In tournament schedules, a break occurs when a team plays two consecutive home or two consecutive away games. Minimizing breaks is important for ensuring competitive fairness and logistical efficiency. This article addresses the problem of minimizing breaks in incomplete round-robin schedules in which each pair of teams plays again each other at most once. The problem of minimizing breaks is a classical problem that was previously thoroughly studied in the context of complete round-robin schedules. Using a graph-theoretic model we analyze structural properties of incomplete round-robin schedules. We derive some bounds on the minimum number of breaks. Then, we propose an algorithm that is able to construct incomplete single round-robin schedules minimizing the number of breaks for given numbers of teams and rounds if the number of rounds is not larger than 3 / 4 of the number of teams.

In round-robin schedules, a break occurs when a team plays two consecutive home or two consecutive away games. Minimizing breaks is important for ensuring competitive fairness and logistical efficiency. This article addresses the problem of minimizing breaks in incomplete round-robin schedules in which each pair of teams plays again each other at most once. The problem of minimizing breaks is a classical problem that was previously thoroughly studied in the context of complete round-robin schedules. Using a graph-theoretic model we analyze structural properties of incomplete round-robin schedules. We derive some bounds on the minimum number of breaks. Then, we propose an algorithm that is able to construct incomplete single round-robin schedules minimizing the number of breaks for given numbers of teams and rounds if the number of rounds is not larger than 3/4 of the number of teams.

ABSTRACT Effective planning of human resources is critical in designing an efficient home healthcare system. In this study, we present a novel home healthcare staffing, routing, and scheduling problem inspired by a real‐world application. The proposed problem addresses a set of patients, with varying daily visit requirements, being served by a set of caregivers with different qualification levels over a multi‐day multi‐shift planning horizon. The problem aims to minimize the number of extra shifts for caregivers, maximize the allocation of caregivers to emergencies, and minimize the sum of route durations over the planning horizon. These objectives are optimized hierarchically while considering a set of restrictions, including time windows, skill matching, synchronization, care continuity, and labor regulations. To tackle the problem, we introduce a mixed‐integer linear programming model. The model is then extended and two sets of valid inequalities are incorporated to enhance its tightness. Computational experiments are conducted on a set of 20 instances. The results highlight the efficiency of the proposed extension in increasing both the number of instances that can be solved to optimality and the number of instances for which a feasible solution is found.

This paper tackles the 2-Dimensional Guillotine Cutting Stock Problem with Stack Constraints. The problem asks for the cutting of a set of items with the minimum amount of raw material. The cutting patterns are subject to a number of constraints, including a new realistic constraint, regarding item precedence, which has just been introduced in the literature. In this case, the items are organized in stacks, where each stack represents a customer request and defines the order in which the items must be cut. That is, if item i precedes item j within a stack, then i must be cut before j. However, there is no precedence constraint between items in different stacks. This constraint comes from applications where items must be stacked and shipped in the exact order that they will be used by the customer, thus avoiding the risk of damaging fragile items (as is the case in the glass industry) or the cost of moving heavy items (as is the case in the steel industry). We propose two constructive heuristics extended from the literature for the problem, in addition to a dynamic programming based heuristic that uses as a subroutine an exact pseudo-polynomial time algorithm developed for the Rectangular Knapsack Problem with Batch Constraints. Computational experiments, performed on three sets of realistic instances, showed that the dynamic programming based heuristic found solutions with smaller optimally gaps in all instances evaluated.

Given a directed acyclic graph G=(V,A) and two vertices u,v∈V, the reachability problem is to answer if there is a path from u to v in the graph. In the context of very large graphs, with millions of vertices and a series of queries to be answered, it is not practical to search the graph for each query. On the other hand, the storage of the full transitive closure of the graph is also impractical due to its O(|V|2) size. Scalable approaches aim to create indices used to prune the search during its execution. Negative indices may be able to determine (in constant time) that a query has a negative answer while positive indices may determine (again in constant time) that a query has a positive answer. In this paper we propose a novel scalable approach called LYNX that uses a large number of topological sorts of G as a negative cut index without degrading the query time. A similar strategy is applied regarding a positive cut index. In addition, LYNX proposes a user-defined index size that enables the user to control the ratio between negative and positive cuts depending on the expected query pattern. We show by computational experiments that LYNX consistently outperforms the state-of-the-art approach in terms of query-time using the same index-size for graphs with high reachability ratio. In intelligent computer systems that rely on frequent tests of connectivity in graphs, LYNX can reduce the time delay experience by end users through a reduced query time. This comes at the expense of an increased setup time whenever the underlying graph is updated.

The exploration of one-factorizations of complete graphs is the foundation of some classical sports scheduling problems. One has to traverse the landscape of such one-factorizations by moving from one of those to a so-called neighbor one-factorization. This approach amounts to modifying locally the coloring associated with a one-factorization. We consider some particular types of modifications and describe various constructions which give one-factorizations which may be modified or not by these techniques. Among those are recoloring of bichromatic cycles, altering of optimally colored subcliques of even size, or recoloring of chordless lanterns.

Graph Theory provides a set of powerful tools (both theorems and algorithms) for problem modeling and solving in numerous domains. Though there are several libraries implementing graph algorithms and targeting different platforms and users, few of those offer parallel implementations. To the best of our knowledge, there is a particular need for an easier to use and extend library, specifically designed to exploit the multicore architecture trend for high performance parallelism. In this paper we describe Magical, a new OpenMP-based C++ multicore graph library. Our focus is to provide an implementation of graph algorithms which is designed for multicore architectures, by means of an easy to use application programming interface. We describe the library design and evaluate its performance by means of a case study concerning a shortest-paths problem.

Metaheuristics are approximate algorithms that are able to find very good solutions to hard combinatorial optimization problems. They do, however, offer a wide range of possibilities for implementations of effective robust parallel algorithms which run in much smaller computation times than their sequential counterparts. We present four slightly differing strategies for the parallelization of an extended GRASP with ILS heuristic for the mirrored traveling tournament problem. Computational results on widely used benchmark instances, using a varying number of processors, illustrate the effectiveness and the scalability of the different strategies. These low communication cost parallel heuristics not only find solutions faster, but also produce better quality solutions than the best known sequential algorithm.