Daniele Pretolani

A directed hypergraph model for random time dependent shortest paths

Abstract We consider routing problems in dynamic networks where arc travel times are both random and time dependent. The problem of finding the best route to a fixed destination is formulated in terms of shortest hyperpaths on a suitable time-expanded directed hypergraph. The latter problem can be solved in linear time, with respect to the size of the hypergraph, for several definitions of hyperpath length. Different criteria for ranking routes can be modeled by suitable definitions of hyperpath length. We also show that the problem becomes intractable if a constraint on the route structure is imposed.

Finding the K shortest hyperpaths

The K shortest paths problem has been extensively studied for many years. Efficient methods have been devised, and many practical applications are known. Shortest hyperpath models have been proposed for several problems in different areas, for example in relation with routing in dynamic networks. However, the K shortest hyperpaths problem has not yet been investigated. In this paper we present procedures for finding the K shortest hyperpaths in a directed hypergraph. This is done by extending existing algorithms for K shortest loopless paths. Computational experiments on the proposed procedures are performed, and applications in transportation, planning and combinatorial optimization are discussed.

Max Horn SAT and the minimum cut problem in directed hypergraphs

In this paper we consider the Maximum Horn Satisfiability problem, which is reduced to the problem of finding a minimum cardinality cut on a directed hypergraph. For the latter problem, we propose different IP formulations, related to three different definitions of hyperpath weight. We investigate the properties of their linear relaxations, showing that they define a hierarchy. The weakest relaxation is shown to be equivalent to the relaxation of a well known IP formulation of Max Horn SAT, and to a max-flow problem on hypergraphs. The tightest relaxation, which is a disjunctive programming problem, is shown to have integer optimum. The intermediate relaxation consists in a set covering problem with a possible exponential number of constraints. This latter relaxation provides an approximation of the convex hull of the integer solutions which, as proven by the experimental results given, is much tighter than the one known in the literature.

Efficient labelling algorithms for the maximum noncrossing matching problem

Abstract Consider a bipartite graph; lets suppose we draw the origin nodes and the destination nodes arranged in two columns, and the edges as straight-line segments. A noncrossing matching is a subset of edges such that no two of them intersect. Several algorithms for the problem of finding the noncrossing matching of maximum cardinality are proposed. Moreover an extension to weighted graphs is considered.

Finding the K shortest hyperpaths using reoptimization

We present some reoptimization techniques for computing (shortest) hyperpath weights in a directed hypergraph. These techniques are exploited to improve the worst-case computational complexity (as well as the practical performance) of an algorithm finding the K shortest hyperpaths in acyclic hypergraphs.

An aggregate label setting policy for the multi-objective shortest path problem

We consider label setting algorithms for the multi-objective shortest path problem with any number of sum and bottleneck objectives. We propose a weighted sum aggregate ordering of the labels, specifically tailored to combine sum and bottleneck objectives. We show that the aggregate order leads to a consistent reduction of solution times (up to two-thirds) with respect to the classical lexicographic order.

Easy Cases of Probabilistic Satisfiability

The Probabilistic Satisfiability problem (PSAT) can be considered as a probabilistic counterpart of the classical SAT problem. In a PSAT instance, each clause in a CNF formula is assigned a probability of being true; the problem consists in checking the consistency of the assigned probabilities. Actually, PSAT turns out to be computationally much harder than SAT, e.g., it remains difficult for some classes of formulas where SAT can be solved in polynomial time. A column generation approach has been proposed in the literature, where the pricing sub-problem reduces to a Weighted Max-SAT problem on the original formula. Here we consider some easy cases of PSAT, where it is possible to give a compact representation of the set of consistent probability assignments. We follow two different approaches, based on two different representations of CNF formulas. First we consider a representation based on directed hypergraphs. By extending a well-known integer programming formulation of SAT and Max-SAT, we solve the case in which the hypergraph does not contain cycles; a linear time algorithm is provided for this case. Then we consider the co-occurrence graph associated with a formula. We provide a solution method for the case in which the co-occurrence graph is a partial 2-tree, and we show how to extend this result to partial k-trees with k>2.

Lower bounds for the quadratic semi-assignment problem

Abstract This paper presents a class of lower bounds for the Quadratic Semi-Assignment Problem (QSAP). These bounds are based on recent results on polynomially solvable cases, in particular we will consider the QSAPs whose quadratic cost coefficients define a reducible graph . The idea is to decompose the problem into several subproblems, each defined on a reducible subgraph. Lagrangean decomposition technique is used to improve the results. Several lower bounds are computationally compared on several types of test problems.

Shortest paths in piecewise continuous time-dependent networks

We consider a shortest path problem, where the travel times on the arcs may vary with time and waiting at any node is allowed. Simple adaptations of the Dijkstra algorithm may fail to solve the problem, when discontinuities exist. We propose a new Dijkstra-like algorithm that overcomes these difficulties.

ON SOME PATH PROBLEMS ON ORIENTED HYPERGRAPHS.

The BF-graphs form a particular class of Directed Hypergraphs. For this important family, different applications are known in data bases and artificial intelligence domain. They may also be used to describe the behavior of concurrent systems. We present here a theoretical analysis of several hyperpath problems in BF-graphs, with emphasis on the acyclic BF-graphs. After briefly exposing the basic concepts of directed hypergraphs, we presesnt an algorithm for finding a BF-path. We next discuss the problem of finding a hyperpath cover, and present a polynomial solution for two constrained hyperpath problems.