An edge-labeled graph (ELG) G=(V,E,L)$G=(V,E,L)$ is a graph such that V$V$ is the set of vertices, E$E$ is the set of edges, L$L$ is the set of labels (colors), and each edge e is an element of E$e \in E$ has a label associated. Given an ELG G=(V,E,L)$G=(V,E,L)$, the goal of the minimum labeling global cut problem (MLGCP) is to find a subset L 'subset of L$L<^>{\prime } \subseteq L$ such that the removal of all edges with labels in L '$L<^>{\prime }$ disconnects G$G$ and |L '|$|L<^>{\prime }|$ is minimum. This work proposes three new mathematical formulations for the MLGCP, namely PART, VC, and TE as well as branch-and-cut algorithms to solve them. Additionally, a theoretical study was carried out on the MLGCP input graph, leading to the concept of chromatic closure, used in preprocessing algorithms for this model PART. Finally, a comprehensive polyhedral investigation of the model is performed. The computational experiments showed that the PART$PART$ model, adopting the chromatic closure concept and its branch-and-cut algorithm, can solve small to average-sized instances in reasonable times.

Clustering algorithms are used to partition datasets associated with various real-world applications. However, in addition to the adopted algorithm, the obtained partition depends on the data distribution. Consequently, applying a single algorithm can lead to producing a poor-quality partition. Cluster Ensemble (CE) is an alternative to produce a good quality partition, as it combines different dataset partitions into a single consensus partition. According to the literature, in general, the consensus partition is less sensitive to noise when compared to that produced by a single algorithm. This work proposes a CE algorithm (BRKGA-CE) that combines: (i) three different strategies for producing base partitions; (ii) the BRKGA metaheuristic; (iii) the mean silhouette index, and (iv) an iterative method applied in the final phase of BRKGA-CE that allocates each object into a cluster. The core idea of BRKGA-CE is to find the representative objects of each cluster in order to maximize the mean silhouette index. To evaluate BRKGA-CE, computational experiments were carried out with 20 datasets and the main algorithms from the literature, where two well-known external validation indices (& Nscr;& Mscr;& Iscr; and & Ascr;& Rscr;) and hypothesis tests were applied. As a result, BRKGA-CE presented good-quality solutions, in the three strategies compared to the other algorithms.

Broadcasting is an essential operation in distributed systems, with a wide range of applications. This study is focused on solving the Weighted Minimum Broadcast Time (WMBT), a problem that extends the classical Minimum Broadcast Time problem (MBT) by incorporating costs associated with each communication operation. We propose five contributions to the WMBT: (i) an integer linear programming model, (ii) two greedy algorithms, (iii) two Biased Random-Key Genetic Algorithms (BRKGAs), (iv) a lower bound algorithm, (v) a reduction rule to decrease an instance size, and (vi) a method to create instances with known optimal solutions. Our novel approaches are compared with state-of-the-art methods using large-scale synthetic instances. The experimental results demonstrate the effectiveness of our proposals. The greedy algorithms attains the best known solutions in a significant number of instances, while the two BRKGAs further enhance this performance, surpassing the greedy algorithms in many of the tested instances.

A natural constraint in real-world applications is avoiding conflicting elements in problem solutions. Let = (, ) be a graph where each edge E has a positive integer weight (), and let = (, ) be a conflict graph such that subset of and each edge = 12 E represents a conflict between two edges 1, 2 E . In the MINIMUM CONFLICT-FREE SPANNING TREE (MCFST) problem, we are asked to find a spanning tree avoiding pairs of conflicting edges (if such a tree exists) with minimum cost. In contrast to the polynomial-time solvability of MINIMUM SPANNING TREE, to determine whether an instance (, ) of MCFST admits a feasible solution is an NP- complete problem. In this paper, we present a multivariate complexity analysis of MCFST by considering particular classes of graphs and . We show that the problem of determining whether an instance (, ) of MCFST has a feasible solution is NP-complete even if is a bipartite planar subcubic graph, and is a disjoint union of paths with three vertices. Contrastingly, we show that when is complete and is bipartite, then a solution for (, ) can be found in linear time, while the problem of finding an optimal solution is NP-hard. Also, we present (in)approximability results for MCFST on complete graphs , and a parameterized algorithm regarding the distance from the conflict graph to a hereditary graph class T for which MCFST on E T is polynomial-time solvable.

The main complexity classes of the Parameterized Intractability Theory are based on weighted Boolean circuit satisfiability problems and organized into a hierarchy so-called W-hierarchy. The W-hierarchy enables fine-grained complexity analyses of parameterized problems that are unlikely to belong to the FPT class. In this paper, we introduce the Th-hierarchy, a natural generalization of the W-hierarchy defined by unweighted threshold circuit satisfiability problems. Investigating the relationship between Th-hierarchy and W-hierarchy, we discuss the complexity of transforming Threshold circuits into Boolean circuits, and observe that sorting networks are powerful tools to handle such transformations. First, we show that these hierarchies collapse at the last level (W[P][Formula: see text][Formula: see text][Formula: see text]Th[P]). After that, we present a time complexity analysis of an AKS sorting network construction, which supports some of our results. Finally, we prove that Th[[Formula: see text]] [Formula: see text] W[SAT] for every [Formula: see text]. As a by-product, our studies suggest that it is relevant to consider a new class based on logarithmic depth circuits in the W-hierarchy.

Avoiding conflicting elements is a natural constraint that appears in several graph problems making them more challenging and close to real applications. Minimum Conflict-Free Spanning Tree (MCFST) is a variant of the classic Minimum Spanning Tree (MST) problem, where we are asked to find (if any) the spanning tree avoiding pairs of conflicting edges (conflict-free) of minimum cost. Although it is well known that MST is polynomial-time solvable, the MCFST problem is N P -hard. In this paper, we present a GRASP with adaptive memory (GRASP-AM) for Minimum Conflict-Free Spanning Tree . Adaptive memory (AM) is used in the constructive phase to decide which set of edges generates good solutions. Furthermore, we show how to implement the local search adopted in the GRASP-AM efficiently. Experimental results on a well-known benchmark indicate that our proposal outperforms the best existing heuristic for the problem. In particular, our GRASP-AM was able to find all known optimal solutions and to improve best-known solutions.