quadratic_assignment(method=’faq’)#
- scipy.optimize.quadratic_assignment(A, B, method='faq', options=None)
Solve the quadratic assignment problem (approximately).
This function solves the Quadratic Assignment Problem (QAP) and the Graph Matching Problem (GMP) using the Fast Approximate QAP Algorithm (FAQ) [1].
Quadratic assignment solves problems of the following form:
\[\begin{split}\min_P & \ {\ \text{trace}(A^T P B P^T)}\\ \mbox{s.t. } & {P \ \epsilon \ \mathcal{P}}\\\end{split}\]where \(\mathcal{P}\) is the set of all permutation matrices, and \(A\) and \(B\) are square matrices.
Graph matching tries to maximize the same objective function. This algorithm can be thought of as finding the alignment of the nodes of two graphs that minimizes the number of induced edge disagreements, or, in the case of weighted graphs, the sum of squared edge weight differences.
Note that the quadratic assignment problem is NP-hard. The results given here are approximations and are not guaranteed to be optimal.
- Parameters:
- A2-D array, square
The square matrix \(A\) in the objective function above.
- B2-D array, square
The square matrix \(B\) in the objective function above.
- methodstr in {‘faq’, ‘2opt’} (default: ‘faq’)
The algorithm used to solve the problem. This is the method-specific documentation for ‘faq’. ‘2opt’ is also available.
- Returns:
- resOptimizeResult
OptimizeResult
containing the following fields.- col_ind1-D array
Column indices corresponding to the best permutation found of the nodes of B.
- funfloat
The objective value of the solution.
- nitint
The number of Frank-Wolfe iterations performed.
See also
For documentation for the rest of the parameters, see
scipy.optimize.quadratic_assignment
- Options:
- ——-
- maximizebool (default: False)
Maximizes the objective function if
True
.- partial_match2-D array of integers, optional (default: None)
Fixes part of the matching. Also known as a “seed” [2].
Each row of partial_match specifies a pair of matched nodes: node
partial_match[i, 0]
of A is matched to nodepartial_match[i, 1]
of B. The array has shape(m, 2)
, wherem
is not greater than the number of nodes, \(n\).- rng{None, int,
numpy.random.Generator
, numpy.random.RandomState
}, optionalIf seed is None (or np.random), the
numpy.random.RandomState
singleton is used. If seed is an int, a newRandomState
instance is used, seeded with seed. If seed is already aGenerator
orRandomState
instance then that instance is used.- P02-D array, “barycenter”, or “randomized” (default: “barycenter”)
Initial position. Must be a doubly-stochastic matrix [3].
If the initial position is an array, it must be a doubly stochastic matrix of size \(m' \times m'\) where \(m' = n - m\).
If
"barycenter"
(default), the initial position is the barycenter of the Birkhoff polytope (the space of doubly stochastic matrices). This is a \(m' \times m'\) matrix with all entries equal to \(1 / m'\).If
"randomized"
the initial search position is \(P_0 = (J + K) / 2\), where \(J\) is the barycenter and \(K\) is a random doubly stochastic matrix.- shuffle_inputbool (default: False)
Set to True to resolve degenerate gradients randomly. For non-degenerate gradients this option has no effect.
- maxiterint, positive (default: 30)
Integer specifying the max number of Frank-Wolfe iterations performed.
- tolfloat (default: 0.03)
Tolerance for termination. Frank-Wolfe iteration terminates when \(\frac{||P_{i}-P_{i+1}||_F}{\sqrt{m')}} \leq tol\), where \(i\) is the iteration number.
Notes
The algorithm may be sensitive to the initial permutation matrix (or search “position”) due to the possibility of several local minima within the feasible region. A barycenter initialization is more likely to result in a better solution than a single random initialization. However, calling
quadratic_assignment
several times with different random initializations may result in a better optimum at the cost of longer total execution time.References
[1]J.T. Vogelstein, J.M. Conroy, V. Lyzinski, L.J. Podrazik, S.G. Kratzer, E.T. Harley, D.E. Fishkind, R.J. Vogelstein, and C.E. Priebe, “Fast approximate quadratic programming for graph matching,” PLOS one, vol. 10, no. 4, p. e0121002, 2015, DOI:10.1371/journal.pone.0121002
[2]D. Fishkind, S. Adali, H. Patsolic, L. Meng, D. Singh, V. Lyzinski, C. Priebe, “Seeded graph matching”, Pattern Recognit. 87 (2019): 203-215, DOI:10.1016/j.patcog.2018.09.014
[3]“Doubly stochastic Matrix,” Wikipedia. https://en.wikipedia.org/wiki/Doubly_stochastic_matrix
Examples
As mentioned above, a barycenter initialization often results in a better solution than a single random initialization.
>>> from numpy.random import default_rng >>> rng = default_rng() >>> n = 15 >>> A = rng.random((n, n)) >>> B = rng.random((n, n)) >>> res = quadratic_assignment(A, B) # FAQ is default method >>> print(res.fun) 46.871483385480545 # may vary
>>> options = {"P0": "randomized"} # use randomized initialization >>> res = quadratic_assignment(A, B, options=options) >>> print(res.fun) 47.224831071310625 # may vary
However, consider running from several randomized initializations and keeping the best result.
>>> res = min([quadratic_assignment(A, B, options=options) ... for i in range(30)], key=lambda x: x.fun) >>> print(res.fun) 46.671852533681516 # may vary
The ‘2-opt’ method can be used to further refine the results.
>>> options = {"partial_guess": np.array([np.arange(n), res.col_ind]).T} >>> res = quadratic_assignment(A, B, method="2opt", options=options) >>> print(res.fun) 46.47160735721583 # may vary