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All solvers support the following:" } ] }, { "__type": "DefList", "__tag": 4033, "children": [ { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "maximize" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "maximize" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "partial_match" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "partial_match" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "rng" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "rng" } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "For method-specific options, see " }, { "__type": "CrossRef", "__tag": 4002, "value": "show_options('quadratic_assignment')", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.optimize._optimize:show_options" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Quadratic assignment solves problems of the following form:" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\min_P & \\ {\\ \\text{trace}(A^T P B P^T)}\\\\\n\\mbox{s.t. } & {P \\ \\epsilon \\ \\mathcal{P}}\\\\" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\mathcal{P}" }, { "__type": "Text", "__tag": 4046, "value": " is the set of all permutation matrices, and " }, { "__type": "InlineMath", "__tag": 4057, "value": "A" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "B" }, { "__type": "Text", "__tag": 4046, "value": " are square matrices." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Graph matching tries to " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "maximize" } ] }, { "__type": "Text", "__tag": 4046, "value": " 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." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Note that the quadratic assignment problem is NP-hard. The results given here are approximations and are not guaranteed to be optimal." } ] } ], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "/scipy/optimize/_qap.py", "item_line": 14, "item_type": "function", "aliases": [ "scipy.optimize.quadratic_assignment" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy.optimize import quadratic_assignment\nrng = np.random.default_rng()\nA = np.array([[0, 80, 150, 170], [80, 0, 130, 100],\n [150, 130, 0, 120], [170, 100, 120, 0]])\nB = np.array([[0, 5, 2, 7], [0, 0, 3, 8],\n [0, 0, 0, 3], [0, 0, 0, 0]])\nres = quadratic_assignment(A, B, options={'rng': rng})\nprint(res)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nThe see the relationship between the returned ``col_ind`` and ``fun``,\nuse ``col_ind`` to form the best permutation matrix found, then evaluate\nthe objective function :math:`f(P) = trace(A^T P B P^T )`.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "perm = res['col_ind']\nP = np.eye(len(A), dtype=int)[perm]\nfun = np.trace(A.T @ P @ B @ P.T)\nprint(fun)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nAlternatively, to avoid constructing the permutation matrix explicitly,\ndirectly permute the rows and columns of the distance matrix.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "fun = np.trace(A.T @ B[perm][:, perm])\nprint(fun)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nAlthough not guaranteed in general, ``quadratic_assignment`` happens to\nhave found the globally optimal solution.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from itertools import permutations\nperm_opt, fun_opt = None, np.inf\nfor perm in permutations([0, 1, 2, 3]):\n perm = np.array(perm)\n fun = np.trace(A.T @ B[perm][:, perm])\n if fun < fun_opt:\n fun_opt, perm_opt = fun, perm\nprint(np.array_equal(perm_opt, res['col_ind']))\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nHere is an example for which the default method,\n:ref:`'faq' `, does not find the global optimum.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "A = np.array([[0, 5, 8, 6], [5, 0, 5, 1],\n [8, 5, 0, 2], [6, 1, 2, 0]])\nB = np.array([[0, 1, 8, 4], [1, 0, 5, 2],\n [8, 5, 0, 5], [4, 2, 5, 0]])\nres = quadratic_assignment(A, B, options={'rng': rng})\nprint(res)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nIf accuracy is important, consider using :ref:`'2opt' `\nto refine the solution.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "guess = np.array([np.arange(len(A)), res.col_ind]).T\nres = quadratic_assignment(A, B, method=\"2opt\",\n options = {'rng': rng, 'partial_guess': guess})\nprint(res)\n", "execution_status": "success" } ], "title": [], "level": 0, "target": null }, "see_also": [], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "A", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "B", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "method", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "faq" }, { "__type": "SigParam", "__tag": 4030, "name": "options", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "quadratic_assignment" }, "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] \"2-opt,\" Wikipedia.", " https://en.wikipedia.org/wiki/2-opt" ], "qa": "scipy.optimize._qap:quadratic_assignment", "arbitrary": [], "local_refs": [ "A", "B", "method", "options", "res" ] }