On the Maximum Quadratic Assignment Problem
Quadratic assignment is a basic problem in combinatorial optimization that generalizes several other problems such as traveling salesman, linear arrangement, dense k subgraph, and clustering with given sizes. The input to the quadratic assignment problem consists of two n x n symmetric nonnegative m...
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| Vydané v: | Mathematics of operations research Ročník 34; číslo 4; s. 859 - 868 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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Linthicum
INFORMS
01.11.2009
Institute for Operations Research and the Management Sciences |
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| ISSN: | 0364-765X, 1526-5471 |
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| Abstract | Quadratic assignment is a basic problem in combinatorial optimization that generalizes several other problems such as traveling salesman, linear arrangement, dense k subgraph, and clustering with given sizes. The input to the quadratic assignment problem consists of two n x n symmetric nonnegative matrices and . Given matrices W, D , and a permutation , the objective function is . In this paper, we study the maximum quadratic assignment problem , where the goal is to find a permutation that maximizes . We give an -approximation algorithm, which is the first nontrivial approximation guarantee for this problem. The above guarantee also holds when the matrices W, D are asymmetric. An indication of the hardness of maximum quadratic assignment is that it contains as a special case the dense k subgraph problem, for which the best-known approximation ratio is (Feige et al. [Feige, U., G. Kortsarz, D. Peleg. 2001. The dense k -subgraph problem. Algorithmica 29 (3) 410–421]).
When one of the matrices W, D satisfies triangle inequality , we obtain a -approximation algorithm. This improves over the previously best-known approximation guarantee of four (Arkin et al. [Arkin, E. M., R. Hassin, M. Sviridenko. 2001. Approximating the maximum quadratic assignment problem. Inform. Processing Lett. 77 13–16]) for this special case of maximum quadratic assignment.
The performance guarantee for maximum quadratic assignment with triangle inequality can be proved relative to an optimal solution of a natural linear programming relaxation that has been used earlier in branch-and-bound approaches (see, eg., Adams and Johnson [Adams, W. P., T. A. Johnson. 1994. Improved linear programming-based lower bounds for the quadratic assignment problem. DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 16 43–77]). It can also be shown that this linear program (LP) has an integrality gap of for general maximum quadratic assignment. |
|---|---|
| AbstractList | Quadratic assignment is a basic problem in combinatorial optimization that generalizes several other problems such as traveling salesman, linear arrangement, dense k subgraph, and clustering with given sizes. The input to the quadratic assignment problem consists of two n x n symmetric normegative matrices W = ([w.sub.i, j) and D = ([d.sub.i,j]). Given matrices W, D, and a permutation [pi]: [n] [right arrow] [n], the objective function is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this paper, we study the maximum quadratic assignment problem, where the goal is to find a permutation [pi] that maximizes Q(pi). We give an o([square root of ([square root of (n)])-approximation algorithm, which is the first nontrivial approximation guarantee for this problem. The above guarantee also holds when the matrices W, D are asymmetric. An indication of the hardness of maximum quadratic assignment is that it contains as a special case the dense k subgraph problem, for which the best-known approximation ratio is [approximately equal to][n.sup.1/3] (Feige et al. [Feige, U., G. Kortsarz, D. Peleg. 2001. The dense k-subgraph problem. Algorithmica 29(3) 410-421]). When one of the matrices W, D satisfies triangle inequality, we obtain a 2e/(e-1) [approximately equal to] 3.16-approximation algorithm. This improves over the previously best-known approximation guarantee of four (Arkin et al. [Arkin, E. M., R. Hassin, M. Sviridenko. 2001. Approximating the maximum quadratic assignment problem. Inform. Processing Lett. 77 13-16]) for this special case of maximum quadratic assignment. The performance guarantee for maximum quadratic assignment with triangle inequality can be proved relative to an optimal solution of a natural linear programming relaxation that has been used earlier in branch-and-bound approaches (see, eg., Adams and Johnson [Adams, W. R, T. A. Johnson. 1994. Improved linear programming-based lower bounds for the quadratic assignment problem. DIMACS Set Discrete Math. Theoret. Comput. Sci. 16 43-77]). It can also be shown that this linear program (LP) has an integrality gap of [??]([square root of (n)]) for general maximum quadratic assignment. Key words: approximation algorithms; linear programming relaxation MSC2000 subject classification: Primary: 90C27, 90C59, 68W25; secondary: 68W40, 68W20 OR/MS subject classification: Primary: Analysis of algorithms--suboptimal algorithms; secondary: networks/graphs--heuristics History: Received November 8, 2008; revised July 15, 2009. Published online in Articles in Advance October 20, 2009. DOI 10.1287/moor.1090.0418 Quadratic assignment is a basic problem in combinatorial optimization that generalizes several other problems such as traveling salesman, linear arrangement, dense k subgraph, and clustering with given sizes. The input to the quadratic assignment problem consists of two n x n symmetric nonnegative matrices and . Given matrices W, D , and a permutation , the objective function is . In this paper, we study the maximum quadratic assignment problem , where the goal is to find a permutation that maximizes . We give an -approximation algorithm, which is the first nontrivial approximation guarantee for this problem. The above guarantee also holds when the matrices W, D are asymmetric. An indication of the hardness of maximum quadratic assignment is that it contains as a special case the dense k subgraph problem, for which the best-known approximation ratio is (Feige et al. [Feige, U., G. Kortsarz, D. Peleg. 2001. The dense k -subgraph problem. Algorithmica 29 (3) 410–421]). When one of the matrices W, D satisfies triangle inequality , we obtain a -approximation algorithm. This improves over the previously best-known approximation guarantee of four (Arkin et al. [Arkin, E. M., R. Hassin, M. Sviridenko. 2001. Approximating the maximum quadratic assignment problem. Inform. Processing Lett. 77 13–16]) for this special case of maximum quadratic assignment. The performance guarantee for maximum quadratic assignment with triangle inequality can be proved relative to an optimal solution of a natural linear programming relaxation that has been used earlier in branch-and-bound approaches (see, eg., Adams and Johnson [Adams, W. P., T. A. Johnson. 1994. Improved linear programming-based lower bounds for the quadratic assignment problem. DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 16 43–77]). It can also be shown that this linear program (LP) has an integrality gap of for general maximum quadratic assignment. Quadratic assignment is a basic problem in combinatorial optimization that generalizes several other problems such as traveling salesman, linear arrangement, dense k subgraph, and clustering with given sizes. The input to the quadratic assignment problem consists of two n × n symmetric nonnegative matrices [Formula: see text] and [Formula: see text]. Given matrices W, D, and a permutation [Formula: see text], the objective function is [Formula: see text]. In this paper, we study the maximum quadratic assignment problem, where the goal is to find a permutation π that maximizes [Formula: see text]. We give an [Formula: see text]-approximation algorithm, which is the first nontrivial approximation guarantee for this problem. The above guarantee also holds when the matrices W, D are asymmetric. An indication of the hardness of maximum quadratic assignment is that it contains as a special case the dense k subgraph problem, for which the best-known approximation ratio is [Formula: see text] (Feige et al. [Feige, U., G. Kortsarz, D. Peleg. 2001. The dense k-subgraph problem. Algorithmica 29(3) 410–421]). When one of the matrices W, D satisfies triangle inequality, we obtain a [Formula: see text]-approximation algorithm. This improves over the previously best-known approximation guarantee of four (Arkin et al. [Arkin, E. M., R. Hassin, M. Sviridenko. 2001. Approximating the maximum quadratic assignment problem. Inform. Processing Lett. 77 13–16]) for this special case of maximum quadratic assignment. The performance guarantee for maximum quadratic assignment with triangle inequality can be proved relative to an optimal solution of a natural linear programming relaxation that has been used earlier in branch-and-bound approaches (see, eg., Adams and Johnson [Adams, W. P., T. A. Johnson. 1994. Improved linear programming-based lower bounds for the quadratic assignment problem. DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 16 43–77]). It can also be shown that this linear program (LP) has an integrality gap of [Formula: see text] for general maximum quadratic assignment. Quadratic assignment is a basic problem in combinatorial optimization that generalizes several other problems such as traveling salesman, linear arrangement, dense k subgraph, and clustering with given sizes. The input to the quadratic assignment problem consists of two n x n symmetric normegative matrices W = ([w.sub.i, j) and D = ([d.sub.i,j]). Given matrices W, D, and a permutation [pi]: [n] [right arrow] [n], the objective function is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this paper, we study the maximum quadratic assignment problem, where the goal is to find a permutation [pi] that maximizes Q(pi). We give an o([square root of ([square root of (n)])-approximation algorithm, which is the first nontrivial approximation guarantee for this problem. The above guarantee also holds when the matrices W, D are asymmetric. An indication of the hardness of maximum quadratic assignment is that it contains as a special case the dense k subgraph problem, for which the best-known approximation ratio is [approximately equal to][n.sup.1/3] (Feige et al. [Feige, U., G. Kortsarz, D. Peleg. 2001. The dense k-subgraph problem. Algorithmica 29(3) 410-421]). Quadratic assignment is a basic problem in combinatorial optimization that generalizes several other problems such as traveling salesman, linear arrangement, dense k subgraph, and clustering with given sizes. The input to the quadratic assignment problem consists of two n x n symmetric nonnegative matrices $W = (w_{i,j} )\,and\,D = (d_{i,j} )$ . Given matrices W, D, and a permutation π: [n] → [n], the objective function is $Q(\pi ): = \Sigma _{i,j[n],i\# j} w_{i,j} .d_{\pi (i),\pi (j)} $ . In this paper, we study the maximum quadratic assignment problem, where the goal is to find a permutation π that maximizes Q(π). We give an Õ $(\sqrt n )$ -approximation algorithm, which is the first nontrivial approximation guarantee for this problem. The above guarantee also holds when the matrices W, D are asymmetric. An indication of the hardness of maximum quadratic assignment is that it contains as a special case the dense subgraph problem, for which the best-known approximation ratio is ≈n⅓ (Feige et al. [Feige, U., G. Kortsarz, D. Peleg. 2001. The dense k-subgraph problem. Algorithmica 29(3) 410-421]). When one of the matrices W, D satisfies triangle inequality, we obtain a 2e/(e— 1) ≈3.16-approximation algorithm. This improves over the previously best-known approximation guarantee of four (Arkin et al. [Arkin, E. M., R. Hassin, M. Sviridenko. 2001. Approximating the maximum quadratic assignment problem. Inform. Processing Lett. 77 13-16]) for this special case of maximum quadratic assignment. The performance guarantee for maximum quadratic assignment with triangle inequality can be proved relative to an optimal solution of a natural linear programming relaxation that has been used earlier in branch-and-bound approaches (see, eg., Adams and Johnson [Adams, W. P., T. A. Johnson. 1994. Improved linear programming-based lower bounds for the quadratic assignment problem. DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 16 43-77]). It can also be shown that this linear program (LP) has an integrality gap of ${\rm{\Omega (}}\sqrt n {\rm{)}}$ for general maximum quadratic assignment. Quadratic assignment is a basic problem in combinatorial optimization that generalizes several other problems such as traveling salesman, linear arrangement, dense k subgraph, and clustering with given sizes. The input to the quadratic assignment problem consists of two n × n symmetric nonnegative matrices W = (w^sub i, j^) and D = (d^sub i, j^). Given matrices W, D, and a permutation π : [n] [arrow right] [n], the objective function is Q(π) := Σ^sub i, j∈[n], i≠j^ w^sub i, j^ * d^sub π(i),π(j)^. In this paper, we study the maximum quadratic assignment problem, where the goal is to find a permutation π that maximizes Q(π). We give e an ...-approximation algorithm, which is the first nontrivial approximation guarantee for this problem. The above guarantee also holds when the matrices W, D are asymmetric. An indication of the hardness of maximum quadratic assignment is that it contains as a special case the dense k subgraph problem, for which the best-known approximation ratio is [asymptotically =]n^sup 1/3^ (Feige et al. [Feige, U., G. Kortsarz, D. Peleg. 2001. The dense k-subgraph problem. Algorithmica 29(3) 410 421]). When one of the matrices W, D satisfies triangle inequality, we obtain a 2e/(e-1) [asymptotically =] 3.16-approximation algorithm. This improves over the previously best-known approximation guarantee of four (Arkin et al. [Arkin, E. M., R. Hassin, M. Sviridenko. 2001. Approximating the maximum quadratic assignment problem. Inform. Processing Lett. 77 13-16]) for this special case of maximum quadratic assignment. The performance guarantee for maximum quadratic assignment with triangle inequality can be proved relative to an optimal solution of a natural linear programming relaxation that has been used earlier in branch-and-bound approaches (see, eg., Adams and Johnson [Adams, W. P., T. A. Johnson. 1994. Improved linear programming-based lower bounds for the quadratic assignment problem. DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 16 43-77]). It can also be shown that this linear program (LP) has an integrality gap of ... for general maximum quadratic assignment. [PUBLICATION ABSTRACT] Quadratic assignment is a basic problem in combinatorial optimization that generalizes several other problems such as traveling salesman, linear arrangement, dense k subgraph, and clustering with given sizes. The input to the quadratic assignment problem consists of two n × n symmetric nonnegative matrices \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $W=(w_{i, j})$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $D=(d_{i, j})$ \end{document} . Given matrices W, D , and a permutation \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\pi: [n] \rightarrow [n]$ \end{document} , the objective function is \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $Q(\pi):= \sum_{i, j \in [n],\, i \ne j} w_{i, j} \cdot d_{\pi(i), \pi(j)}$ \end{document} . In this paper, we study the maximum quadratic assignment problem , where the goal is to find a permutation π that maximizes \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $Q(\pi)$ \end{document} . We give an \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\tilde{O}(\sqrt{n})$ \end{document} -approximation algorithm, which is the first nontrivial approximation guarantee for this problem. The above guarantee also holds when the matrices W, D are asymmetric. An indication of the hardness of maximum quadratic assignment is that it contains as a special case the dense k subgraph problem, for which the best-known approximation ratio is \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\approx n^{1/3}$ \end{document} (Feige et al. [Feige, U., G. Kortsarz, D. Peleg. 2001. The dense k -subgraph problem. Algorithmica 29 (3) 410-421]). When one of the matrices W, D satisfies triangle inequality , we obtain a \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $2e/(e-1) \approx 3.16$ \end{document} -approximation algorithm. This improves over the previously best-known approximation guarantee of four (Arkin et al. [Arkin, E. M., R. Hassin, M. Sviridenko. 2001. Approximating the maximum quadratic assignment problem. Inform. Processing Lett. 77 13-16]) for this special case of maximum quadratic assignment. The performance guarantee for maximum quadratic assignment with triangle inequality can be proved relative to an optimal solution of a natural linear programming relaxation that has been used earlier in branch-and-bound approaches (see, eg., Adams and Johnson [Adams, W. P., T. A. Johnson. 1994. Improved linear programming-based lower bounds for the quadratic assignment problem. DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 16 43-77]). It can also be shown that this linear program (LP) has an integrality gap of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\tilde{\Omega}(\sqrt{n})$ \end{document} for general maximum quadratic assignment. |
| Audience | Academic |
| Author | Nagarajan, Viswanath Sviridenko, Maxim |
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| SubjectTerms | Algorithms Analysis Approximation Approximation algorithms Approximation theory Approximations Branch & bound algorithms Combinatorial optimization Integrality Linear programming linear programming relaxation Mathematical permutation Mathematics Maximum value Objective functions Quadratic programming Studies Triangle inequalities Vertices |
| Title | On the Maximum Quadratic Assignment Problem |
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