We describe a few applications of semide nite programming in combinatorial optimization.

The Quadratic Assignment Problem (QAP) consists of assigning n facilities to n locations so as to minimize the total weighted cost of interactions between facilities. The QAP arises in many diverse settings, is known to be NP-hard, and can be solved to optimality only for fairly small size instances (typically, n < 25). Neighborhood search algorithms are the most popular heuristic algorithms to solve larger size instances of the QAP. The most extensively used neighborhood structure for the QAP is the 2-exchange neighborhood. This neighborhood is obtained by swapping the locations of two facilities and thus has size O(n²). Previous efforts to explore larger size neighborhoods (such as 3-exchange or 4-exchange neighborhoods) were not very successful, as it took too long to evaluate the larger set of neighbors. In this paper, we propose very largescale neighborhood (VLSN) search algorithms where the size of the neighborhood is very large and we propose a novel search procedure to heuristically enumerate good neighbors. Our search procedure relies on the concept of improvement graph which allows us to evaluate neighbors much faster than the existing methods. We present extensive computational results of our algorithms on standard benchmark instances. These investigations reveal that very large-scale neighborhood search algorithms give consistently better solutions compared the popular 2-exchange neighborhood algorithms considering both the solution time and solution accuracy.

We describe MW -- a software framework that allows users to quickly and easily parallelize scientific computations using the masterworker paradigm on the computational grid. MW provides both a "top level" interface to application software and a "bottom level" interface to existing grid computing toolkits. Both interfaces are briefly described. We conclude with a case study, where the necessary Grid services are provided by the Condor high-throughput computing system, and the MW-enabled application code is used to solve a combinatorial optimization problem of unprecedented complexity. This work was supported in part by Grants No. CDA-9726385 and CDA-9623632 from the National Science Foundation. y Department of Electrical and Computer Engineering, Northwestern University, and Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, goux@mcs.anl.gov z Computer Sciences Department, University of Wisconsin - Madison, 1210 West Dayton Street, Madison, WI 53706, fsanjeevk,yodermeg@cs.wisc.edu x Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, linderot@mcs.anl.gov 1 1

Abstract not found

We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems.

Lagrangian duality underlies many efficient algorithms for convex minimization problems. A key ingredient is strong duality. Lagrangian relaxation also provides lower bounds for nonconvex problems, where the quality of the lower bound depends on the duality gap. Quadratically constrained quadratic programs (QQPs) provide important examples of nonconvex programs. For the simple case of one quadratic constraint (the trust region subproblem) strong duality holds. In addition, necessary and sufficient (strengthened) second order optimality conditions exist. However, these duality results already fail for the two trust region subproblem. Surprisingly, there are classes of more complex, nonconvex QQPs where strong duality holds. One example is the special case of orthogonality constraints, which arise naturally in relaxations for the quadratic assignment problem (QAP). In this paper we show that strong duality also holds for a relaxation of QAP where the orthogonality constraint is replaced ...

A heuristic technique that combines a genetic algorithm with a Tabu Search algorithm is applied to the Quadratic Assignment Problem (QAP). The hybrid algorithm improves the results obtained through the application of each of these algorithms separately. The QAP is a NP-hard problem and instances of size n> 15 are still considered intractable. The results of our experiments suggest that CHC combined with TS (CHC+TS), and a TS with elitist backtracking algorithm are able to obtain good near optimal solutions within 0.75 % of the best-known solutions. CHC+TS produces the best-known solution in 12 of the 16 QAPLIB problems tested, where n ranges from 10 to 256. 1

In this paper, we propose two exact algorithms for the GQAP (generalized quadratic assignment problem). In this problem, given M facilities and N locations, the facility space requirements, the location available space, the facility installation costs, the flows between facilities, and the distance costs between locations, one must assign each facility to exactly one location so that each location has sufficient space for all facilities assigned to it and the sum of the products of the facility flows by the corresponding distance costs plus the sum of the installation costs is minimized. This problem generalizes the well-known quadratic assignment problem (QAP). Both exact algorithms combine a previously proposed branch-and-bound scheme with a new Lagrangean relaxation procedure over a known RLT (Reformulation-Linearization Technique) formulation. We also apply transformational lower bounding techniques to improve the performance of the new procedure. We report detailed experimental results where 19 out of 21 instances with up to 35 facilities are solved in up to a few days of running time. Six of these instances were open.

Quadratic Assignment is a basic problem in combinatorial optimization, which 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 non-negative matrices W = (wi,j) and D = (di,j). Given matrices W, D, and a permutation π: [n] → [n], the objective function is Q(π). = � i,j∈[n],i�=j wi,j · dπ(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 an Õ(√n) approximation algorithm, which is the first non-trivial 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 ≈ n1/3 (Feige et al. [8]). 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 bestknown approximation guarantee of 4 (Arkin et al. [4]) 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 [1]). It can also be shown that this LP has an integrality gap of ˜ Ω ( √ n) for general Maximum Quadratic Assignment.

of algorithms