Results 1 
3 of
3
Very LargeScale Neighborhood Search for the Quadratic Assignment Problem
 DISCRETE APPLIED MATHEMATICS
, 2002
"... 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 NPhard, and can be solved to optimality only for fairly small size instances ..."
Abstract

Cited by 108 (11 self)
 Add to MetaCart
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 NPhard, 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 2exchange 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 3exchange or 4exchange 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 largescale neighborhood search algorithms give consistently better solutions compared the popular 2exchange neighborhood algorithms considering both the solution time and solution accuracy.
zapproximations
 Journal of Algorithms
, 2001
"... Approximation algorithms for NPhard optimization problems have been widely studied for over three decades. Most of these measure the quality of the solution produced by taking the ratio of the cost of the solution produced by the algorithm to the cost of an optimal solution. In certain cases, this ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
Approximation algorithms for NPhard optimization problems have been widely studied for over three decades. Most of these measure the quality of the solution produced by taking the ratio of the cost of the solution produced by the algorithm to the cost of an optimal solution. In certain cases, this ratio may not be very meaningful for example, if the ratio of the worst solution to the best solution is at most some constant ff, then an approximation algorithm with factor ff may in fact yield the worst solution! To overcome this hurdle (among others), several authors have independently suggested the use of a different measure which we call zapproximation. An algorithm is an ff zapproximation if it runs in polynomial time, and produces a solution whose distance from the optimal one is at most ff times the distance between the optimal solution and the worst possible solution. The results known so far about zapproximations are either of the inapproximability type or rather straightforward observations. We design polynomial time algorithms for several fundamental discrete optimization problems, in particular we obtain a zapproximation factor of 1 2 for the directed traveling salesman problem (TSP) (with no triangle inequality assumption). For the undirected TSP this improves to
TSP Heuristics: Domination Analysis and Complexity
, 2001
"... We show that the 2Opt and 3Opt heuristics for the traveling salesman problem (TSP) on a complete graph K n produce a solution no worse than the average cost of a tour in K n in a polynomial number of iterations. As a consequence, the domination numbers of the 2 Opt and 3 Opt, CarlierVillon, Short ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
We show that the 2Opt and 3Opt heuristics for the traveling salesman problem (TSP) on a complete graph K n produce a solution no worse than the average cost of a tour in K n in a polynomial number of iterations. As a consequence, the domination numbers of the 2 Opt and 3 Opt, CarlierVillon, Shortest Path Ejection Chain, and LinKernighan heuristics are all at least (n 2)! 2 . The domination number of the Christodes heuristic is shown to be no more than d n 2 e!, and for the Double Tree heuristic and a variation of the Christodes heuristic the domination number is shown to be one (even if the edge costs satisfy the triangle inequality). Further, unless P=NP, no polynomial time approximation algorithm exists for the TSP with domination number at least (n 1)! k for any constant k or with domination number at least (n 1)! ( k k+1 (n + r))! 1 for any constant r and any constant k such that k 1 mod (n + r): The complexity of nding the value of the median tour and of similar problems is also studied. Key words: Domination Analysis, Approximation Algorithms, Traveling Salesman Problem, Computational Complexity Email addresses: punnen@unbsj.ca (Abraham Punnen), fmargot@ms.uky.edu (Francois Margot), kabadi@unb.ca (Santosh Kabadi). 1 Work completed while visiting University of Kentucky. Work supported by the NSERC grant OPG 0170381 2 Work supported by the NSERC grant OPG 0008085 March, 2001 1