Roberto Cordone’s research while affiliated with University of Milan and other places

The enumeration of legal transition paths leading to a target state (or set of states) is of paramount importance in the control of discrete event systems, but is hindered by the state explosion problem. A method is proposed in this paper, in the context of Petri nets, to calculate and enumerate firing count vectors for which there exists at least an admissible transition sequence leading to a given target marking. The method is shown to improve the approach based on singular complementary transition invariants proposed by Kostin and combines an integer linear programming formulation that finds the shortest minimal solution and a branching procedure that realizes a partition of the solution set. The enumeration can be restricted to minimal solutions or extended to non-minimal ones. Moreover, the approach is extended by adding a further constraint that the target transition sequences should pass by intermediate markings (in a specific order or not). Finally, source, target and via markings can be replaced by sets of markings. Some analytical examples are discussed in detail to show the effectiveness of the proposed approach.

Fortification-interdiction games are tri-level adversarial games where two opponents act in succession to protect, disrupt and simply use an infrastructure for a specific purpose. Many such games have been formulated and tackled in the literature through specific algorithmic methods, however very few investigations exist on the completeness of such fortification problems in order to locate them rigorously in the polynomial hierarchy. We clarify the completeness status of several well-known fortification problems, such as the Tri-level Interdiction Knapsack Problem with unit fortification and attack weights, the Max-flow Interdiction Problem and Shortest Path Interdiction Problem with Fortification, the Multi-level Critical Node Problem with unit weights, as well as a well-studied electric grid defence planning problem. For all of these problems, we prove their completeness either for the $\Sigma^p_2$ or the $\Sigma^p_3$ class of the polynomial hierarchy. We also prove that the Multi-level Fortification-Interdiction Knapsack Problem with an arbitrary number of protection and interdiction rounds and unit fortification and attack weights is complete for any level of the polynomial hierarchy, therefore providing a useful basis for further attempts at proving the completeness of protection-interdiction games at any level of said hierarchy.

The Weighted Safe Set Problem requires to partition an undirected graph into two families of connected components, respectively denoted as safe and unsafe, in such a way that each safe component dominates the unsafe adjacent components with respect to a weight function. We introduce some improvements to an existing exact approach that produce a significant reduction in the effort required to find the optimum or in the gap between the optimum and the best solution obtained within a given time limit. The first improvement consists of a relaxation that is weaker than the original one, but allows to adopt a more effective branching strategy and stronger variable fixing procedures. The second one is the integration of a dedicated heuristic in the exact approach. The experimental results show a strong average reduction of the computational time and the number of branching nodes. This also mitigates the anticipated termination of the algorithm due to the exhaustion of the memory on the larg est benchmark instances.

Constructive methods are one of the main families of heuristic approaches to combinatorial optimization problems. They usually start from an empty set and subsequently add elements until a promising feasible solution is found. Their advantages are simplicity in design, analysis and implementation, and an intuitive structure. In general, though not always, they also have a limited computational complexity. An obvious counterpart is offered by destructive methods, which start from the whole ground set in which the solutions are embedded, and subsequently remove elements until they reach a promising feasible solution. This chapter describes the systematic development of a number of constructive and destructive methods for the MaxSum and the MaxMinSum diversity problems and their application, both as stand-alone approaches and to initialize an improvement metaheuristic.

The Weighted Safe Set Problem aims to determine in an undirected graph a subset of vertices such that the weights of the connected components they induce exceed the weights of the adjacent components induced by the complementary subset. We tackle the problem with various approaches based on randomised constructive and destructive procedures, comparing them with each other and with the only other existing heuristic approach, that is a randomised destructive heuristic. The experiments concern all the available benchmark instances (random graphs up to 60 vertices), but also new benchmark instances with larger size and different topologies, namely random, small-world, regular, planar, grid and real-world graphs up to 300 vertices. Dense random graphs, in fact, appear to become progressively easier to solve as their size grows. On the other instances, the best performance is obtained by combining a randomised constructive procedure, a deterministic destructive procedure, and delayed termination conditions that allow a deeper exploration of the feasible region.

The Weighted Safe Set Problem requires to partition an undirected graph into two families of connected components, respectively denoted as safe and unsafe, in such a way that each safe component dominates the unsafe adjacent components with respect to a weight function. We introduce a combinatorial branch and bound approach, whose main strength is a refined relaxation that combines graph manipulations and the solution of an auxiliary problem. We also propose fixing procedures to reduce the number of branching nodes. The algorithm solves all weighted instances available in the literature and most unweighted ones, up to 50 vertices, with computational times orders of magnitude smaller than the competing algorithms. In order to investigate the limits of the approach, we introduce a benchmark of graphs with 60 vertices, solving to optimality the denser instances.

In this paper we study the problem of partitioning a tree with n weighted vertices into p connected components. For each component, we measure its gap, that is, the difference between the maximum and the minimum weight of its vertices, with the aim of minimizing the sum of such differences. We present an O(n3p2) time and O(n3p) space algorithm for this problem. Then, we generalize it, requiring a minimum of ϵ≥1 nodes in each connected component, and provide an O(n3p2ϵ2) time and O(n3pϵ) space algorithm to solve this new problem version. We provide a refinement of our analysis involving the topology of the tree and an improvement of the algorithms for the special case in which the weights of the vertices have a heap structure. All presented algorithms can be straightforwardly extended to other similar objective functions. Actually, for the problem of minimizing the maximum gap with a minimum number of nodes in each component, we propose an algorithm which is independent of ϵ and requires O(n2lognp2) time and O(n2p) space.

We study the problem of satisfying the maximum number of distance geometry constraints with minimum experimental error. This models the determination of the shape of proteins from atomic distance data which are obtained from nuclear magnetic resonance experiments and exhibit experimental and systematic errors. Experimental errors are represented by interval constraints on Euclidean distances. Systematic errors occur from a misassignment of distances to wrong atomic pairs: we represent such errors by maximizing the number of satisfiable distance constraints. We present many mathematical programming formulations, as well as a “matheuristic” algorithm based on reformulations, relaxations, restrictions and refinement. We show that this algorithm works on protein graphs with hundreds of atoms and thousands of distances.

Graph partitioning is a widely studied problem in the literature with several applications in real life contexts. In this paper we study the problem of partitioning a graph, with weights at its vertices, into p connected components. For each component of the partition we measure the difference between the maximum and the minimum weight of a vertex in the component. We consider two objective functions to minimize, one measuring the maximum of such differences among all the components in the partition, and the other measuring the sum of the differences between the maximum and the minimum weight of a vertex in each component. We focus our analysis on tree graphs and provide polynomial time algorithms for solving these optimization problems on such graphs. In particular, we present an O(n2logn) time algorithm for the min–max version of the problem on general trees and several, more efficient polynomial algorithms for some trees with a special structure, such as spiders and caterpillars. Finally, we present NP-hardness and approximation results on general graphs for both the objective functions.