The hardness of approximation: gap location
Computational Complexity
Approximation algorithms for minimum tree partition
Discrete Applied Mathematics
Approximation algorithms for min-sum p-clustering
Discrete Applied Mathematics
P-Complete Approximation Problems
Journal of the ACM (JACM)
Approximating the maximum quadratic assignment problem
Information Processing Letters
Approximation algorithms
Approximation algorithms for maximization problems arising in graph partitioning
Journal of Algorithms
Relations between average case complexity and approximation complexity
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
SIAM Journal on Discrete Mathematics
Information Processing Letters
Ruling Out PTAS for Graph Min-Bisection, Dense k-Subgraph, and Bipartite Clique
SIAM Journal on Computing
Approximating the minimum quadratic assignment problems
ACM Transactions on Algorithms (TALG)
Approximation algorithms for maximum dispersion
Operations Research Letters
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Minimum congestion mapping in a cloud
Proceedings of the 30th annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing
Maximizing polynomials subject to assignment constraints
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
An O(n4) Algorithm for the QAP Linearization Problem
Mathematics of Operations Research
Hi-index | 0.00 |
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 $W=(w_{i, j})$ and $D=(d_{i, j})$. Given matrices W, D, and a permutation $\pi: [n] \rightarrow [n]$, the objective function is $Q(\pi):= \sum_{i, j \in [n], i \ne j} w_{i, j} \cdot d_{\pi(i), \pi(j)}$. In this paper, we study the maximum quadratic assignment problem, where the goal is to find a permutation π that maximizes $Q(\pi)$. We give an $\tilde{O}(\sqrt{n})$-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 k subgraph problem, for which the best-known approximation ratio is $\approx n^{1/3}$ (Feige et al. [Feige, U., G. Kortsarz, D. Peleg. 2001. The dense k-subgraph problem. Algorithmica29(3) 410--421]). When one of the matrices W, D satisfies triangle inequality, we obtain a $2e/(e-1) \approx 3.16$-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 $\tilde{\Omega}(\sqrt{n})$ for general maximum quadratic assignment.