<std::vector<T>>& cost, const std::vector<int>& row_to_col) {
T result = T();
for (int row = 0; row < int(row_to_col.size()); row++) {
if (row_to_col[row] != -1) result += cost[row][row_to_col[row]];
}
return result;
}
} // namespace detail
template <class T>
HungarianResult<T> hungarian_min(const std::vector<std::vector<T>>& cost) {
int row_count = int(cost.size());
int col_count = row_count == 0 ? 0 : int(cost[0].size());
for (const auto& row : cost) assert(int(row.size()) == col_count);
HungarianResult<T> result;
result.cost = T();
result.row_to_col.assign(row_count, -1);
result.col_to_row.assign(col_count, -1);
if (row_count == 0 || col_count == 0) return result;
bool transposed = row_count > col_count;
int n = transposed ? col_count : row_count;
int m = transposed ? row_count : col_count;
T inf = std::numeric_limits<T>::max() / T(4);
std::vector<T> u(n + 1, T()), v(m + 1, T()), minv(m + 1);
std::vector<int> p(m + 1, 0), way(m + 1, 0);
auto value = [&](int i, int j) -> T {
return transposed ? cost[j][i] : cost[i][j];
};
for (int i = 1; i <= n; i++) {
p[0] = i;
int j0 = 0;
std::fill(minv.begin(), minv.end(), inf);
std::vector<char> used(m + 1, false);
do {
used[j0] = true;
int i0 = p[j0];
int j1 = 0;
T delta = inf;
for (int j = 1; j <= m; j++) {
if (used[j]) continue;
T cur = value(i0 - 1, j - 1) - u[i0] - v[j];
if (cur < minv[j]) {
minv[j] = cur;
way[j] = j0;
}
if (minv[j] < delta) {
delta = minv[j];
j1 = j;
}
}
for (int j = 0; j <= m; j++) {
if (used[j]) {
u[p[j]] += delta;
v[j] -= delta;
} else {
minv[j] -= delta;
}
}
j0 = j1;
} while (p[j0] != 0);
do {
int j1 = way[j0];
p[j0] = p[j1];
j0 = j1;
} while (j0 != 0);
}
for (int j = 1; j <= m; j++) {
if (p[j] == 0) continue;
int i = p[j] - 1;
int matched = j - 1;
if (transposed) {
int row = matched;
int col = i;
result.row_to_col[row] = col;
result.col_to_row[col] = row;
} else {
int row = i;
int col = matched;
result.row_to_col[row] = col;
result.col_to_row[col] = row;
}
}
result.cost = detail::assignment_cost(cost, result.row_to_col);
return result;
}
template <class T>
HungarianResult<T> hungarian_max(const std::vector<std::vector<T>>& cost) {
std::vector<std::vector<T>> negated = cost;
for (auto& row : negated) {
for (auto& x : row) x = -x;
}
auto result = hungarian_min(negated);
result.cost = detail::assignment_cost(cost, result.row_to_col);
return result;
}
template <class T>
HungarianResult<T> hungarian(const std::vector<std::vector<T>>& cost) {
return hungarian_min(cost);
}
} // namespace opt
} // namespace m1une
#line 1 "optimization/integer_lp.hpp"
#line 6 "optimization/integer_lp.hpp"
#include <cmath>
#line 10 "optimization/integer_lp.hpp"
#line 1 "optimization/simplex.hpp"
#line 9 "optimization/simplex.hpp"
namespace m1une {
namespace opt {
enum class SimplexStatus {
Optimal,
Infeasible,
Unbounded,
};
template <class T>
struct SimplexResult {
SimplexStatus status;
T objective_value;
std::vector<T> variables;
bool is_optimal() const { return status == SimplexStatus::Optimal; }
bool is_infeasible() const { return status == SimplexStatus::Infeasible; }
bool is_unbounded() const { return status == SimplexStatus::Unbounded; }
};
namespace detail {
template <class T>
T simplex_abs(T x) {
return x < T() ? -x : x;
}
template <class T>
struct SimplexTableau {
int constraint_count;
int variable_count;
T eps;
std::vector<int> basis;
std::vector<int> nonbasis;
std::vector<std::vector<T>> table;
SimplexTableau(const std::vector<std::vector<T>>& a, const std::vector<T>& b,
const std::vector<T>& c, T epsilon)
: constraint_count(int(b.size())),
variable_count(int(c.size())),
eps(epsilon),
basis(constraint_count),
nonbasis(variable_count + 1),
table(constraint_count + 2, std::vector<T>(variable_count + 2, T())) {
for (int i = 0; i < constraint_count; i++) {
for (int j = 0; j < variable_count; j++) table[i][j] = a[i][j];
}
for (int i = 0; i < constraint_count; i++) {
basis[i] = variable_count + i;
table[i][artificial_col()] = T(-1);
table[i][rhs_col()] = b[i];
}
for (int j = 0; j < variable_count; j++) {
nonbasis[j] = j;
table[objective_row()][j] = -c[j];
}
nonbasis[artificial_col()] = artificial_id();
table[auxiliary_row()][artificial_col()] = T(1);
}
int objective_row() const { return constraint_count; }
int auxiliary_row() const { return constraint_count + 1; }
int artificial_col() const { return variable_count; }
int rhs_col() const { return variable_count + 1; }
int artificial_id() const { return -1; }
T normalize(T x) const {
return simplex_abs(x) <= eps ? T() : x;
}
bool less_with_tie(int row, int lhs, int rhs) const {
if (table[row][lhs] < table[row][rhs] - eps) return true;
if (table[row][rhs] < table[row][lhs] - eps) return false;
return nonbasis[lhs] < nonbasis[rhs];
}
bool better_leaving_row(int lhs, int rhs, int entering_col) const {
T lhs_ratio = table[lhs][rhs_col()] / table[lhs][entering_col];
T rhs_ratio = table[rhs][rhs_col()] / table[rhs][entering_col];
if (lhs_ratio < rhs_ratio - eps) return true;
if (rhs_ratio < lhs_ratio - eps) return false;
return basis[lhs] < basis[rhs];
}
void pivot(int leaving_row, int entering_col) {
T inverse = T(1) / table[leaving_row][entering_col];
for (int i = 0; i < constraint_count + 2; i++) {
if (i == leaving_row) continue;
for (int j = 0; j < variable_count + 2; j++) {
if (j == entering_col) continue;
table[i][j] -= table[leaving_row][j] * table[i][entering_col] * inverse;
}
}
for (int j = 0; j < variable_count + 2; j++) {
if (j != entering_col) table[leaving_row][j] *= inverse;
}
for (int i = 0; i < constraint_count + 2; i++) {
if (i != leaving_row) table[i][entering_col] *= -inverse;
}
table[leaving_row][entering_col] = inverse;
std::swap(basis[leaving_row], nonbasis[entering_col]);
}
bool run_simplex(int row) {
while (true) {
int entering_col = -1;
for (int j = 0; j <= variable_count; j++) {
if (nonbasis[j] == artificial_id()) continue;
if (entering_col == -1 || less_with_tie(row, j, entering_col)) entering_col = j;
}
if (entering_col == -1 || table[row][entering_col] >= -eps) return true;
int leaving_row = -1;
for (int i = 0; i < constraint_count; i++) {
if (table[i][entering_col] <= eps) continue;
if (leaving_row == -1 || better_leaving_row(i, leaving_row, entering_col)) {
leaving_row = i;
}
}
if (leaving_row == -1) return false;
pivot(leaving_row, entering_col);
}
}
bool make_feasible() {
int leaving_row = 0;
for (int i = 1; i < constraint_count; i++) {
if (table[i][rhs_col()] < table[leaving_row][rhs_col()]) leaving_row = i;
}
if (constraint_count == 0 || table[leaving_row][rhs_col()] >= -eps) return true;
pivot(leaving_row, artificial_col());
if (!run_simplex(auxiliary_row())) return false;
if (table[auxiliary_row()][rhs_col()] < -eps) return false;
for (int i = 0; i < constraint_count; i++) {
if (basis[i] != artificial_id()) continue;
int entering_col = -1;
for (int j = 0; j <= variable_count; j++) {
if (nonbasis[j] == artificial_id()) continue;
if (simplex_abs(table[i][j]) <= eps) continue;
if (entering_col == -1 || nonbasis[j] < nonbasis[entering_col]) entering_col = j;
}
if (entering_col != -1) pivot(i, entering_col);
}
return true;
}
SimplexStatus solve(std::vector<T>& variables, T& objective_value) {
if (!make_feasible()) return SimplexStatus::Infeasible;
if (!run_simplex(objective_row())) return SimplexStatus::Unbounded;
variables.assign(variable_count, T());
for (int i = 0; i < constraint_count; i++) {
if (0 <= basis[i] && basis[i] < variable_count) {
variables[basis[i]] = normalize(table[i][rhs_col()]);
}
}
objective_value = normalize(table[objective_row()][rhs_col()]);
return SimplexStatus::Optimal;
}
};
} // namespace detail
template <class T>
SimplexResult<T> simplex_maximize(const std::vector<std::vector<T>>& a, const std::vector<T>& b,
const std::vector<T>& c, T eps = T(1e-10)) {
static_assert(std::is_floating_point_v<T>, "simplex requires a floating-point type");
assert(int(a.size()) == int(b.size()));
for (const auto& row : a) assert(int(row.size()) == int(c.size()));
assert(eps > T());
SimplexResult<T> result;
result.status = SimplexStatus::Infeasible;
result.objective_value = std::numeric_limits<T>::quiet_NaN();
result.variables.assign(c.size(), T());
detail::SimplexTableau<T> solver(a, b, c, eps);
result.status = solver.solve(result.variables, result.objective_value);
if (result.status == SimplexStatus::Infeasible) {
result.objective_value = std::numeric_limits<T>::quiet_NaN();
} else if (result.status == SimplexStatus::Unbounded) {
result.objective_value = std::numeric_limits<T>::infinity();
}
return result;
}
template <class T>
SimplexResult<T> simplex_minimize(const std::vector<std::vector<T>>& a, const std::vector<T>& b,
const std::vector<T>& c, T eps = T(1e-10)) {
std::vector<T> negated = c;
for (T& x : negated) x = -x;
auto result = simplex_maximize(a, b, negated, eps);
if (result.status == SimplexStatus::Optimal) {
result.objective_value = -result.objective_value;
} else if (result.status == SimplexStatus::Unbounded) {
result.objective_value = -std::numeric_limits<T>::infinity();
}
return result;
}
template <class T>
SimplexResult<T> simplex(const std::vector<std::vector<T>>& a, const std::vector<T>& b,
const std::vector<T>& c, T eps = T(1e-10)) {
return simplex_maximize(a, b, c, eps);
}
} // namespace opt
} // namespace m1une
#line 12 "optimization/integer_lp.hpp"
namespace m1une {
namespace opt {
enum class IntegerLpStatus {
Optimal,
Infeasible,
Unbounded,
};
template <class T>
struct IntegerLpResult {
IntegerLpStatus status;
T objective_value;
std::vector<T> variables;
bool is_optimal() const { return status == IntegerLpStatus::Optimal; }
bool is_infeasible() const { return status == IntegerLpStatus::Infeasible; }
bool is_unbounded() const { return status == IntegerLpStatus::Unbounded; }
};
namespace detail {
template <class T>
struct IntegerLpSolver {
using Real = long double;
struct Node {
std::vector<std::vector<Real>> a;
std::vector<Real> b;
};
int variable_count;
bool maximize;
Real eps;
std::vector<T> objective;
std::vector<Real> relaxation_objective;
Node initial_node;
bool has_incumbent = false;
T best_value = T();
std::vector<T> best_variables;
IntegerLpSolver(const std::vector<std::vector<T>>& a, const std::vector<T>& b,
const std::vector<T>& c, bool is_maximize, Real epsilon)
: variable_count(int(c.size())),
maximize(is_maximize),
eps(epsilon),
objective(c),
relaxation_objective(c.size(), Real()),
initial_node() {
initial_node.a.assign(a.size(), std::vector<Real>(variable_count, Real()));
initial_node.b.assign(b.size(), Real());
for (int i = 0; i < int(a.size()); i++) {
for (int j = 0; j < variable_count; j++) initial_node.a[i][j] = Real(a[i][j]);
initial_node.b[i] = Real(b[i]);
}
Real sign = maximize ? Real(1) : Real(-1);
for (int j = 0; j < variable_count; j++) relaxation_objective[j] = sign * Real(c[j]);
}
Real abs_value(Real x) const {
return x < Real() ? -x : x;
}
bool better_value(T lhs, T rhs) const {
return maximize ? lhs > rhs : lhs < rhs;
}
bool can_prune_by_bound(Real relaxation_value) const {
if (!has_incumbent) return false;
Real signed_best = maximize ? Real(best_value) : -Real(best_value);
return relaxation_value <= signed_best + eps;
}
T evaluate(const std::vector<T>& variables) const {
T result = T();
for (int i = 0; i < variable_count; i++) result += objective[i] * variables[i];
return result;
}
bool round_solution(const std::vector<Real>& real_variables, std::vector<T>& variables) const {
variables.assign(variable_count, T());
for (int i = 0; i < variable_count; i++) {
Real value = real_variables[i];
if (value < -eps) return false;
Real rounded = std::round(value);
if (abs_value(value - rounded) > eps) return false;
variables[i] = static_cast<T>(rounded);
}
return true;
}
int find_fractional_variable(const std::vector<Real>& real_variables) const {
int result = -1;
Real best_distance = eps;
for (int i = 0; i < variable_count; i++) {
Real value = real_variables[i];
Real rounded = std::round(value);
Real distance = abs_value(value - rounded);
if (distance > best_distance) {
best_distance = distance;
result = i;
}
}
return result;
}
Node with_upper_bound(const Node& node, int variable, T bound) const {
Node result = node;
result.a.emplace_back(variable_count, Real());
result.a.back()[variable] = Real(1);
result.b.push_back(Real(bound));
return result;
}
Node with_lower_bound(const Node& node, int variable, T bound) const {
Node result = node;
result.a.emplace_back(variable_count, Real());
result.a.back()[variable] = Real(-1);
result.b.push_back(-Real(bound));
return result;
}
void push_branches(std::vector<Node>& stack, const Node& node, int variable, Real value) const {
Real floor_value = std::floor(value);
Real ceil_value = std::ceil(value);
T upper_bound = static_cast<T>(floor_value);
T lower_bound = static_cast<T>(ceil_value);
bool has_upper_branch = upper_bound >= T();
bool prefer_lower_branch = relaxation_objective[variable] >= -eps;
if (prefer_lower_branch) {
if (has_upper_branch) stack.push_back(with_upper_bound(node, variable, upper_bound));
stack.push_back(with_lower_bound(node, variable, lower_bound));
} else {
stack.push_back(with_lower_bound(node, variable, lower_bound));
if (has_upper_branch) stack.push_back(with_upper_bound(node, variable, upper_bound));
}
}
bool has_positive_direction(const Node& node) const {
std::vector<std::vector<Real>> direction_a = node.a;
std::vector<Real> direction_b(node.b.size(), Real());
std::vector<Real> objective_row(variable_count, Real());
for (int i = 0; i < variable_count; i++) objective_row[i] = -relaxation_objective[i];
direction_a.push_back(objective_row);
direction_b.push_back(Real(-1));
std::vector<Real> zero_objective(variable_count, Real());
auto result = simplex_maximize(direction_a, direction_b, zero_objective, eps);
return result.is_optimal();
}
bool find_integer_feasible(const Node& start, std::vector<T>& feasible_variables) const {
std::vector<Node> stack;
stack.push_back(start);
std::vector<Real> zero_objective(variable_count, Real());
while (!stack.empty()) {
Node node = stack.back();
stack.pop_back();
auto relaxation = simplex_maximize(node.a, node.b, zero_objective, eps);
if (relaxation.is_infeasible()) continue;
if (relaxation.is_unbounded()) continue;
if (round_solution(relaxation.variables, feasible_variables)) return true;
int variable = find_fractional_variable(relaxation.variables);
if (variable == -1) continue;
push_branches(stack, node, variable, relaxation.variables[variable]);
}
return false;
}
void update_incumbent(const std::vector<T>& variables) {
T value = evaluate(variables);
if (!has_incumbent || better_value(value, best_value)) {
has_incumbent = true;
best_value = value;
best_variables = variables;
}
}
IntegerLpResult<T> make_infeasible_result() const {
IntegerLpResult<T> result;
result.status = IntegerLpStatus::Infeasible;
result.objective_value = T();
result.variables.assign(variable_count, T());
return result;
}
IntegerLpResult<T> make_unbounded_result(const std::vector<T>& variables) const {
IntegerLpResult<T> result;
result.status = IntegerLpStatus::Unbounded;
result.objective_value =
maximize ? std::numeric_limits<T>::max() : std::numeric_limits<T>::lowest();
result.variables = variables;
return result;
}
IntegerLpResult<T> make_optimal_result() const {
IntegerLpResult<T> result;
result.status = IntegerLpStatus::Optimal;
result.objective_value = best_value;
result.variables = best_variables;
return result;
}
IntegerLpResult<T> solve() {
std::vector<Node> stack;
stack.push_back(initial_node);
while (!stack.empty()) {
Node node = stack.back();
stack.pop_back();
auto relaxation = simplex_maximize(node.a, node.b, relaxation_objective, eps);
if (relaxation.is_infeasible()) continue;
if (relaxation.is_unbounded()) {
std::vector<T> feasible_variables;
if (has_positive_direction(node) && find_integer_feasible(node, feasible_variables)) {
return make_unbounded_result(feasible_variables);
}
continue;
}
if (can_prune_by_bound(relaxation.objective_value)) continue;
std::vector<T> integer_variables;
if (round_solution(relaxation.variables, integer_variables)) {
update_incumbent(integer_variables);
continue;
}
int variable = find_fractional_variable(relaxation.variables);
if (variable == -1) continue;
push_branches(stack, node, variable, relaxation.variables[variable]);
}
if (!has_incumbent) return make_infeasible_result();
return make_optimal_result();
}
};
} // namespace detail
template <class T>
IntegerLpResult<T> integer_lp_maximize(const std::vector<std::vector<T>>& a,
const std::vector<T>& b, const std::vector<T>& c,
long double eps = 1e-10L) {
static_assert(std::is_integral_v<T> && std::is_signed_v<T>,
"integer_lp requires a signed integer type");
assert(int(a.size()) == int(b.size()));
for (const auto& row : a) assert(int(row.size()) == int(c.size()));
assert(eps > 0);
detail::IntegerLpSolver<T> solver(a, b, c, true, eps);
return solver.solve();
}
template <class T>
IntegerLpResult<T> integer_lp_minimize(const std::vector<std::vector<T>>& a,
const std::vector<T>& b, const std::vector<T>& c,
long double eps = 1e-10L) {
static_assert(std::is_integral_v<T> && std::is_signed_v<T>,
"integer_lp requires a signed integer type");
assert(int(a.size()) == int(b.size()));
for (const auto& row : a) assert(int(row.size()) == int(c.size()));
assert(eps > 0);
detail::IntegerLpSolver<T> solver(a, b, c, false, eps);
return solver.solve();
}
template <class T>
IntegerLpResult<T> integer_lp(const std::vector<std::vector<T>>& a, const std::vector<T>& b,
const std::vector<T>& c, long double eps = 1e-10L) {
return integer_lp_maximize(a, b, c, eps);
}
} // namespace opt
} // namespace m1une
#line 1 "optimization/k_project_selection.hpp"
#line 10 "optimization/k_project_selection.hpp"
#line 1 "optimization/project_selection.hpp"
#line 9 "optimization/project_selection.hpp"
#line 1 "graph/flow/max_flow.hpp"
#line 9 "graph/flow/max_flow.hpp"
namespace m1une {
namespace flow {
template <class Cap>
struct MaxFlow {
struct Edge {
int from;
int to;
Cap cap;
Cap flow;
};
private:
struct InternalEdge {
int to;
int rev;
Cap cap;
};
struct Position {
int from;
int edge;
};
int _n;
std::vector<Position> _pos;
std::vector<std::vector<InternalEdge>> _g;
Cap highest_label_preflow_push(int s, int t) {
const int dead = 2 * _n;
const int unreachable = _n + 1;
std::vector<Cap> excess(_n, Cap(0));
std::vector<int> state(8 * std::size_t(_n) + 2);
int* height = state.data();
int* height_count = height + _n;
int* current = height_count + dead + 1;
int* queue = current + _n;
int* next = queue + _n;
int* bucket_head = next + _n;
std::vector<char> active(_n, false);
int highest = -1;
long long work = 0;
const long long arc_count =
2LL * static_cast<long long>(_pos.size());
const long long work_limit = std::max(1LL, 4 * arc_count + _n);
auto activate = [&](int v) {
if (v == s || v == t || active[v] || excess[v] == Cap(0) ||
height[v] >= dead) {
return;
}
active[v] = true;
next[v] = bucket_head[height[v]];
bucket_head[height[v]] = v;
highest = std::max(highest, height[v]);
};
auto rebuild_buckets = [&]() {
std::fill(bucket_head, bucket_head + dead + 1, -1);
std::fill(active.begin(), active.end(), false);
highest = -1;
for (int v = 0; v < _n; v++) activate(v);
};
auto global_relabel = [&]() {
std::fill(height, height + _n, unreachable);
std::fill(height_count, height_count + dead + 1, 0);
std::fill(current, current + _n, 0);
int head = 0;
int tail = 0;
height[t] = 0;
height[s] = _n;
queue[tail++] = t;
while (head != tail) {
int v = queue[head++];
for (const auto& e : _g[v]) {
if (e.to == s || height[e.to] != unreachable) continue;
const auto& reverse = _g[e.to][e.rev];
if (reverse.cap == Cap(0)) continue;
height[e.to] = height[v] + 1;
queue[tail++] = e.to;
}
}
for (int v = 0; v < _n; v++) height_count[height[v]]++;
rebuild_buckets();
work = 0;
};
auto gap = [&](int empty_height) {
for (int v = 0; v < _n; v++) {
if (v == s || v == t || height[v] <= empty_height ||
height[v] >= _n) {
continue;
}
height_count[height[v]]--;
height[v] = unreachable;
height_count[height[v]]++;
current[v] = 0;
}
rebuild_buckets();
};
auto relabel = [&](int v) -> bool {
int old_height = height[v];
int new_height = dead;
work += int(_g[v].size());
for (const auto& e : _g[v]) {
if (e.cap != Cap(0)) {
new_height = std::min(new_height, height[e.to] + 1);
}
}
height_count[old_height]--;
height[v] = std::min(new_height, dead);
height_count[height[v]]++;
current[v] = 0;
if (old_height < _n && height_count[old_height] == 0) {
gap(old_height);
return true;
}
return false;
};
auto push = [&](int v, InternalEdge& e) {
Cap sent = std::min(excess[v], e.cap);
bool was_zero = excess[e.to] == Cap(0);
e.cap -= sent;
_g[e.to][e.rev].cap += sent;
excess[v] -= sent;
excess[e.to] += sent;
if (was_zero) activate(e.to);
};
auto discharge = [&](int v) {
while (excess[v] != Cap(0) && height[v] < dead) {
if (current[v] == int(_g[v].size())) {
if (relabel(v)) return;
continue;
}
auto& e = _g[v][current[v]];
work++;
if (e.cap != Cap(0) && height[v] == height[e.to] + 1) {
push(v, e);
} else {
current[v]++;
}
}
activate(v);
};
for (auto& e : _g[s]) {
if (e.to == s || e.cap == Cap(0)) continue;
Cap sent = e.cap;
e.cap = Cap(0);
_g[e.to][e.rev].cap += sent;
excess[e.to] += sent;
}
global_relabel();
while (highest >= 0) {
if (bucket_head[highest] == -1) {
highest--;
continue;
}
int v = bucket_head[highest];
bucket_head[highest] = next[v];
if (!active[v] || height[v] != highest) continue;
active[v] = false;
discharge(v);
if (work >= work_limit) global_relabel();
}
return excess[t];
}
public:
MaxFlow() : MaxFlow(0) {}
explicit MaxFlow(int n) : _n(n), _g(n) {
assert(0 <= n);
}
int size() const {
return _n;
}
int edge_count() const {
return int(_pos.size());
}
void reserve_edges(int edge_count) {
assert(0 <= edge_count);
_pos.reserve(edge_count);
if (_n == 0 || edge_count == 0 ||
2 * std::size_t(edge_count) < std::size_t(_n)) {
return;
}
const std::size_t average_degree =
(3 * std::size_t(edge_count) + std::size_t(_n) - 1)
/ std::size_t(_n);
for (auto& edges : _g) edges.reserve(average_degree);
}
void reserve_edges(int edge_count, const std::vector<int>& degrees) {
assert(0 <= edge_count);
assert(int(degrees.size()) == _n);
_pos.reserve(edge_count);
for (int v = 0; v < _n; v++) {
assert(0 <= degrees[v]);
_g[v].reserve(degrees[v]);
}
}
int add_edge(int from, int to, Cap cap) {
assert(0 <= from && from < _n);
assert(0 <= to && to < _n);
assert(Cap(0) <= cap);
int id = int(_pos.size());
int from_id = int(_g[from].size());
int to_id = int(_g[to].size());
if (from == to) to_id++;
_pos.push_back(Position{from, from_id});
_g[from].push_back(InternalEdge{to, to_id, cap});
_g[to].push_back(InternalEdge{from, from_id, Cap(0)});
return id;
}
int add_undirected_edge(int first, int second, Cap cap) {
static_assert(std::numeric_limits<Cap>::is_signed);
assert(0 <= first && first < _n);
assert(0 <= second && second < _n);
assert(Cap(0) <= cap);
assert(cap <= std::numeric_limits<Cap>::max() / Cap(2));
int id = int(_pos.size());
int first_id = int(_g[first].size());
int second_id = int(_g[second].size());
if (first == second) second_id++;
_pos.push_back(Position{first, ~first_id});
_g[first].push_back(InternalEdge{second, second_id, cap});
_g[second].push_back(InternalEdge{first, first_id, cap});
return id;
}
Edge get_edge(int i) const {
assert(0 <= i && i < int(_pos.size()));
const auto& position = _pos[i];
int from = position.from;
bool undirected = position.edge < 0;
int idx = undirected ? ~position.edge : position.edge;
const auto& e = _g[from][idx];
const auto& re = _g[e.to][e.rev];
if (undirected) {
return Edge{
from,
e.to,
(e.cap + re.cap) / Cap(2),
(re.cap - e.cap) / Cap(2)
};
}
return Edge{from, e.to, e.cap + re.cap, re.cap};
}
std::vector<Edge> edges() const {
std::vector<Edge> result;
result.reserve(_pos.size());
for (int i = 0; i < int(_pos.size()); i++) result.push_back(get_edge(i));
return result;
}
void change_edge(int i, Cap new_cap, Cap new_flow) {
assert(0 <= i && i < int(_pos.size()));
assert(Cap(0) <= new_cap);
auto& position = _pos[i];
int from = position.from;
bool undirected = position.edge < 0;
int idx = undirected ? ~position.edge : position.edge;
auto& e = _g[from][idx];
auto& re = _g[e.to][e.rev];
if (undirected) {
assert(new_cap <= std::numeric_limits<Cap>::max() / Cap(2));
assert(-new_cap <= new_flow && new_flow <= new_cap);
e.cap = new_cap - new_flow;
re.cap = new_cap + new_flow;
} else {
assert(Cap(0) <= new_flow && new_flow <= new_cap);
e.cap = new_cap - new_flow;
re.cap = new_flow;
}
}
Cap max_flow(int s, int t) {
assert(0 <= s && s < _n);
assert(0 <= t && t < _n);
assert(s != t);
return highest_label_preflow_push(s, t);
}
Cap max_flow_push_relabel(int s, int t) {
assert(0 <= s && s < _n);
assert(0 <= t && t < _n);
assert(s != t);
return highest_label_preflow_push(s, t);
}
Cap max_flow_dinic(int s, int t) {
return max_flow(s, t, std::numeric_limits<Cap>::max());
}
Cap max_flow(int s, int t, Cap flow_limit) {
assert(0 <= s && s < _n);
assert(0 <= t && t < _n);
assert(s != t);
std::vector<int> work(3 * std::size_t(_n));
int* level = work.data();
int* iter = level + _n;
int* queue = iter + _n;
auto bfs = [&]() -> bool {
std::fill(level, level + _n, -1);
int head = 0;
int tail = 0;
level[s] = 0;
queue[tail++] = s;
while (head != tail) {
int v = queue[head++];
for (const auto& e : _g[v]) {
if (level[e.to] != -1 || e.cap == Cap(0)) continue;
level[e.to] = level[v] + 1;
if (e.to == t) return true;
queue[tail++] = e.to;
}
}
return level[t] != -1;
};
auto dfs = [&](auto&& self, int v, Cap up) -> Cap {
if (v == s) return up;
Cap result = Cap(0);
const int current_level = level[v];
auto& edges = _g[v];
const int edge_count = int(edges.size());
for (int& i = iter[v]; i < edge_count; i++) {
auto& e = edges[i];
if (level[e.to] + 1 != current_level) continue;
auto& reverse = _g[e.to][e.rev];
if (reverse.cap == Cap(0)) continue;
Cap d = self(
self,
e.to,
std::min(up - result, reverse.cap)
);
if (d == Cap(0)) continue;
e.cap += d;
reverse.cap -= d;
result += d;
if (result == up) return result;
}
level[v] = _n;
return result;
};
Cap flow = 0;
while (flow < flow_limit && bfs()) {
std::fill(iter, iter + _n, 0);
flow += dfs(dfs, t, flow_limit - flow);
}
return flow;
}
std::vector<bool> min_cut(int s) const {
assert(0 <= s && s < _n);
std::vector<bool> visited(_n, false);
std::vector<int> queue(_n);
int head = 0;
int tail = 0;
visited[s] = true;
queue[tail++] = s;
while (head != tail) {
int v = queue[head++];
for (const auto& e : _g[v]) {
if (e.cap == Cap(0) || visited[e.to]) continue;
visited[e.to] = true;
queue[tail++] = e.to;
}
}
return visited;
}
};
} // namespace flow
} // namespace m1une
#line 11 "optimization/project_selection.hpp"
namespace m1une {
namespace opt {
template <class T>
struct ProjectSelectionResult {
bool feasible;
T max_gain;
std::vector<bool> selected;
bool is_feasible() const {
return feasible;
}
};
template <class T>
class ProjectSelection {
static_assert(std::is_integral_v<T> && std::is_signed_v<T>);
struct Arc {
int from;
int to;
T cap;
};
static constexpr int source = -1;
static constexpr int sink = -2;
int _project_count;
int _vertex_count;
T _offset = T();
T _finite_cap_sum = T();
std::vector<Arc> _arcs;
std::vector<std::pair<int, int>> _hard_arcs;
void assert_project(int project) const {
(void)project;
assert(0 <= project && project < _project_count);
}
void assert_vertex(int vertex) const {
(void)vertex;
assert(0 <= vertex && vertex < _vertex_count);
}
void add_offset(T value) {
if (value > T()) {
assert(_offset <= std::numeric_limits<T>::max() - value);
} else if (value < T()) {
assert(_offset >= std::numeric_limits<T>::lowest() - value);
}
_offset += value;
}
T nonnegative_difference(T large, T small) const {
assert(small <= large);
if (small < T()) {
assert(large <= std::numeric_limits<T>::max() + small);
}
return large - small;
}
void add_arc(int from, int to, T cap) {
assert(cap >= T());
if (from == to) return;
assert(cap <= std::numeric_limits<T>::max() - _finite_cap_sum);
_finite_cap_sum += cap;
_arcs.push_back(Arc{from, to, cap});
}
void add_hard_arc(int from, int to) {
if (from == to) return;
_hard_arcs.emplace_back(from, to);
}
void add_vertex_gain(int vertex, T gain_if_selected, T gain_if_unselected) {
assert_vertex(vertex);
if (gain_if_selected >= gain_if_unselected) {
add_offset(gain_if_selected);
add_arc(source, vertex,
nonnegative_difference(gain_if_selected, gain_if_unselected));
} else {
add_offset(gain_if_unselected);
add_arc(vertex, sink,
nonnegative_difference(gain_if_unselected, gain_if_selected));
}
}
int add_auxiliary_vertex() {
return _vertex_count++;
}
public:
ProjectSelection() : ProjectSelection(0) {}
explicit ProjectSelection(int project_count)
: _project_count(project_count), _vertex_count(project_count) {
assert(project_count >= 0);
}
int size() const {
return _project_count;
}
void add_gain(int project, T gain_if_selected) {
add_gain(project, gain_if_selected, T());
}
void add_gain(int project, T gain_if_selected, T gain_if_unselected) {
assert_project(project);
add_vertex_gain(project, gain_if_selected, gain_if_unselected);
}
void add_penalty(int selected_project, int unselected_project, T penalty) {
assert_project(selected_project);
assert_project(unselected_project);
add_arc(selected_project, unselected_project, penalty);
}
void add_penalty_if_different(int project_a, int project_b, T penalty) {
assert_project(project_a);
assert_project(project_b);
add_arc(project_a, project_b, penalty);
add_arc(project_b, project_a, penalty);
}
void add_gain_if_same(int project_a, int project_b, T gain) {
assert(gain >= T());
add_offset(gain);
add_penalty_if_different(project_a, project_b, gain);
}
void add_hard_implication(int selected_project, int required_project) {
assert_project(selected_project);
assert_project(required_project);
add_hard_arc(selected_project, required_project);
}
void force_selected(int project) {
assert_project(project);
add_hard_arc(source, project);
}
void force_unselected(int project) {
assert_project(project);
add_hard_arc(project, sink);
}
void add_gain_if_all_selected(const std::vector<int>& projects, T gain) {
assert(gain >= T());
for (int project : projects) assert_project(project);
if (projects.empty()) {
add_offset(gain);
return;
}
if (projects.size() == 1) {
add_vertex_gain(projects[0], gain, T());
return;
}
if (projects.size() == 2) {
add_vertex_gain(projects[0], gain, T());
add_arc(projects[0], projects[1], gain);
return;
}
int auxiliary = add_auxiliary_vertex();
add_vertex_gain(auxiliary, gain, T());
for (int project : projects) add_hard_arc(auxiliary, project);
}
void add_gain_if_all_unselected(const std::vector<int>& projects, T gain) {
assert(gain >= T());
for (int project : projects) assert_project(project);
if (projects.empty()) {
add_offset(gain);
return;
}
if (projects.size() == 1) {
add_vertex_gain(projects[0], T(), gain);
return;
}
if (projects.size() == 2) {
add_vertex_gain(projects[0], T(), gain);
add_arc(projects[1], projects[0], gain);
return;
}
int auxiliary = add_auxiliary_vertex();
add_vertex_gain(auxiliary, T(), gain);
for (int project : projects) add_hard_arc(project, auxiliary);
}
ProjectSelectionResult<T> solve() const {
int s = _vertex_count;
int t = s + 1;
flow::MaxFlow<T> max_flow(_vertex_count + 2);
auto vertex_id = [&](int vertex) {
if (vertex == source) return s;
if (vertex == sink) return t;
return vertex;
};
for (const auto& arc : _arcs) {
max_flow.add_edge(vertex_id(arc.from), vertex_id(arc.to), arc.cap);
}
T hard_cap = T();
if (!_hard_arcs.empty()) {
assert(_finite_cap_sum < std::numeric_limits<T>::max());
hard_cap = _finite_cap_sum + T(1);
for (auto [from, to] : _hard_arcs) {
max_flow.add_edge(vertex_id(from), vertex_id(to), hard_cap);
}
}
T cut_cost =
_hard_arcs.empty() ? max_flow.max_flow(s, t) : max_flow.max_flow(s, t, hard_cap);
ProjectSelectionResult<T> result;
result.feasible = _hard_arcs.empty() || cut_cost < hard_cap;
result.max_gain = T();
result.selected.assign(_project_count, false);
if (!result.feasible) return result;
assert(_offset >= std::numeric_limits<T>::lowest() + cut_cost);
result.max_gain = _offset - cut_cost;
auto source_side = max_flow.min_cut(s);
for (int project = 0; project < _project_count; project++) {
result.selected[project] = source_side[project];
}
return result;
}
};
} // namespace opt
} // namespace m1une
#line 12 "optimization/k_project_selection.hpp"
namespace m1une {
namespace opt {
template <class T>
struct KProjectSelectionResult {
bool feasible;
T max_gain;
std::vector<int> values;
bool is_feasible() const {
return feasible;
}
};
template <class T>
class KProjectSelection {
static_assert(std::is_integral_v<T> && std::is_signed_v<T>);
static_assert(sizeof(T) <= sizeof(long long));
using Wide = __int128_t;
std::vector<int> _value_counts;
std::vector<int> _first_threshold;
ProjectSelection<T> _binary;
T _constant = T();
static int threshold_count(const std::vector<int>& value_counts) {
assert(value_counts.size() <=
std::size_t(std::numeric_limits<int>::max()));
long long count = 0;
for (int value_count : value_counts) {
assert(value_count >= 1);
count += value_count - 1;
assert(count <= std::numeric_limits<int>::max());
}
return int(count);
}
static std::vector<int> repeated_value_counts(
int project_count,
int value_count
) {
assert(project_count >= 0);
assert(value_count >= 1);
return std::vector<int>(project_count, value_count);
}
void assert_project(int project) const {
(void)project;
assert(0 <= project && project < size());
}
int threshold(int project, int value) const {
assert_project(project);
(void)value;
assert(1 <= value && value < _value_counts[project]);
return _first_threshold[project] + value - 1;
}
static T narrow(Wide value) {
assert(Wide(std::numeric_limits<T>::lowest()) <= value);
assert(value <= Wide(std::numeric_limits<T>::max()));
return T(value);
}
void add_constant(T gain) {
_constant = narrow(Wide(_constant) + gain);
}
void add_threshold_gain(int project, int value, Wide gain) {
if (gain == 0) return;
_binary.add_gain(threshold(project, value), narrow(gain));
}
public:
KProjectSelection() : KProjectSelection(std::vector<int>()) {}
explicit KProjectSelection(std::vector<int> value_counts)
: _value_counts(std::move(value_counts)),
_first_threshold(_value_counts.size()),
_binary(threshold_count(_value_counts)) {
int first = 0;
for (int project = 0; project < size(); project++) {
_first_threshold[project] = first;
first += _value_counts[project] - 1;
}
for (int project = 0; project < size(); project++) {
for (int value = 2; value < _value_counts[project]; value++) {
_binary.add_hard_implication(
threshold(project, value),
threshold(project, value - 1)
);
}
}
}
KProjectSelection(int project_count, int value_count)
: KProjectSelection(repeated_value_counts(project_count, value_count)) {}
int size() const {
return int(_value_counts.size());
}
int value_count(int project) const {
assert_project(project);
return _value_counts[project];
}
void add_gain(int project, const std::vector<T>& gains) {
assert_project(project);
assert(int(gains.size()) == _value_counts[project]);
add_constant(gains[0]);
for (int value = 1; value < _value_counts[project]; value++) {
add_threshold_gain(
project,
value,
Wide(gains[value]) - gains[value - 1]
);
}
}
void add_gain(
int project_a,
int project_b,
const std::vector<std::vector<T>>& gains
) {
assert_project(project_a);
assert_project(project_b);
assert(project_a != project_b);
const int count_a = _value_counts[project_a];
const int count_b = _value_counts[project_b];
assert(int(gains.size()) == count_a);
for (const auto& row : gains) assert(int(row.size()) == count_b);
add_constant(gains[0][0]);
for (int value_a = 1; value_a < count_a; value_a++) {
add_threshold_gain(
project_a,
value_a,
Wide(gains[value_a][0]) - gains[value_a - 1][0]
);
}
for (int value_b = 1; value_b < count_b; value_b++) {
add_threshold_gain(
project_b,
value_b,
Wide(gains[0][value_b]) - gains[0][value_b - 1]
);
}
for (int value_a = 1; value_a < count_a; value_a++) {
for (int value_b = 1; value_b < count_b; value_b++) {
Wide mixed =
Wide(gains[value_a][value_b])
- gains[value_a - 1][value_b]
- gains[value_a][value_b - 1]
+ gains[value_a - 1][value_b - 1];
assert(mixed >= 0);
T gain = narrow(mixed);
if (gain == T()) continue;
int threshold_a = threshold(project_a, value_a);
int threshold_b = threshold(project_b, value_b);
_binary.add_gain(threshold_a, gain);
_binary.add_penalty(threshold_a, threshold_b, gain);
}
}
}
void force_value(int project, int value) {
assert_project(project);
assert(0 <= value && value < _value_counts[project]);
force_value_at_least(project, value);
force_value_at_most(project, value);
}
void force_value_at_least(int project, int lower_bound) {
assert_project(project);
assert(0 <= lower_bound && lower_bound < _value_counts[project]);
if (lower_bound > 0) {
_binary.force_selected(threshold(project, lower_bound));
}
}
void force_value_at_most(int project, int upper_bound) {
assert_project(project);
assert(0 <= upper_bound && upper_bound < _value_counts[project]);
if (upper_bound + 1 < _value_counts[project]) {
_binary.force_unselected(threshold(project, upper_bound + 1));
}
}
KProjectSelectionResult<T> solve() const {
auto binary_result = _binary.solve();
KProjectSelectionResult<T> result;
result.feasible = binary_result.feasible;
result.max_gain = T();
result.values.assign(size(), 0);
if (!result.feasible) return result;
result.max_gain = narrow(Wide(_constant) + binary_result.max_gain);
for (int project = 0; project < size(); project++) {
for (int value = 1; value < _value_counts[project]; value++) {
if (!binary_result.selected[threshold(project, value)]) break;
result.values[project] = value;
}
}
return result;
}
};
} // namespace opt
} // namespace m1une
#line 9 "optimization/all.hpp"
#line 9 "verify/optimization/project_selection.test.cpp"
using ProjectSelection = m1une::opt::ProjectSelection<long long>;
void test_basic() {
ProjectSelection solver(4);
solver.add_gain(0, 8);
solver.add_gain(1, 5);
solver.add_gain(2, -4);
solver.add_gain(3, 3, 1);
solver.add_penalty(0, 1, 10);
solver.add_penalty_if_different(1, 2, 2);
solver.add_gain_if_same(0, 3, 4);
solver.add_gain_if_all_selected(std::vector<int>{0, 1, 2}, 7);
solver.add_gain_if_all_unselected(std::vector<int>{1, 2}, 6);
auto result = solver.solve();
assert(result.is_feasible());
long long expected = std::numeric_limits<long long>::lowest();
for (int mask = 0; mask < (1 << 4); mask++) {
auto selected = [&](int project) {
return ((mask >> project) & 1) != 0;
};
long long gain = 0;
gain += selected(0) ? 8 : 0;
gain += selected(1) ? 5 : 0;
gain += selected(2) ? -4 : 0;
gain += selected(3) ? 3 : 1;
if (selected(0) && !selected(1)) gain -= 10;
if (selected(1) != selected(2)) gain -= 2;
if (selected(0) == selected(3)) gain += 4;
if (selected(0) && selected(1) && selected(2)) gain += 7;
if (!selected(1) && !selected(2)) gain += 6;
if (gain > expected) {
expected = gain;
}
}
assert(result.max_gain == expected);
long long returned_mask = 0;
for (int project = 0; project < 4; project++) {
if (result.selected[project]) returned_mask |= 1LL << project;
}
auto selected = [&](int project) {
return ((returned_mask >> project) & 1) != 0;
};
long long returned_gain = 0;
returned_gain += selected(0) ? 8 : 0;
returned_gain += selected(1) ? 5 : 0;
returned_gain += selected(2) ? -4 : 0;
returned_gain += selected(3) ? 3 : 1;
if (selected(0) && !selected(1)) returned_gain -= 10;
if (selected(1) != selected(2)) returned_gain -= 2;
if (selected(0) == selected(3)) returned_gain += 4;
if (selected(0) && selected(1) && selected(2)) returned_gain += 7;
if (!selected(1) && !selected(2)) returned_gain += 6;
assert(returned_gain == expected);
}
void test_hard_constraints() {
ProjectSelection solver(4);
solver.add_gain(0, 10);
solver.add_gain(1, -2);
solver.add_gain(2, 7);
solver.add_gain(3, 3);
solver.add_hard_implication(0, 1);
solver.add_hard_implication(1, 2);
solver.force_selected(0);
solver.force_unselected(3);
auto result = solver.solve();
assert(result.is_feasible());
assert(result.max_gain == 15);
assert(result.selected[0]);
assert(result.selected[1]);
assert(result.selected[2]);
assert(!result.selected[3]);
ProjectSelection impossible(1);
impossible.force_selected(0);
impossible.force_unselected(0);
assert(!impossible.solve().is_feasible());
}
void test_empty_groups_and_repeated_solve() {
ProjectSelection solver(0);
solver.add_gain_if_all_selected({}, 3);
solver.add_gain_if_all_unselected({}, 4);
assert(solver.size() == 0);
auto first = solver.solve();
auto second = solver.solve();
assert(first.is_feasible() && second.is_feasible());
assert(first.max_gain == 7 && second.max_gain == 7);
assert(first.selected.empty() && second.selected.empty());
}
void test_unary_against_bruteforce() {
for (int n = 1; n <= 8; n++) {
ProjectSelection solver(n);
std::vector<long long> selected_gain(n);
std::vector<long long> unselected_gain(n);
for (int i = 0; i < n; i++) {
selected_gain[i] = (i * 7 + n * 3) % 13 - 6;
unselected_gain[i] = (i * 5 + n * 2) % 11 - 5;
solver.add_gain(i, selected_gain[i], unselected_gain[i]);
}
auto result = solver.solve();
long long expected = std::numeric_limits<long long>::lowest();
for (int mask = 0; mask < (1 << n); mask++) {
long long gain = 0;
for (int i = 0; i < n; i++) {
gain += ((mask >> i) & 1) ? selected_gain[i] : unselected_gain[i];
}
if (gain > expected) expected = gain;
}
assert(result.is_feasible());
assert(result.max_gain == expected);
}
}
void test_mixed_models_against_bruteforce() {
for (int n = 1; n <= 7; n++) {
for (int test = 0; test < 24; test++) {
ProjectSelection solver(n);
std::vector<long long> selected_gain(n);
std::vector<long long> unselected_gain(n);
for (int i = 0; i < n; i++) {
selected_gain[i] = (test * 7 + i * 5 + n) % 15 - 7;
unselected_gain[i] = (test * 3 + i * 11 + n * 2) % 13 - 6;
solver.add_gain(i, selected_gain[i], unselected_gain[i]);
}
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
if ((test + i * 3 + j * 5) % 7 != 0) continue;
long long penalty = (test + i + j) % 6;
solver.add_penalty(i, j, penalty);
}
}
for (int i = 0; i < n; i++) {
for (int j = i + 1; j < n; j++) {
if ((test + i + j) % 5 == 0) {
solver.add_penalty_if_different(i, j, (test + i * 2 + j) % 5);
}
if ((test + i * 2 + j * 3) % 8 == 0) {
solver.add_gain_if_same(i, j, (test + i + j * 2) % 6);
}
}
}
std::vector<int> all_selected;
std::vector<int> all_unselected;
for (int i = 0; i < n; i++) {
if ((test + i) % 3 != 0) all_selected.push_back(i);
if ((test + i * 2) % 4 != 0) all_unselected.push_back(i);
}
long long all_selected_gain = test % 7;
long long all_unselected_gain = (test * 2 + n) % 8;
solver.add_gain_if_all_selected(all_selected, all_selected_gain);
solver.add_gain_if_all_unselected(all_unselected, all_unselected_gain);
for (int i = 0; i < n; i++) {
int j = (i + 1) % n;
if ((test + i * 4) % 11 == 0) solver.add_hard_implication(i, j);
}
if (test % 9 == 0) solver.force_selected(test % n);
if (test % 10 == 0) solver.force_unselected((test * 3 + 1) % n);
bool found = false;
long long expected = std::numeric_limits<long long>::lowest();
for (int mask = 0; mask < (1 << n); mask++) {
auto selected = [&](int project) {
return ((mask >> project) & 1) != 0;
};
bool feasible = true;
for (int i = 0; i < n; i++) {
int j = (i + 1) % n;
if ((test + i * 4) % 11 == 0 && selected(i) && !selected(j)) {
feasible = false;
}
}
if (test % 9 == 0 && !selected(test % n)) feasible = false;
if (test % 10 == 0 && selected((test * 3 + 1) % n)) feasible = false;
if (!feasible) continue;
long long gain = 0;
for (int i = 0; i < n; i++) {
gain += selected(i) ? selected_gain[i] : unselected_gain[i];
}
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
if ((test + i * 3 + j * 5) % 7 == 0 && selected(i) && !selected(j)) {
gain -= (test + i + j) % 6;
}
}
}
for (int i = 0; i < n; i++) {
for (int j = i + 1; j < n; j++) {
if ((test + i + j) % 5 == 0 && selected(i) != selected(j)) {
gain -= (test + i * 2 + j) % 5;
}
if ((test + i * 2 + j * 3) % 8 == 0 && selected(i) == selected(j)) {
gain += (test + i + j * 2) % 6;
}
}
}
bool every_selected = true;
for (int project : all_selected) every_selected &= selected(project);
if (every_selected) gain += all_selected_gain;
bool every_unselected = true;
for (int project : all_unselected) every_unselected &= !selected(project);
if (every_unselected) gain += all_unselected_gain;
found = true;
if (gain > expected) expected = gain;
}
auto result = solver.solve();
assert(result.is_feasible() == found);
if (found) assert(result.max_gain == expected);
}
}
}
int main() {
m1une::utilities::FastInput fast_input;
m1une::utilities::FastOutput fast_output;
test_basic();
test_hard_constraints();
test_empty_groups_and_repeated_solve();
test_unary_against_bruteforce();
test_mixed_models_against_bruteforce();
long long a, b;
fast_input >> a >> b;
fast_output << a + b << '\n';
}
verify/optimization/project_selection.test.cpp