Giorgio Gallo

A fast parametric maximum flow algorithm and applications

The classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter, and the goal is to find the value of the parameter such that the corresponding maximum flow or minimum cut satisfies some side condition. Finding the desired parameter value requires solving a sequence of related maximum flow problems. In this paper it is shown that the recent maximum flow algorithm of Goldberg and Tarjan can be extended to solve an important class of such parametric maximum flow problems, at the cost of only a constant factor in its worst-case time bound. Faster algorithms for a variety of combinatorial optimization problems follow from the result.

Directed hypergraphs and applications

This paper deals with directed hypergraphs as a tool to model and solve some calsses of problems arising in Operations Research and in Computer Science. Concepts such as connectivity, paths and cuts are defined. An extension of the amin duality results to a special class of hypergraphs is presented. Algorithms to perform visits of hypergraphs and to find optimal paths are studied in detail. Some applications arising in propositional logic, An-Or graphs, relational data bases and transportation analysis are presented. (A)

On some matching problems arising in vehicle scheduling models

Two bipartite matching problems arising in Vehicle Scheduling are considered: the capacitated matching and the multicommodity matching. For the former, given a reasonable cost structure, we can exhibit a polynomial time algorithm, while the general case is conjectured to be NP-hard. The latter problem is shown to be NP-hard. A heuristic algorithm based on Lagrangean relaxation for the capacitated version of the multicommodity matching is also presented together with experimental results.

Network models for vehicle and crew scheduling

Abstract Network models and methods have proven to be rather successful in many application fields. In this survey their use in the solution of Vehicle Scheduling and Crew Scheduling problems in Mass Transit settings is reviewed. An attempt is made to encompass in a unified framework some of the most relevant models on the subject to be found in the literature.

Algorithms for testing the satisfiability of propositional formulae

Abstract The best-known algorithm for the satisfiability problem in the case of propositional formulae ( SAT ) is the implicit enumeration version of the Davis–Putnam algorithm, as described in Lovelands book. We review this algorithm, which we call DPL , and some recent variants. As it is often the case with enumerative algorithms for decision type problems, neither DPL nor its variants incorporate any effective device to prune the search tree. We investigate the idea of using the solution of related subproblems as such a pruning device. It is well known that HORN-SAT , the satisfiability problem in the case of Horn clauses, is easy: in fact an algorithm, DG , has been proposed by Dowling and Gallier, which solves HORN-SAT in linear time. Here we show by means of a set of experiments that the efficiency of DG is not only theoretical but practical as well. In fact we show that on a set of randomly generated problems the complexity of DG grows almost linearly with the problem size, while, for instance, the complexity of the unit resolution approach grows almost quadratically. Then we propose two relaxation schemes which map instances of SAT into instances of HORN-SAT ; such relaxation schemes are used to derive two new enumerative algorithms for SAT : HORN 1 and HORN 2. These algorithms have been compared experimentally with DPL and with its variants. Our results show that HORN 2 outperforms the other algorithms; in particular it runs several times faster than DPL on almost all the test problems we have generated.

Discrete facility location and routing of obnoxious activities

The problem of simultaneously locating obnoxious facilities and routing obnoxious materials between a set of built-up areas and the facilities is addressed.Obnoxious facilities are those facilities which cause exposure to people as well as to the environment i.e. dump sites, chemical industrial plants, electric power supplier networks, nuclear reactors and so on. A discrete combined location-routing model, which we refer to as Obnoxious Facility Location and Routing model (OFLR), is defined. OFLR is a NP-hard problem for which a Lagrangean heuristic approach is presented. The Lagrangean relaxation proposed allows to decompose OFLR into a Location subproblem and a Routing subproblem; such subproblems are then strengthened by adding suitable inequalities. Based on this Lagrangean relaxation two simple Lagrangean heuristics are provided. An effective Branch and Bound algorithm is then presented, which aims at reducing the gap between the above mentioned lower and upper bounds. Our Branch and Bound exploits the information gathered while going down in the enumeration tree in order to solve efficiently the subproblems related to other nodes. This is accomplished by using a bundle method to solve at each node the Lagrangean dual. Some variants of the proposed Branch and Bound method are defined in order to identify the best strategy for different classes of instances. A comparison of computational results relative to these variants is presented.

Bilinear programming: An exact algorithm

The Bilinear Programming Problem is a structured quadratic programming problem whose objective function is, in general, neither convex nor concave. Making use of the formal linearity of a dual formulation of the problem, we give a necessary and sufficient condition for optimality, and an algorithm to find an optimal solution.

A multi-level bottleneck assignment approach to the bus drivers' rostering problem

Abstract The problem of finding a work assignment for drivers in a given time horizon, in such a way as to have an even distribution of the workload, is considered. This problem is formulated as a Multi-level Bottleneck Assignment Problem (MBA). The MBA problem is studied: it is shown that it is NP-complete and an asymptotically optimal algorithm is presented. Some computational results are illustrated which prove the efficiency of the algorithm.

An algorithm for the min concave cost flow problem

Abstract A method is presented to solve that class of network flow problems, which may be formulated as one source - multiple destination minimum cost flow problems with concave costs. The global optimum is searched using a branch and bound procedure, in which the enumeration scheme is based on a characterization of the optimal solution set, while linear relaxations of the original problem provide lower bounds.

A combined transportation and scheduling problem

Abstract We consider a problem in which a given good has to be delivered from some origins (say production plants), to some destinations (say nodes at which the transportation mode is changed, or simply customers), during a workday, by means of a given fleet of trucks, at minimum cost. For the purpose of solution the problem is split into two levels, where, at the first level, the decision concerns the planning of trips in order to deliver goods, while, at the second level, the vehicles needed to operate the trips have to be scheduled. The solution approach presented here is based on Lagrangean Decomposition and makes use of a new algorithm for the approximate solution of the Lagrangean Dual. Computational results from a set of real life problems are presented