Marco Trubian

Applying tabu search to the job-shop scheduling problem

In this paper, we apply the tabu-search technique to the job-shop scheduling problem, a notoriously difficult problem in combinatorial optimization. We show that our implementation of this method dominates both a previous approach with tabu search and the other heuristics based on iterative improvements.

Channel assignment problem in cellular systems: a new model and a tabu search algorithm

The channel assignment in cellular systems has the task of planning the reuse of available frequencies in a spectrum efficient way. A classical approach to frequency assignment problems, when applied to the frequency planning of cellular networks, does not enable this task to be performed in an efficient way, since it does not consider the cumulative effect of interferers. We propose a new model for the channel assignment problem in narrow-band cellular networks, which accounts for the cumulative effect of interferers. In this model, the service area is partitioned into regions and the propagation characteristics are assigned by means of the levels received in each region by the considered base stations (BSs). The objective is to maximize the sum of traffic loads offered by regions in which the ratio between the received power and the sum of powers received from interfering transmissions is above a threshold value. We also present an algorithm, based on tabu search (TS) techniques, to solve this problem. This algorithm has been tested on some instances obtained by using a simple radio channel model and on a real world instance.

Timed high-level nets

Petri nets have been widely used for modeling and analyzing concurrent systems. Several reasons contribute to their success: the simplicity of the model, the immediate graphical representation, the easy modeling of asynchronous aspects, the possibility of reasoning about important properties such as reachability, liveness, boundedness. However, the original model fails in representing two important features: complex functional aspects, such as conditions which rule the flow of control, and time. Due to that, two different classes of extensions of Petri nets have been proposed: high-level nets and timed Petri nets. High-level nets allow the representation of functional aspects in full details, but do not provide a means for representing time; on the other hand, timed Petri nets have been thought for time representation, but they do not provide a means for representing detailed functinal aspects. Thus, these two important aspects cannot be mastered together. In particular, it is difficult to express relationships between time and functional aspects.This paper investigates the relationships between high-level nets and timed Petri nets, thus extending a first set of results published in a previous paper, where a unifying Petri net based model for time representation has been proposed. It first recalls how time can be represented in a Petri net extension called ER nets, and assesses its generality. It then investigates the relationships of ER nets with the best known high-level nets. In particular it shows the overall equivalence of ER nets, Colored Petri nets and Predicate/Transition nets, and extends the mechanism for time representation introduced in ER nets to both Colored Petri nets and Predicate/Transition nets. It also shows that these models cannot be simplified without significantly constraining the timing aspects that can be modeled.

Solution of large weighted equicut problems

Given a weighted undirected graph, the equicut problem consists of finding a partition of the vertex set into two subsets of equal cardinality such that the sum of the weights of the edges belonging to the cut defined by the partition is minimized. The problem is NP-hard and has several practical applications. In recent years a number of algorithms based on metaheuristic techniques have been proposed. In this work we first present a survey of the algorithms from the literature, then we propose a new tabu search algorithm and compare it with the other heuristics through extensive computational experiments on several classes of graphs with up to 4000 nodes and 320 000 edges. The results show that our approach easily determines the optimal solution for small graphs and its average performances are greatly superior to those of the other approximating algorithms.

Solving the feedback vertex set problem on undirected graphs

Abstract Feedback vertex problems consist of removing a minimal number of vertices of a directed or undirected graph in order to make it acyclic. The problem is known to be NP -complete. In this paper we consider the variant on undirected graphs. The polyhedral structure of the feedback vertex set polytope is studied. We prove that this polytope is full dimensional and show that some inequalities are facet defining. We describe a new large class of valid constraints, the subset inequalities. A branch-and-cut algorithm for the exact solution of the problem is then outlined, and separation algorithms for the inequalities studied in the paper are proposed. A local search heuristic is described next. Finally, we create a library of 1400 randomly generated instances with the geometric structure suggested by the applications, and we computationally compare the two algorithmic approaches on our library.

Data-dependent bounds for the general and the asymmetric Stacker-Crane problems

The Stacker-Crane Problem (SCP) is a sequencing problem, arising in scheduling and transportation, that consists of finding the minimum cost cycle on a mixed graph with oriented arcs and unoriented edges: feasible solutions must traverse all the arcs. Approximation algorithms are known to provide a fixed worst-case bound if the triangle inequality holds. We consider the worst-case performance of approximation algorithms for the SCP when the triangle inequality can be violated (General SCP) and for a similar problem formulated on a complete digraph (Asymmetric SCP).

New bounds for optimum traffic assignment in satellite communication

Abstract The use of satellites to exchange information between distant places on the earth is a well assessed technique, but each satellite has high cost, so it is important to utilize it efficiently. A satellite receives requests of transmission between pairs of earth stations, which can be represented by a traffic matrix. One of the methods used for managing these communications is the satellite switched time-division-multiple-access system which requires to partition the traffic matrix in submatrices and transmits the information of each submatrix in a single time slot. Therefore a crucial decision for an efficient use of the satellite is how to partition the given traffic matrix so that the total time required to transmit the information is minimized. In the literature this problem has been addressed mainly in the special case in which the number of possible contemporary transmissions is equal to the number of rows and columns in the traffic matrix. In this paper we consider the general case in which the size of the matrix is larger than the number of contemporary transmissions. Our scope is to provide effective heuristic algorithms for finding good approximating solutions and to give tight lower bounds on the optimal solution value, in order to evaluate the quality of the heuristics. In this paper we assume that a satellite has l receiving and transmitting antennas, and we are given a traffic matrix D to be transmitted by interconnecting pairs of receiving–transmitting antennas, through an on board switch. We also assume that l is strictly smaller than the number of rows and columns of D, that no preemption of the communications is allowed, and that changing the configuration of the switch requires a negligible time. We ask for a set of switch configurations that minimizes the total time occurring for transmitting the entire traffic matrix. We present some new lower bounds on the optimum solution value and a new technique to combine bounds which obtains a dominating value. We then present five heuristics: the first two are obtained modifying algorithms from the literature; two others are obtained with standard techniques; the last algorithm is an implementation of a new and promising tabu search method which is called exploring tabu search. Extensive computational experiments compare the performances of the heuristics and that of the lower bound, on randomly generated instances.

A worst-case analysis of two approximate algorithms for the asymmetric travelling salesman problem

Abstract Two algorithms are presented for the asymmetric travelling salesman problem. The worst case performance of both of them is bounded by quantities which are independent on the size of the problem (number of vertices) but depend on the weights associated to the arcs. The second algorithm improves a bound previously obtained under identical hypothesis.

A note on the approximation of the asymmetric traveling salesman problem

Abstract We show that some asymmetric traveling salesman problem (ATSP) instances are approximable within bounds equal to 3 and 9/5, when they satisfy sufficient conditions more restrictive than the triangle inequality, very simple to test and nicely structured: they only depend on a measure of satisfaction of the triangle inequality and a measure of the graph asymmetry. We discuss the applicability of such conditions and we present two preprocessing linear programs to reformulate ATSP instances into equivalent ones achieving data-dependent bounds by the same approximation algorithms.

A branch and price algorithm for the SS/TDMA problem with cardinality constraint

In this paper, we consider a square nxn traffic matrix D and a satellite having l@?n receiving and transmitting antennas. Transmissions are allowed by interconnecting pairs of receiving-transmitting antennas, through an on-board switch. We also assume that no preemption of the communications is allowed, and that changing the configuration of the switch requires a negligible time. We ask for a set of switch configurations that minimizes the total time occurring for transmitting the entire traffic matrix. In the literature, an exact method has been already proposed for the SS/TDMA optimization problem without cardinality constraint (i.e. for instances having l=n), but only heuristics have been proposed for the general case. In this paper, from a known MILP formulation we derive an extended formulation, and we use a column generation method to obtain its LP-relaxation. We address the derived pricing problem through a polynomial time algorithm based on solvers for the k-cardinality assignment problem. We insert this lower bound method in a two-phase branch and price algorithm in order to obtain the optimal solutions. To compare our new exact method with the one proposed in literature to solve the SS/TDMA optimization problem without cardinality constraint, we extend this older method to consider the cardinality constraint. The final computational experiments, comparing our new method with other already known methods, show the efficiency of our algorithms. Scope and purpose: Satellites are the commonly used devices to wirelessly connect two distant earth points. Since satellites are very expansive devices, it is important to schedule the communications requests in an efficient manner. A satellite receives transmissions requests between pairs of earth stations, and these requests can be described by a traffic matrix. One method used in practice to satisfy these requests is called Satellite-Switched Time-Division-Multiple-Access (SS/TDMA) which requires to partition the traffic matrix entries in sets called frames and to transmit the elements of each different frame in a single time slot. Each frame must contain at most one communication involving the same earth station and the number of its nonzero entries must be at most equal to the number of satellite antennas. Since a satellite has to complete the transmission of a frame before starting the transmission of the next one, each frame is associated with a completion time equal to the maximum transmission time of its entries. A crucial decision, to minimize the total satellite used time, is how to partition the given traffic matrix. Our aim is to provide an effective exact algorithm for finding optimal solutions to the SS/TDMA problem and to compare it with other methods already known in the literature.