scipy quadratic assignment

Quadratic assignment solves problems of the following form:

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.

The default method 'faq' <optimize.qap-faq> uses the Fast Approximate QAP algorithm ; it typically offers the best combination of speed and accuracy. Method '2opt' <optimize.qap-2opt> can be computationally expensive, but may be a useful alternative, or it can be used to refine the solution returned by another method.

The square matrix $A$ in the objective function above.

The square matrix $B$ in the objective function above.

The algorithm used to solve the problem. 'faq' <optimize.qap-faq> (default) and '2opt' <optimize.qap-2opt> are available.

A dictionary of solver options. All solvers support the following:

partial_match

If seed is None (or :None:None:`np.random` ), the numpy.random.RandomState singleton is used. If seed is an int, a new RandomState instance is used, seeded with seed . If seed is already a Generator or RandomState instance then that instance is used.

For method-specific options, see show_options('quadratic_assignment') <show_options> .

OptimizeResult containing the following fields.

Approximates solution to the quadratic assignment problem and the graph matching problem.

The see the relationship between the returned col_ind and fun , use col_ind to form the best permutation matrix found, then evaluate the objective function $f(P) = trace(A^T P B P^T )$ .

Alternatively, to avoid constructing the permutation matrix explicitly, directly permute the rows and columns of the distance matrix.

Although not guaranteed in general, quadratic_assignment happens to have found the globally optimal solution.

Here is an example for which the default method, 'faq' <optimize.qap-faq> , does not find the global optimum.

If accuracy is important, consider using '2opt' <optimize.qap-2opt> to refine the solution.

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

Local connectivity graph

Hover to see nodes names; edges to Self not shown, Caped at 50 nodes.

Using a canvas is more power efficient and can get hundred of nodes ; but does not allow hyperlinks; , arrows or text (beyond on hover)

SVG is more flexible but power hungry; and does not scale well to 50 + nodes.

All aboves nodes referred to, (or are referred from) current nodes; Edges from Self to other have been omitted (or all nodes would be connected to the central node "self" which is not useful). Nodes are colored by the library they belong to, and scaled with the number of references pointing them

scipy.optimize.linear_sum_assignment #

Solve the linear sum assignment problem.

The cost matrix of the bipartite graph.

Calculates a maximum weight matching if true.

An array of row indices and one of corresponding column indices giving the optimal assignment. The cost of the assignment can be computed as cost_matrix[row_ind, col_ind].sum() . The row indices will be sorted; in the case of a square cost matrix they will be equal to numpy.arange(cost_matrix.shape[0]) .

for sparse inputs

The linear sum assignment problem [1] is also known as minimum weight matching in bipartite graphs. A problem instance is described by a matrix C, where each C[i,j] is the cost of matching vertex i of the first partite set (a ‘worker’) and vertex j of the second set (a ‘job’). The goal is to find a complete assignment of workers to jobs of minimal cost.

Formally, let X be a boolean matrix where \(X[i,j] = 1\) iff row i is assigned to column j. Then the optimal assignment has cost

where, in the case where the matrix X is square, each row is assigned to exactly one column, and each column to exactly one row.

This function can also solve a generalization of the classic assignment problem where the cost matrix is rectangular. If it has more rows than columns, then not every row needs to be assigned to a column, and vice versa.

This implementation is a modified Jonker-Volgenant algorithm with no initialization, described in ref. [2] .

New in version 0.17.0.

https://en.wikipedia.org/wiki/Assignment_problem

DF Crouse. On implementing 2D rectangular assignment algorithms. IEEE Transactions on Aerospace and Electronic Systems , 52(4):1679-1696, August 2016, DOI:10.1109/TAES.2016.140952

• Analysis of Algorithms
• Backtracking
• Dynamic Programming
• Divide and Conquer
• Geometric Algorithms
• Mathematical Algorithms
• Pattern Searching
• Bitwise Algorithms
• Branch & Bound
• Randomized Algorithms

Quadratic Assignment Problem (QAP)

• Channel Assignment Problem
• Assignment Operators In C++
• Solidity - Assignment Operators
• Job Assignment Problem using Branch And Bound
• Range Minimum Query with Range Assignment
• Transportation Problem | Set 1 (Introduction)
• Transportation Problem | Set 2 (NorthWest Corner Method)
• Transportation Problem | Set 3 (Least Cost Cell Method)
• Transportation Problem | Set 4 (Vogel's Approximation Method)
• Assignment Operators in C
• QA - Placement Quizzes | Profit and Loss | Question 7
• QA - Placement Quizzes | Profit and Loss | Question 4
• QA - Placement Quizzes | Profit and Loss | Question 12
• QA - Placement Quizzes | Profit and Loss | Question 10
• QA - Placement Quizzes | Profit and Loss | Question 8
• QA - Placement Quizzes | Age | Question 4
• QA - Placement Quizzes | Profit and Loss | Question 9
• QA - Placement Quizzes | Profit and Loss | Question 11
• IBM Placement Paper | Quantitative Analysis Set - 5

The Quadratic Assignment Problem (QAP) is an optimization problem that deals with assigning a set of facilities to a set of locations, considering the pairwise distances and flows between them.

The problem is to find the assignment that minimizes the total cost or distance, taking into account both the distances and the flows.

The distance matrix and flow matrix, as well as restrictions to ensure each facility is assigned to exactly one location and each location is assigned to exactly one facility, can be used to formulate the QAP as a quadratic objective function.

The QAP is a well-known example of an NP-hard problem , which means that for larger cases, computing the best solution might be difficult. As a result, many algorithms and heuristics have been created to quickly identify approximations of answers.

There are various types of algorithms for different problem structures, such as:

• Precise algorithms
• Approximation algorithms
• Metaheuristics like genetic algorithms and simulated annealing
• Specialized algorithms

Example: Given four facilities (F1, F2, F3, F4) and four locations (L1, L2, L3, L4). We have a cost matrix that represents the pairwise distances or costs between facilities. Additionally, we have a flow matrix that represents the interaction or flow between locations. Find the assignment that minimizes the total cost based on the interactions between facilities and locations. Each facility must be assigned to exactly one location, and each location can only accommodate one facility.

Facilities cost matrix:

Flow matrix:

To solve the QAP, various optimization techniques can be used, such as mathematical programming, heuristics, or metaheuristics. These techniques aim to explore the search space and find the optimal or near-optimal solution.

The solution to the QAP will provide an assignment of facilities to locations that minimizes the overall cost.

The solution generates all possible permutations of the assignment and calculates the total cost for each assignment. The optimal assignment is the one that results in the minimum total cost.

To calculate the total cost, we look at each pair of facilities in (i, j) and their respective locations (location1, location2). We then multiply the cost of assigning facility1 to facility2 (facilities[facility1][facility2]) with the flow from location1 to location2 (locations[location1][location2]). This process is done for all pairs of facilities in the assignment, and the costs are summed up.

Overall, the output tells us that assigning facilities to locations as F1->L1, F3->L2, F2->L3, and F4->L4 results in the minimum total cost of 44. This means that Facility 1 is assigned to Location 1, Facility 3 is assigned to Location 2, Facility 2 is assigned to Location 3, and Facility 4 is assigned to Location 4, yielding the lowest cost based on the given cost and flow matrices.This example demonstrates the process of finding the optimal assignment by considering the costs and flows associated with each facility and location. The objective is to find the assignment that minimizes the total cost, taking into account the interactions between facilities and locations.

Applications of the QAP include facility location, logistics, scheduling, and network architecture, all of which require effective resource allocation and arrangement.

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DOC: in optimize.quadratic_assignment 2opt method, partial_match and partial_guess need to be sorted #18722

ErikVisseMartindale commented Jun 22, 2023

melissawm commented Jun 22, 2023

Sorry, something went wrong.

yashasvimisra2798 commented Dec 9, 2023

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lucascolley commented Dec 9, 2023

Suzaalx commented Feb 7, 2024 • edited

Lucascolley commented feb 7, 2024, suzaalx commented feb 8, 2024, melissawm commented feb 8, 2024.

This is then extracted and autogenerated when the documentation pages are built.

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