Evolutionary computation of forests with Degree- and Role-Constrained Minimum Spanning Trees

Finding the degree-constrained minimum spanning tree (DCMST) of a graph is a widely studied NP-hard problem. One of its most important applications is network design. Here we deal with a new variant of the DCMST problem, which consists of finding not only the degree- but also the role-constrained minimum spanning tree (DRCMST), i.e., we add constraints to restrict the role of the nodes in the tree to root, intermediate or leaf node. Furthermore, we do not limit the number of root nodes to one, thereby, generally, building a forest of DRCMSTs. The modeling of network design problems can benefit from the possibility of generating more than one tree and determining the role of the nodes in the network. We propose a novel permutation-based representation to encode these forests. In this new representation, one permutation simultaneously encodes all the trees to be built. We simulate a wide variety of DRCMST problems which we optimize using eight different evolutionary computation algorithms encoding individuals of the population using the proposed representation. The algorithms we use are: estimation of distribution algorithm, generational genetic algorithm, steady-state genetic algorithm, covariance matrix adaptation evolution strategy, differential evolution, elitist evolution strategy, non-elitist evolution strategy and particle swarm optimization. The best results are for the estimation of distribution algorithms and both types of genetic algorithms, although the genetic algorithms are significantly faster. -------------------------------------------------------------------------------------------------- Trabajo publicado en: Antón Sánchez, Laura; Bielza Lozoya, Maria Concepcion y Larrañaga Múgica, Pedro (2017). Network Design through Forests with Degree- and Role-constrained Minimum Spanning Trees. "Journal of Heuristics ", v. 23 (n. 1); pp. 31-51. ------------------------------------------

Memory lower bounds for deterministic self-stabilization

In the context of self-stabilization, a \emph{silent} algorithm guarantees that the register of every node does not change once the algorithm has stabilized. At the end of the 90's, Dolev et al. [Acta Inf. '99] showed that, for finding the centers of a graph, for electing a leader, or for constructing a spanning tree, every silent algorithm must use a memory of $\Omega(\log n)$ bits per register in $n$-node networks. Similarly, Korman et al. [Dist. Comp. '07] proved, using the notion of proof-labeling-scheme, that, for constructing a minimum-weight spanning trees (MST), every silent algorithm must use a memory of $\Omega(\log^2n)$ bits per register. It follows that requiring the algorithm to be silent has a cost in terms of memory space, while, in the context of self-stabilization, where every node constantly checks the states of its neighbors, the silence property can be of limited practical interest. In fact, it is known that relaxing this requirement results in algorithms with smaller space-complexity. In this paper, we are aiming at measuring how much gain in terms of memory can be expected by using arbitrary self-stabilizing algorithms, not necessarily silent. To our knowledge, the only known lower bound on the memory requirement for general algorithms, also established at the end of the 90's, is due to Beauquier et al.~[PODC '99] who proved that registers of constant size are not sufficient for leader election algorithms. We improve this result by establishing a tight lower bound of $\Theta(\log \Delta+\log \log n)$ bits per register for self-stabilizing algorithms solving $(\Delta+1)$-coloring or constructing a spanning tree in networks of maximum degree~$\Delta$. The lower bound $\Omega(\log \log n)$ bits per register also holds for leader election

The edge slide graph of the n-dimensional cube : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū, New Zealand

The goal of this thesis is to understand the spanning trees of the n-dimensional cube Qn by understanding their edge slide graph. An edge slide is a move that “slides” an edge of a spanning tree of Qn across a two-dimensional face, and the edge slide graph is the graph on the spanning trees of Qn with an edge between two trees if they are connected by an edge slide. Edge slides are a restricted form of an edge move, in which the edges involved in the move are constrained by the structure of Qn, and the edge slide graph is a subgraph of the tree graph of Qn given by edge moves. The signature of a spanning tree of Qn is the n-tuple (a1; : : : ; an), where ai is the number of edges in the ith direction. The signature of a tree is invariant under edge slides and is therefore constant on connected components. We say that a signature is connected if the trees with that signature lie in a single connected component, and disconnected otherwise. The goal of this research is to determine which signatures are connected. Signatures can be naturally classified as reducible or irreducible, with the reducible signatures being further divided into strictly reducible and quasi-irreducible signatures. We determine necessary and sufficient conditions for (a1; : : : ; an) to be a signature of Qn, and show that strictly reducible signatures are disconnected. We conjecture that strict reducibility is the only obstruction to connectivity, and present substantial partial progress towards an inductive proof of this conjecture. In particular, we reduce the inductive step to the problem of proving under the inductive hypothesis that every irreducible signature has a “splitting signature” for which the upright trees with that signature and splitting signature all lie in the same component. We establish this step for certain classes of signatures, but at present are unable to complete it for all. Hall’s Theorem plays an important role throughout the work, both in characterising the signatures, and in proving the existence of certain trees used in the arguments

On morphological hierarchical representations for image processing and spatial data clustering

Hierarchical data representations in the context of classi cation and data clustering were put forward during the fties. Recently, hierarchical image representations have gained renewed interest for segmentation purposes. In this paper, we briefly survey fundamental results on hierarchical clustering and then detail recent paradigms developed for the hierarchical representation of images in the framework of mathematical morphology: constrained connectivity and ultrametric watersheds. Constrained connectivity can be viewed as a way to constrain an initial hierarchy in such a way that a set of desired constraints are satis ed. The framework of ultrametric watersheds provides a generic scheme for computing any hierarchical connected clustering, in particular when such a hierarchy is constrained. The suitability of this framework for solving practical problems is illustrated with applications in remote sensing

Setting Parameters by Example

We introduce a class of "inverse parametric optimization" problems, in which one is given both a parametric optimization problem and a desired optimal solution; the task is to determine parameter values that lead to the given solution. We describe algorithms for solving such problems for minimum spanning trees, shortest paths, and other "optimal subgraph" problems, and discuss applications in multicast routing, vehicle path planning, resource allocation, and board game programming.Comment: 13 pages, 3 figures. To be presented at 40th IEEE Symp. Foundations of Computer Science (FOCS '99

Developing Efficient Metaheuristics for Communication Network Problems by using Problem-specific Knowledge

Metaheuristics, such as evolutionary algorithms or simulated annealing, are widely applicable heuristic optimization strategies that have shown encouraging results for a large number of difficult optimization problems. To show high performance, metaheuristics need to be adapted to the properties of the problem at hand. This paper illustrates how efficient metaheuristics can be developed for communication network problems by utilizing problem-specific knowledge for the design of a high-quality problem representation. The minimum communication spanning tree (MCST) problem finds a communication spanning tree that connects all nodes and satisfies their communication requirements for a minimum total cost. An investigation into the properties of the problem reveals that optimum solutions are similar to the minimum spanning tree (MST). Consequently, a problem-specific representation, the link biased (LB) encoding, is developed, which represents trees as a list of floats. The LB encoding makes use of the knowledge that optimum solutions are similar to the MST, and encodes trees that are similar to the MST with a higher probability. Experimental results for different types of metaheuristics show that metaheuristics using the LB-encoding efficiently solve existing MCST problem instances from the literature, as well as randomly generated MCST problems of different sizes and types