"""
Copyright 2013 Steven Diamond
This file is part of CVXPY.
CVXPY is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
CVXPY is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with CVXPY. If not, see <http://www.gnu.org/licenses/>.
"""
from .assign import Assign
import numpy as np
import cvxpy as cvx
# TODO change to cycle.
class Tour(Assign):
""" A permutation matrix that describes a single cycle.
e.g. 1->3->5->2->4->1
"""
def __init__(self, n, *args, **kwargs):
super(Tour, self).__init__(rows=n, cols=n, *args, **kwargs)
# Compute projection with maximal weighted matching.
def _project(self, matrix):
if self.is_scalar():
return 1
else:
matrix = matrix.copy()
# Greedy algorithm.
# Fix largest entry that still could be a tour.
# Recurse.
tour = np.zeros(self.shape[0], dtype=int) - 1
result = np.zeros(self.shape)
for i in range(self.shape[0]):
while True:
idx = np.argmax(matrix)
row, col = list(zip(*np.unravel_index([idx], self.shape)))[0]
# Check that consistent with tour.
tour[row] = col
if self._no_cycles(tour):
result[row, col] = 1
matrix[row,:] = -np.inf
matrix[:,col] = -np.inf
break
else:
matrix[row, col] = -np.inf
tour[row] = -1
assert not (matrix == -np.inf).all()
return result
def _no_cycles(self, tour):
"""Return true if the tour has no cycles.
"""
for i in range(self.shape[0]):
visited = []
cur = i
while True:
visited.append(cur)
cur = tour[cur]
if cur in visited:
return len(visited) == self.shape[0]
elif cur == -1:
break
return True
def _neighbors(self, matrix):
"""Swap a->b->c->d to a->c->b->d
"""
neighbors_list = []
idxs = np.argmax(matrix, axis=1)
for a in range(self.shape[0]):
new_mat = matrix.copy()
b = idxs[a]
c = idxs[b]
d = idxs[c]
new_mat[a,c] = 1
new_mat[a,b] = 0
new_mat[b,d] = 1
new_mat[b,c] = 0
new_mat[c,b] = 1
new_mat[c,d] = 0
neighbors_list += [new_mat]
return neighbors_list
def relax(self):
"""Convex relaxation.
"""
constr = super(Tour, self).relax()
# Ensure it's a tour.
n = self.shape[0]
# Diagonal == 0 constraint.
constr += [cvx.diag(self) == 0]
# Spectral constraint.
mat_val = np.cos(2*np.pi/n)*np.eye(n) + 4*np.ones((n,n))/n
return constr + [mat_val >> (self + self.T)/2]
