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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 ..."
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Cited by 108 (11 self)
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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.
A Study of Exponential Neighborhoods for the Travelling Salesman Problem and for the Quadratic Assignment Problem
, 1998
"... This paper deals with socalled exponential neighborhoods for combinatorial optimization problems, i.e. with large sets of feasible solutions whose size grows exponentially with the input length. We are especially interested in exponential neighborhoods over which the TSP (respectively, the QAP) can ..."
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Cited by 22 (2 self)
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This paper deals with socalled exponential neighborhoods for combinatorial optimization problems, i.e. with large sets of feasible solutions whose size grows exponentially with the input length. We are especially interested in exponential neighborhoods over which the TSP (respectively, the QAP) can be solved in polynomial time, and we investigate combinatorial and algorithmical questions related to such neighborhoods. First, we perform a careful study of exponential neighborhoods for the TSP. We investigate neighborhoods that can be defined in a simple way via assignments, matchings in bipartite graphs, partial orders, trees and other combinatorial structures. We identify several properties of these combinatorial structures that lead to polynomial time optimization algorithms, and we also provide variants that slightly violate these properties and lead to NPcomplete optimization problems. Whereas it is relatively easy to find exponential neighborhoods over which the TSP can be solved...
The Structure of Circular Decomposable Metrics
 in: Proc. 4th ESA, Lect. Notes Comput. Sci. 1136 (SpringerVerlag
, 1996
"... Introduction Given a hard combinatorial optimization problem, a natural research direction is to find specially structured cases which can be solved more easily. These special cases may be useful in their own right or they may be used as a surrogate for finding approximate solutions in more general ..."
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Cited by 12 (1 self)
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Introduction Given a hard combinatorial optimization problem, a natural research direction is to find specially structured cases which can be solved more easily. These special cases may be useful in their own right or they may be used as a surrogate for finding approximate solutions in more general cases. There have been a number of special cases found for the Traveling Salesman Problem (TSP). In the TSP, there are n points and a distance function D[i; j] that maps pairs of points into nonnegative values representing the distances between the points. The objective is to find a permutation ß that minimizes P n i=1 D[ß(i); ß(i+ 1)] +D[ß(n); ß(0)]. We will work with symmetric distance functions, where D[i; j] = D[j;
On approximating restricted cycle covers
 In Workshop on Approximation and Online Algorithms (WAOA
, 2005
"... A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An Lcycle cover is a cycle cover in which the length of every cycle is in the set L. The weight of a cycle cover of an edgeweighted graph is the sum of the weights of its edges. We come close to settli ..."
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Cited by 7 (4 self)
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A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An Lcycle cover is a cycle cover in which the length of every cycle is in the set L. The weight of a cycle cover of an edgeweighted graph is the sum of the weights of its edges. We come close to settling the complexity and approximability of computing Lcycle covers. On the one hand, we show that for almost all L, computing Lcycle covers of maximum weight in directed and undirected graphs is APXhard. Most of our hardness results hold even if the edge weights are restricted to zero and one. On the other hand, we show that the problem of computing Lcycle covers of maximum weight can be approximated within a factor of 2 for undirected graphs and within a factor of 8/3 in the case of directed graphs. This holds for arbitrary sets L. 1
Approximation algorithms for multicriteria traveling salesman problems
, 2006
"... In multicriteria optimization, several objective functions are to be optimized. Since the different objective functions are usually in conflict with each other, one cannot consider only one particular solution as optimal. Instead, the aim is to compute socalled Pareto curves. Since Pareto curves c ..."
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Cited by 7 (4 self)
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In multicriteria optimization, several objective functions are to be optimized. Since the different objective functions are usually in conflict with each other, one cannot consider only one particular solution as optimal. Instead, the aim is to compute socalled Pareto curves. Since Pareto curves cannot be computed efficiently in general, we have to be content with approximations to them. We are concerned with approximating Pareto curves of multicriteria traveling salesman problems (TSP). We provide algorithms for computing approximate Pareto curves for the symmetric TSP with triangle inequality ( ∆STSP), symmetric and asymmetric TSP with strengthened triangle inequality (∆(γ)STSP and ∆(γ)ATSP), and symmetric and asymmetric TSP with weights one and two (STSP(1, 2) and ATSP(1, 2)). We design a deterministic polynomialtime algorithm that computes (1 + γ + ε)approximate Pareto curves for multicriteria ∆(γ)STSP for γ ∈ [ 1, 1]. We also present two randomized approximation algo2 rithms for multicriteria ∆(γ)STSP achieving approximation ratios of
Solving the Aerial Fleet Refueling Problem using Group Theoretic Tabu Search
, 2004
"... The Aerial Fleet Refueling Problem (AFRP) is concerned with the efficient and effective use of a heterogeneous set of tanker (refueling) aircraft, located at diverse geographical locations, in the required operations associated with the deployment of a diverse fleet of military aircraft to a foreign ..."
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Cited by 6 (3 self)
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The Aerial Fleet Refueling Problem (AFRP) is concerned with the efficient and effective use of a heterogeneous set of tanker (refueling) aircraft, located at diverse geographical locations, in the required operations associated with the deployment of a diverse fleet of military aircraft to a foreign theater of activity. Typically, the “receiving ” aircraft must traverse great distances over large bodies of water and/or over other inhospitable environs where no ground based refueling resources exist. Lacking the ability to complete their flights without refueling, each receiving aircraft must be serviced one or more times during their deployment flights by means of inflight refueling provided by one of the available tanker aircraft. The receiving aircraft, aggregated into receiver groups (RGs) that fly together, have stipulated departure and destination bases and each RG’s arrival time is bounded by a stated desired earliest and latest time. The excellence of a suggested solution to this very challenging decision making problem is measured relative to a rigorously defined hierarchical multicriteria objective function. This paper describes how the AFRP for the Air Mobility Command (AMC), Scott AFB, IL is efficiently solved using Group Theoretic Tabu Search (GTTS). GTTS uses the symmetric group on n letters (Sn) and applies it to this problem using the Java TM language.
ON APPROXIMATING MULTICRITERIA TSP
 SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE
"... We present approximation algorithms for almost all variants of the multicriteria traveling salesman problem (TSP), whose performances are independent of the number k of criteria and come close to the approximation ratios obtained for TSP with a single objective function. We present randomized appro ..."
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Cited by 6 (2 self)
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We present approximation algorithms for almost all variants of the multicriteria traveling salesman problem (TSP), whose performances are independent of the number k of criteria and come close to the approximation ratios obtained for TSP with a single objective function. We present randomized approximation algorithms for multicriteria maximum traveling salesman problems (MaxTSP). For multicriteria MaxSTSP, where the edge weights have to be symmetric, we devise an algorithm that achieves an approximation ratio of 2/3 − ε. For multicriteria MaxATSP, where the edge weights may be asymmetric, we present an algorithm with an approximation ratio of 1/2 − ε. Our algorithms work for any fixed number k of objectives. To get these ratios, we introduce a decomposition technique for cycle covers. These decompositions are optimal in the sense that no decomposition can always yield more than a fraction of 2/3 and 1/2, respectively, of the weight of a cycle cover. Furthermore, we present a deterministic algorithm for bicriteria MaxSTSP that achieves an approximation ratio of 61/243 ≈ 1/4. Finally, we present a randomized approximation algorithm for the asymmetric multicriteria
Linear Time DynamicProgramming Algorithms for New Classes of Restricted TSPs: A Computational Study
, 2001
"... this paper we discuss an implementation of the dynamicprogramming algorithm for the general case when the integer k is replaced with cityspecific integers k(j), j = 1, . . . , n ..."
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Cited by 4 (0 self)
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this paper we discuss an implementation of the dynamicprogramming algorithm for the general case when the integer k is replaced with cityspecific integers k(j), j = 1, . . . , n
Time Complexity and LinearTime Approximation of the Ancient Two Machine Flow Shop
, 1997
"... We consider the scheduling problems F2 jjC max and F2 j nowait j Cmax , i.e. makespan minimization in a two machine #ow shop, with and without no wait in process. For both problems solution algorithms based on sorting with O#n log n# running time are known, where n denotes the number of jobs #Joh ..."
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Cited by 3 (0 self)
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We consider the scheduling problems F2 jjC max and F2 j nowait j Cmax , i.e. makespan minimization in a two machine #ow shop, with and without no wait in process. For both problems solution algorithms based on sorting with O#n log n# running time are known, where n denotes the number of jobs #Johnson 1954, Gilmore & Gomory 1964#. We prove that no asymptotically faster algorithms can solve these problems. This is done by establishing# n log n#lower bounds in the algebraic computation tree model of computation. Moreover, we develop for every "#0 approximation algorithms with linear running time O#n log 1 " # that deliver feasible schedules whose makespan is at most 1 + " times the optimum makespan.
Structure and Applications of Totally Decomposable Metrics
, 1997
"... this paper is the idea of a cut ..."