Gintaras Palubeckis

Multistart Tabu Search Strategies for the Unconstrained Binary Quadratic Optimization Problem

This paper describes and experimentally compares five rather different multistart tabu search strategies for the unconstrained binary quadratic optimization problem: a random restart procedure, an application of a deterministic heuristic to specially constructed subproblems, an application of a randomized procedure to the full problem, a constructive procedure using tabu search adaptive memory, and an approach based on solving perturbed problems. In the solution improvement phase a modification of a standard tabu search implementation is used. A computational trick applied to this modification – mapping of the current solution to the zero vector – allowed to significantly reduce the time complexity of the search. Computational results are provided for the 25 largest problem instances from the OR-Library and, in addition, for the 18 randomly generated larger and more dense problems. For 9 instances from the OR-Library new best solutions were found.

Iterated tabu search for the maximum diversity problem

In this paper, we deal with the maximum diversity problem (MDP), which asks to select a specified number of elements from a given set so that the sum of distances between the selected elements is as large as possible. We develop an iterated tabu search (ITS) algorithm for solving this problem. We also present a steepest ascent algorithm, which is well suited in application settings where solutions of satisfactory quality are required to be provided very quickly. Computational results for problem instances involving up to 5000 elements show that the ITS algorithm is a very attractive alternative to the existing approaches. In particular, we demonstrate the outstanding performance of ITS on the MDP instances taken from the literature. For 69 such instances, new best solutions were found.

A heuristic-based branch and bound algorithm for unconstrained quadratic zero-one programming

In this paper we describe a branch and bound algorithm for solving the unconstrained quadratic 0–1 programming problem. The salient features of it are the use of quadratic programming heuristics in the transformation of subproblems and exploiting some classes of facets of the polytope related to the quadratic problem in deriving upper bounds on the objective function. We develop facet selection procedures that form a basis of the bound computation algorithm. We present computational experience on four series of randomly generated problems and 14 real instances of a quadratic problem arising in design automation. We remark that the same ideas can also be applied to some other combinatorial optimization problems.ZusammenfassungIn diesem Artikel beschreiben wir einen “Branch and Bound”-Algorithmus zur Lösung von quadratischen Optimierungsaufgaben in 0–1 Variablen und ohne Restriktionen. Das Verfahren verwendet Heuristiken zur Transformation von Teilproblemen. Zur Bestimmung oberer Schranken für die Zielfunktion werden gewiesse Klassen von Facetten des zugehörigen Polyeders verwendet. Weiters werden Auswahlalgorithmen für Facetten angegeben, die die Grundlage der Schrankenberechnungen bilden. Es werden Rechenergebnisse für vier zufällig generierte Aufgabenserien vorgestellt, wie auch von 14 realen Anwendungen aus dem Bereich der automatisierten Projektierung. Es wird betont, daß dieselben Ideen auch auf andere kombinatorische Optimierungsprobleme angewandt werden können.

Fast simulated annealing for single-row equidistant facility layout

Given n facilities and a flow matrix, the single-row equidistant facility layout problem (SREFLP) is to find a one-to-one assignment of n facilities to n locations equally spaced along a straight line so as to minimize the sum of the products of the flows and distances between facilities. We develop a simulated annealing (SA) algorithm for solving this problem. The algorithm provides a possibility to employ either merely pairwise interchanges of facilities or merely insertion moves or both of them. It incorporates an innovative method for computing gains of both types of moves. Experimental analysis shows that this method is significantly faster than traditional approaches. To further speed up SA, we propose a two-mode technique when for high temperatures, at each iteration, only the required gain is calculated and, for lower temperatures, the gains of all possible moves are maintained from iteration to iteration. We experimentally compare SA against the iterated tabu search (ITS) algorithm from the literature. Computational results are reported for SREFLP instances with up to 300 facilities. The results show that the performance of our SA implementation is dramatically better than that of the ITS heuristic.

A branch-and-bound algorithm for the single-row equidistant facility layout problem

In this paper, we deal with the single-row equidistant facility layout problem (SREFLP), which asks to find a one-to-one assignment of n facilities to n locations equally spaced along a straight line so as to minimize the sum of the products of the flows and distances between facilities. We develop a branch-and-bound algorithm for solving this problem. The lower bound is computed first by performing transformation of the flow matrix and then applying the well-known Gilmore–Lawler bounding technique. The algorithm also incorporates a dominance test which allows to drastically reduce redundancy in the search process. The test is based on the use of a tabu search procedure designed to solve the SREFLP. We provide computational results for problem instances of size up to 35 facilities. For a number of instances, the optimal value of the objective function appeared to be smaller than the best value reported in the literature.

Fast local search for single row facility layout

Given n facilities of prescribed lengths and a flow matrix, the single row facility layout problem (SRFLP) is to arrange the facilities along a straight line so as to minimize the total arrangement cost, which is the sum of the products of the flows and center-to-center distances between facilities. We propose interchange and insertion neighborhood exploration (NE) procedures with time complexity O(n2), which is an improvement over O(n3)-time NE procedures from the literature. Numerical results show that, for large SRFLP instances, our insertion-based local search (LS) algorithm is two orders of magnitude faster than the best existing LS techniques. As a case study, we embed this LS algorithm into the variable neighborhood search (VNS) framework. We report computational results for SRFLP instances of size up to 300 facilities. They indicate that our VNS implementation offers markedly better performance than the variant of VNS that uses a recently proposed O(n3)-time insertion-based NE procedure.

An Algorithm for Construction of Test Cases for the Quadratic Assignment Problem

In this paper we present an algorithm for generating quadratic assignment problem (QAP) instances with known provably optimal solution. The flow matrix of such instances is constructed from the matrices corresponding to special graphs whose size may reach the dimension of the problem. In this respect, the algorithm generalizes some existing algorithms based on the iterative selection of triangles only. The set of instances which can be produced by the algorithm is NP-hard. Using multi-start descent heuristic for the QAP, we compare experimentally such test cases against those created by several existing generators and against Nugent-type problems from the QAPLIB as well.

Single row facility layout using multi-start simulated annealing

The single row facility layout problem (SRFLP) is considered.A fast multi-start simulated annealing algorithm is proposed.Both interchange and insertion move gains are computed in linear time.A technique to accelerate computation for lower temperatures is applied.Numerical results are reported for SRFLP instances of size up to 1000 facilities. In the single row facility layout problem (SRFLP), we are given a set of n facilities, their lengths, and a flow cost matrix. The problem asks to arrange the facilities along a straight line so as to minimize the sum of the products of the flow costs and center-to-center distances between facilities. We develop a multi-start simulated annealing (MSA) algorithm for solving this problem. The algorithm employs two move types, pairwise interchanges of facilities and insertions. We propose O ( n ) -time procedures for computing gains of both types of moves. They make our algorithm very fast compared to the traditional approaches, which are based on a straightforward procedure for obtaining the move gain in O ( n 2 ) time. When the temperature during the cooling process drops to a certain level, the computation is accelerated with the use of an additional technique which is reminiscent of a local search algorithm (it takes O ( 1 ) time to compute the move gain and O ( n 2 ) time to perform the required calculations when a move is accepted). We report computational results for SRFLP instances of size up to 1000 facilities. The results demonstrate the superiority of the MSA algorithm over the state-of-the-art methods. The source code implementing MSA is made publicly available as a benchmark for future comparisons.

On the recursive largest first algorithm for graph colouring

Abstract We present a reduction from the problem of colouring the vertices of the graph to the maximum independent set problem (MISP) showing that for an n-vertex graph G the sum of the chromatic number of G and the independence number of a graph derived from the complement of G is equal to n. We investigate the applicability of the MISP greedy algorithm to the graph colouring problem. We prove that the well-known Leighton RLF algorithm and an algorithm obtained by modifying the MISP greedy heuristic behave in exactly the same way.

A new bounding procedure and an improved exact algorithm for the Max-2-SAT problem

Given a CNF (conjunctive normal form) Boolean expression with clauses of size at most two, the Max-2-SAT problem asks to find a truth assignment to the Boolean variables that makes true the maximum number of clauses. In this paper, we describe an innovative upper bound computation procedure which is centered around the use of equations and inequalities that are satisfied by all solutions to the problem. The procedure is incorporated in a branch-and-bound algorithm for Max-2-SAT. An initial solution to the problem is provided by an iterated tabu search heuristic. We present computational experience on the Max-2-SAT benchmark instances from the Max-SAT Evaluation 2007. The results show that the developed branch-and-bound algorithm is very competitive with the best previously reported techniques.