The possible existence of a regular Moore graph of diameter 2 and degree 57 with the maximum number 3250 of vertices has been an open question for over 65 years. One approach to a construction focuses on the set of permutations that describe the 1-factors that give the adjacencies between leaf vertices of pairs of branches of a tree. Most of these permutations are derangements, that is they are permutations with no fixed points. As many products of 2, 3, or 4 of these derangements must also be derangements, it is tempting to use a group of derangements, that is a group of permutations in which every non-identity element is a derangement. The first case to consider is when the group of derangements is a cyclic group of permutations. In this paper it is proved that a construction using only a cyclic group of permutations is impossible. This leaves only the possibility of using some other group of derangements, or a set of derangements that do not form a group. The prospects for extending the work to these cases is considered at the end of the paper.

A stochastic version of the 0-1 Knapsack Problem recently introduced in the literature and named the 0-1 Time-Bomb Knapsack Problem is the topic of the present work. In this problem, in addition to profit and weight, each item is characterized by a probability of exploding, and therefore destroying all the contents of the knapsack, incase it is loaded. The optimization aims at maximizing the expected profit of the selected items, which takes into account also the probabilities of explosion, while fulfilling the capacity constraint. The problem has real-world applications in logistics and cloud computing. In this work, two model-based algorithms are introduced. They are based on partial linearizations of a non-linear model describing the problem. Extensive computational results on the instances available in the literature are presented to position the new methods as the best-performing ones, while comparing against those previously proposed.

A variant of the well-known Set Covering Problem is studied in this paper, where subsets of a collection have to be selected, and pairwise conflicts among subsets of items exist. The selection of each subset has a cost, and the inclusion of conflicting subsets is associated with a penalty to be paid. The problem, which can be used to model real applications, looks for a selection of subsets that cover the original collection, while minimizing the sum of covering and penalty costs. In this paper we consider a compact mixed integer linear program and we solve it with an open-source solver. Computational results on the benchmark instances commonly used in the literature of the problem are reported. The results indicate that the new approach we propose is capable of good results, both in terms of lower and upper bounds, although not matching the state-of-the-art on average. The new approach was, however, able to improve 9 best-known heuristic solutions.

The hazardous orienteering problem is the topic of this study. It is a variant of the well-studied orienteering problem, where a vehicle, given a maximum mission time, has to select and visit customers out of a set of requests, aiming at maximizing the total profit associated with the customers selected. In the hazardous version of the problem we consider, the customers are associated with parcels that have to be collected, and some of them might explode during the transportation, after having been picked up. The probability of explosion depends on the characteristics of the parcel and on the time spent by the dangerous parcel itself on the vehicle. If an explosion happens, the collected profit is totally lost. The target becomes then to select the tour that maximizes the expected profit, taking into account the probability of catastrophic events happening. In this article, we propose an exact solving approach based on a mixed integer linear programming model which is dynamically modified by adding new constraints. The computational results substantially improve the state-of-the-art for the problem.

The Minimum Spanning Tree with Conflicting Edge Pairs is a generalization that adds conflict constraints to a classical optimization problem on graphs used to model several real-world applications. In recent years, several heuristic and exact approaches have been proposed to tackle this problem. In this paper, we present a mixed-integer linear program not previously applied to this problem, and we solve it with an open-source solver. Computational results for the benchmark instances commonly adopted in the literature of the problem are reported. The results indicate that the approach we propose obtains results aligned with those of the much more sophisticated approaches available, notwithstanding it being much simpler to implement. During the experimental campaign, six instances were closed for the first time, with nine improved best-known lower bounds and sixteen improved best-known upper bounds over a total of two hundred thirty instances considered.

A variant of the well-known Shortest Path Problem is studied in this paper, where pairs of conflicting arcs are provided, and for each conflicting pair a penalty is paid once neither or both of the arcs are selected. This configures a set of soft-constraints. The problem, which can be used to model real applications, looks for a path from a given origin to a given destination that minimizes the cost of the arcs traversed plus the penalties incurred. In this paper, we consider a compact mixed integer linear program representing the problem and we solve it with the open-source solver CP-SAT, part of the Google OR-Tools computational suite. An experimental campaign on the instances available from the literature indicates that the approach we propose achieves results comparable with those of state-of-the-art solvers, notwithstanding it is a compact model, while the other approaches require the generation of dynamic constraints in order for the models to be competitive. Some best-known results have been improved in this study, and some instances have been closed for the first time.

A variant of the well-known Knapsack Problem is studied in this paper. In the classic problem, a set of items is given, with each item characterized by a weight and a profit. A knapsack of a given capacity is provided, and the problem consists of selecting a subset of items such that the total weight does not exceed the capacity of the knapsack, while the total profit is maximized. In the variation considered in the present work, pairs of items are conflicting, and cannot be selected at the same time. The resulting problem, which can be used to model several real applications, is considerably harder to approach than the classic one. In this paper, we consider a mixed-integer linear program representing the problem and we solve it with a state-of-the-art black-box software. A vast experimental procedure on the instances available from the literature, and adopted in the last decade by the community, indicates that the approach we propose achieves results comparable with, and in many cases better than, those of state-of-the-art methods, notwithstanding that the latter are typically based on more complex and problem-specific ideas and algorithms than the idea we propose.

Three optimization problems based on node-colored undirected graphs are the subject of the present study. These problems model real-world applications in several domains, such as cybersecurity, bioinformatics, and social networks, although they have a similar abstract representation. In all of the problems, the goal is to partition the graph into colorful connected components, which means that in each of the connected components, a color can appear in at most one node. The problems are optimized according to different objective functions, leading to different optimal partitions. We propose a compact Mixed Integer Linear Programming formulation for each of the three problems. These models are based on spanning trees, represented through multi-commodity flows. The compact nature of the new linear models is easier to handle than the approaches that previously appeared in the literature. These were based on models with an exponential number of constraints, which, therefore, required complex solving techniques based on the dynamic generation of constraints within a branch-and-cut framework. Computational experiments carried out on the standard benchmark instances for the problems show the potential of the new compact methods, which, once fed into modern state-of-the-art solvers, are able to obtain results better than the previous algorithmic approaches. As an outcome of the experimental campaign, a dozen instances of the different problems considered are closed for the first time.

The Maximum Flow Problem with Conflict Constraints is a generalization that adds conflict constraints to a classical optimization problem on networks used to model several real-world applications. In the last few years several approaches, both heuristic and exact, have been proposed to attack the problem. In this paper we consider a mixed integer linear program and solve it with an open-source solver. Computational results on the benchmark instances commonly used in the literature of the problem are reported. All the 160 instances benchmark instances normally used in the literature are solved to optimality for the first time, with 28 instances closed for the first time. Moreover, in the process, 6 improvements to the best-known heuristic solutions are also found.

The Orienteering Problem is a routing problem aiming at selecting a subset of a given set of customers to be visited within a given time budget, so that a total revenue is maximized. Multiple variants of the problem have been studied. The Probabilistic Orienteering Problem is one of these variants, where customers will require a visit according to a certain given probability. Stochasticity makes the model more practical, but concurrently more difficult to solve. Effective approaches to solve the problem potentially lead to higher quality planning in real-life logistics, thanks to the exploitation of the probabilistic informations that can normally be derived from historical data.In this paper we present an iterative model-based algorithm that solves a sequence of deterministic problems and is able to retrieve and certify optimal solutions if run for sufficient time. Experimental results show that the new approach is performing well when compared against both the exact (proven optimality) and heuristic (high quality solutions) algorithms available in the literature.

A variant of the well-known Assignment Problem is studied in this paper, where pairs of assignments are conflicting, and cannot be selected at the same time. This configures a set of hard constraints. The problem, which models real applications, looks for a complete assignment that minimizes the total cost, while no conflict is violated. In this paper, we consider a previously known mixed integer linear program representing the problem and we solve it with the open-source solver CP-SAT, part of the Google OR-Tools computational suite. An experimental campaign on the instances available from the literature, indicates that the approach we propose achieves results comparable with, those of state-of-the-art solvers, notwithstanding its intrinsic conceptual and implementation simplicity. The solver adopted is also able to provide heuristic solutions quicker and better than the heuristic methods previously discussed in the literature.

The possible existence of a Moore graph of diameter 2 and degree 57 has been an open question for more than six decades. In this paper, certain subgraphs of this graph, referred to as t-subgraphs, are considered. Exploiting symmetry by assuming a cyclic group of permutations representing edges joining leaf nodes of branches of a tree, a tractable constraint model for t-subgraphs is created. This can be solved using the Google OR Tools CP-SAT solver. Larger potential t-subgraphs than those currently known are constructed. This further extends a construction of certain sets of mutually orthogonal Latin rectangles. The implications for non-existence proofs of the Moore graph are considered.

Background: The Clustered Orienteering Problem is an optimization problem faced in last-mile logistics. The aim is, given an available time window, to visit vertices and to collect as much profit as possible in the given time. The vertices to visit have to be selected among a set of service requests. In particular, the vertices belong to clusters, the profits are associated with clusters, and the price relative to a cluster is collected only if all the vertices of a cluster are visited. Any solving methods providing better solutions also imply a new step towards sustainable logistics since companies can rely on more efficient delivery patterns, which, in turn, are associated with an improved urban environment with benefits both to the population and the administration thanks to an optimized and controlled last-mile delivery flow. Methods: In this paper, we propose a constraint programming model for the problem, and we empirically evaluate the potential of the new model by solving it with out-of-the-box software. Results: The results indicate that, when compared to the exact methods currently available in the literature, the new approach proposed stands out. Moreover, when comparing the quality of the heuristic solutions retrieved by the new model with those found by tailored methods, a good performance can be observed. In more detail, many new best-known upper bounds for the cost of the optimal solutions are reported, and several instances are solved to optimality for the first time. Conclusions: The paper provides a new practical and easy-to-implement tool to effectively deal with an optimization problem commonly faced in last-mile logistics.

In the Set Orienteering Problem, a single vehicle, leaving from and returning to a depot, has to serve some customers, each one associated with a given spacial location. Customers are grouped in clusters and a given prize is collected once a customer in a cluster is visited. The prize associated with a cluster can be collected at most once. Travel times among locations are provided, together with a maximum available mission time, which normally makes it impossible to visit all the clusters. The target is to design a route for the vehicle that maximizes the total prize collected within the given time limit. In this study, building on the recent literature, we present new preprocessing rules and a new constraint programming model for the problem. Thanks to the symmetry exploitation carried out by the constraint programming solver, new state-of-the-art results are established.

It has been an open question for 6 decades whether a Moore graph of diameter 2 and degree 57 exists. In this paper the question is posed as an optimization problem and an algorithm is described. The algorithm converges to solutions which are massively short of the number of edges required. This, and other supporting work, tend to suggest that the graph does not exist. The formulation presented is a particularly hard testbed for optimization algorithms. It is left as a challenge to others to develop alternative algorithms that may support the claim, or find solutions with more edges, or even construct the Moore graph.

The paper discusses an enhancement to a recently presented supervised learning algorithm to solve the Maximum Independent Set problem. In particular, it is shown that the algorithm can be improved by simplifying the task learnt by the neural network adopted, with measurable effects on the quality of the solutions provided on unseen instances. Empirical results are presented to validate the idea..

B-coloring is a problem in graph theory at the basis of several real applications and also used to improve solution methods for the classical coloring problem. Enhanced solutions for the classical coloring problem have in turn impacts on several other practical applications in scheduling, timetabling and telecommunications. Namely, given a graph G = (V, E), the b-coloring problem consists of maximizing the number of colors used while assigning a color to every vertex in V such that no pair of adjacent vertices receive the same color and every color has a representative, called a b-vertex. A vertex can be a b-vertex if it is adjacent to vertices colored with all the colors apart from the one assigned to it. In this paper we present a novel Iterative Matheuristic Algorithm based on considerations about the structure of promising solutions and a mathematical programming model. A vast section of computational experiments shows how the approach is able to find high quality solutions for commonly established datasets from the literature. In particular, the method we propose is able to improve the best known heuristic solution for 38 instances of the 137 considered. The optimality of the bounds previously known for another 5 instances has also been proved by running the approach we propose exhaustively.

B-coloring is a problem in graph theory. It can model some real applications, as well as being used to enhance solution methods for the classical graph coloring problem. In turn, improved solutions for the classical coloring problem would impact a larger pool of practical applications in several different fields such as scheduling, timetabling and telecommunications. Given a graph G=(V,E), the b-coloring problem aims to maximize the number of colors used while assigning a color to every vertex in V, preventing adjacent vertices from receiving the same color, with every color represented by a special vertex, called a b-vertex. A vertex can be a b-vertex only if the set of colors assigned to its adjacent vertices includes all the colors, apart from the one assigned to the vertex itself. This work employs methods based on Linear Programming to derive new upper and lower bounds for the problem. In particular, starting from a Mixed Integer Linear Programming model recently presented, upper bounds are obtained through partial linear relaxations of this model, while lower bounds are derived by considering different variations of the original model, modified to target a specific number of colors provided as input. The experimental campaign documented in the paper led to several improvements to the state-of-the-art results.