Alberto Caprara

A Heuristic Method for the Set Covering Problem

We present a Lagrangian-based heuristic for the well-known Set Covering Problem (SCP). The algorithm was initially designed for solving very large scale SCP instances, involving up to 5,000 rows and 1,000,000 columns, arising from crew scheduling in the Italian Railway Company, Ferrovie dello Stato SpA. In 1994 Ferrovie dello Stato SpA, jointly with the Italian Operational Research Society, organized a competition, called FASTER, intended to promote the development of algorithms capable of producing good solutions for these instances, since the classical approaches meet with considerable difficulties in tackling them. The main characteristics of the algorithm we propose are (1) a dynamic pricing scheme for the variables, akin to that used for solving large-scale LPs, to be coupled with subgradient optimization and greedy algorithms, and (2) the systematic use of column fixing to obtain improved solutions. Moreover, we propose a number of improvements on the standard way of defining the step-size and the ascent direction within the subgradient optimization procedure, and the scores within the greedy algorithms. Finally, an effective refining procedure is proposed. Our code won the first prize in the FASTER competition, giving the best solution value for all the proposed instances. The algorithm was also tested on the test instances from the literature: in 92 out of the 94 instances in our test bed we found, within short computing time, the optimal (or the best known) solution. Moreover, among the 18 instances for which the optimum is not known, in 6 cases our solution is better than any other solution found by previous techniques.

Modeling and Solving the Train Timetabling Problem

The train timetabling problem aims at determining a periodic timetable for a set of trains that does not violate track capacities and satisfies some operational constraints. In particular, we concentrate on the problem of a single, one-way track linking two major stations, with a number of intermediate stations in between. Each train connects two given stations along the track (possibly different from the two major stations) and may have to stop for a minimum time in some of the intermediate stations. Trains can overtake each other only in correspondence of an intermediate station, and a minimum time interval between two consecutive departures and arrivals of trains in each station is specified.In this paper, we propose a graph theoretic formulation for the problem using a directed multigraph in which nodes correspond to departures/arrivals at a certain station at a given time instant. This formulation is used to derive an integer linear programming model that is relaxed in a Lagrangian way. A novel feature of our model is that the variables in the relaxed constraints are associated only with nodes (as opposed to arcs) of the aforementioned graph. This allows a considerable speed-up in the solution of the relaxation. The relaxation is embedded within a heuristic algorithm which makes extensive use of the dual information associated with the Lagrangian multipliers. We report extensive computational results on real-world instances provided from Ferrovie dello Stato SpA, the Italian railway company, and from Ansaldo Segnalamento Ferroviario SpA.

Algorithms for the Set Covering Problem

The Set Covering Problem (SCP) is a main model for several important applications, including crew scheduling in railway and mass-transit companies. In this survey, we focus our attention on the most recent and effective algorithms for SCP, considering both heuristic and exact approaches, outlining their main characteristics and presenting an experimental comparison on the test-bed instances of Beasleys OR Library.

Sorting by reversals is difficult

We prove that the problem of sorting a permutation by the minimum number of reversals is NP-hard, thus answering a major question on the complexity of a problem which has widely been studied in the last years. The proof is based on the strong relationship between this problem and the problem of finding the maximum number of edge-disjoint alternating cycles in a suitably-defined bicolored graph. For this latter problem we derive a number of structural properties, that can be used for showing its NP-hardness.

Passenger Railway Optimization

Publisher Summary Railway transportation can be split into passenger transportation and cargo transportation. This chapter discusses the European situation, where the major part of railway transportation consists of passenger transportation without addressing important problems in cargo transportation—such as car blocking, train makeup, train routing, and empty car distribution. The chapter describes several mathematical models and optimization techniques that have been developed for effectively supporting traditional planning processes in passenger railway transportation. A lot of research has been carried out in this area, both of a practical and theoretical nature. The results of this research are starting to be applied in practice. Real-time control is at the other side of the planning spectrum. The current trend in the railway industry is a shift from “planning in detail” to “effective real-time control.” Disturbances and disruptions in the railway operations are inevitable. Therefore, large parts of the operational plans are never carried out.

Algorithms for railway crew management

Crew management is concerned with building the work schedules of crews needed to cover a planned timetable. This is a well-known problem in Operations Research and has been historically associated with airlines and mass-transit companies. More recently, railway applications have also come on the scene, especially in Europe. In practice, the overall crew management problem is decomposed into two subproblems, called crew scheduling and crew rostering. In this paper, we give an outline of different ways of modeling the two subproblems and possible solution methods. Two main solution approaches are illustrated for real-world applications. In particular we discuss in some detail the solution techniques currently adopted at the Italian railway company, Ferrovie dello Stato SpA, for solving crew scheduling and rostering problems.

Sorting Permutations by Reversals and Eulerian Cycle Decompositions

We analyze the strong relationship among three combinatorial problems, namely, the problem of sorting a permutation by the minimum number of reversals (MIN-SBR), the problem of finding the maximum number of edge-disjoint alternating cycles in a breakpoint graph associated with a given permutation (MAX-ACD), and the problem of partitioning the edge set of an Eulerian graph into the maximum number of cycles (MAX-ECD). We first illustrate a nice characterization of breakpoint graphs, which leads to a linear-time algorithm for their recognition. This characterization is used to prove that MAX-ECD and MAX-ACD are equivalent, showing the latter to be NP-hard. We then describe a transformation from MAX-ACD to MIN-SBR, which is therefore shown to be NP-hard as well, answering an outstanding question which has been open for some years. Finally, we derive the worst-case performance of a well-known lower bound for MIN-SBR, obtained by solving MAX-ACD, discussing its implications on approximation algorithms for MIN-SBR.

A Lagrangian heuristic algorithm for a real-world train timetabling problem

The train timetabling problem (TTP) aims at determining an optimal timetable for a set of trains which does not violate track capacities and satisfies some operational constraints.In this paper, we describe the design of a train timetabling system that takes into account several additional constraints that arise in real-world applications. In particular, we address the following issues: • Manual block signaling for managing a train on a track segment between two consecutive stations. • Station capacities, i.e., maximum number of trains that can be present in a station at the same time. • Prescribed timetable for a subset of the trains, which is imposed when some of the trains are already scheduled on the railway line and additional trains are to be inserted. • Maintenance operations that keep a track segment occupied for a given period. We show how to incorporate these additional constraints into a mathematical model for a basic version of the problem, and into the resulting Lagrangian heuristic. Computational results on real-world instances from Rete Ferroviaria Italiana (RFI), the Italian railway infrastructure management company, are presented.

1001 optimal PDB structure alignments: integer programming methods for finding the maximum contact map overlap.

Protein structure comparison is a fundamental problem for structural genomics, with applications to drug design, fold prediction, protein clustering, and evolutionary studies. Despite its importance, there are very few rigorous methods and widely accepted similarity measures known for this problem. In this paper we describe the last few years of developments on the study of an emerging measure, the contact map overlap (CMO), for protein structure comparison. A contact map is a list of pairs of residues which lie in three-dimensional proximity in the proteins native fold. Although this measure is in principle computationally hard to optimize, we show how it can in fact be computed with great accuracy for related proteins by integer linear programming techniques. These methods have the advantage of providing certificates of near-optimality by means of upper bounds to the optimal alignment value. We also illustrate effective heuristics, such as local search and genetic algorithms. We were able to obtain for the first time optimal alignments for large similar proteins (about 1,000 residues and 2,000 contacts) and used the CMO measure to cluster proteins in families. The clusters obtained were compared to SCOP classification in order to validate the measure. Extensive computational experiments showed that alignments which are off by at most 10% from the optimal value can be computed in a short time. Further experiments showed how this measure reacts to the choice of the threshold defining a contact and how to choose this threshold in a sensible way.

Exact Solution of the Quadratic Knapsack Problem

The Quadratic Knapsack Problem (QKP) calls for maximizing a quadratic objective function subject to a knapsack constraint, where all coefficients are assumed to be nonnegative and all variables are binary. The problem has applications in location and hydrology, and generalizes the problem of checking whether a graph contains a clique of a given size. We propose an exact branch-and-bound algorithm for QKP, where upper bounds are computed by considering a Lagrangian relaxation that is solvable through a number of (continuous) knapsack problems. Suboptimal Lagrangian multipliers are derived by using subgradient optimization and provide a convenient reformulation of the problem. We also discuss the relationship between our relaxation and other relaxations presented in the literature. Heuristics, reductions, and branching schemes are finally described. In particular, the processing of each node of the branching tree is quite fast: We do not update the Lagrangian multipliers, and use suitable data structures to compute an upper bound in linear expected time in the number of variables. We report exact solution of instances with up to 400 binary variables, i.e., significantly larger than those solvable by the previous approaches. The key point of this improvement is that the upper bounds we obtain are typically within 1% of the optimum, but can still be derived effectively. We also show that our algorithm is capable of solving reasonable-size Max Clique instances from the literature.