Decorous lower bounds for minimum linear arrangement

Minimum Linear Arrangement is a classical basic combinatorial optimization problem from the 1960s, which turns out to be extremely challenging in practice. In particular, for most of its benchmark instances, even the order of magnitude of the optimal solution value is unknown, as testified by the surveys on the problem that contain tables in which the best known solution value often has one more digit than the best known lower bound value. In this paper, we propose a linear-programming based approach to compute lower bounds on the optimum. This allows us, for the first time, to show that the best known solutions are indeed not far from optimal for most of the benchmark instances

Simplicial decompositions of graphs: a survey of applications

AbstractWe survey applications of simplicial decompositions (decompositions by separating complete subgraphs) to problems in graph theory. Among the areas of application are excluded minor theorems, extremal graph theorems, chordal and interval graphs, infinite graph theory and algorithmic aspects

The Bandwidth minimization problem

Mestrado em Métodos Quantitativos para a Decisão Económica e EmpresarialEsta dissertação tem como objetivo comparar o desempenho de duas heurísticas com a resolução de um modelo exato de programação linear inteira na determinação de soluções admissíveis do problema de minimização da largura de banda para matrizes esparsas simétricas. As heurísticas consideradas foram o algoritmo de Cuthill e McKee e o algoritmo Node Centroid com Hill Climbing. As duas heurísticas foram implementadas em VBA e foram avaliadas tendo por base o tempo de execução e a proximidade do valor das soluções admissíveis obtidas ao valor da solução ótima ou minorante. As soluções ótimas e os minorantes para as diversas instâncias consideradas foram obtidos através da execução do código para múltiplas instâncias e através da resolução do problema de Programação Linear Inteira com recurso ao Excel OpenSolver e ao software de otimização CPLEX. Como inputs das heurísticas foram utilizadas matrizes com dimensão entre 4×4 e 5580×5580, diferentes dispersões de elementos não nulos e diferentes pontos de partida.This dissertation intends to compare the performance of two heuristics with the resolution on the exact linear integer program model on the search for admissible solutions of the bandwidth minimization problem for sparse symmetric matrices. The chosen heuristics were the Cuthill and McKee algorithm and the Node Centroid with Hill Climbing algorithm. Both heuristics were implemented in VBA and they were rated taking into consideration the execution time in seconds, the relative proximity of the value obtained to the value of the optimal solution or lower bound. Optimal solutions and lower bounds were obtained through the execution of the code for several instances and trough the resolution of the integer linear problem using the Excel Add-In OpenSolver and the optimization software CPLEX. The inputs for the heuristics were matrices of dimension between 4×4 and 5580×5580, different dispersion of non-null elements and different initialization parameters.info:eu-repo/semantics/publishedVersio

Ordering a sparse graph to minimize the sum of right ends of edges

Motivated by a warehouse logistics problem we study mappings of the vertices of a graph onto prescribed points on the real line that minimize the sum (or equivalently, the average) of the coordinates of the right ends of all edges. We focus on graphs whose edge numbers do not exceed the vertex numbers too much, that is, graphs with few cycles. Intuitively, dense subgraphs should be placed early in the ordering, in order to finish many edges soon. However, our main “calculation trick” is to compare the objective function with the case when (almost) every vertex is the right end of exactly one edge. The deviations from this case are described by “charges” that can form “dipoles”. This reformulation enables us to derive polynomial algorithms and NP-completeness results for relevant special cases, and FPT results

Cubic Planar Graphs That Cannot Be Drawn On Few Lines

For every integer l, we construct a cubic 3-vertex-connected planar bipartite graph G with O(l^3) vertices such that there is no planar straight-line drawing of G whose vertices all lie on l lines. This strengthens previous results on graphs that cannot be drawn on few lines, which constructed significantly larger maximal planar graphs. We also find apex-trees and cubic bipartite series-parallel graphs that cannot be drawn on a bounded number of lines

Geometric Embeddability of Complexes Is ??-Complete

We show that the decision problem of determining whether a given (abstract simplicial) k-complex has a geometric embedding in ?^d is complete for the Existential Theory of the Reals for all d ? 3 and k ? {d-1,d}. Consequently, the problem is polynomial time equivalent to determining whether a polynomial equation system has a real solution and other important problems from various fields related to packing, Nash equilibria, minimum convex covers, the Art Gallery Problem, continuous constraint satisfaction problems, and training neural networks. Moreover, this implies NP-hardness and constitutes the first hardness result for the algorithmic problem of geometric embedding (abstract simplicial) complexes. This complements recent breakthroughs for the computational complexity of piece-wise linear embeddability

A theory of flow network typings and its optimization problems

Many large-scale and safety critical systems can be modeled as flow networks. Traditional approaches for the analysis of flow networks are whole-system approaches in that they require prior knowledge of the entire network before an analysis is undertaken, which can quickly become intractable as the size of network increases. In this thesis we study an alternative approach to the analysis of flow networks, which is modular, incremental and order-oblivious. The formal mechanism for realizing this compositional approach is an appropriately defined theory of network typings. Typings are formalized differently depending on how networks are specified and which of their properties is being verified. We illustrate this approach by considering a particular family of flow networks, called additive flow networks. In additive flow networks, every edge is assigned a constant gain/loss factor which is activated provided a non-zero amount of flow enters that edge. We show that the analysis of additive flow networks, more specifically the max-flow problem, is NP-hard, even when the underlying graph is planar. The theory of network typings gives rise to different forms of graph decomposition problems. We focus on one problem, which we call the graph reassembling problem. Given an abstraction of a flow network as a graph G = (V,E), one possible definition of this problem is specified in two steps: (1) We cut every edge of G into two halves to obtain a collection of |V| one-vertex components, and (2) we splice the two halves of all the edges, one edge at a time, in some order that minimizes the complexity of constructing a typing for G, starting from the typings of its one-vertex components. One optimization is minimizing “maximum” edge-boundary degree of components encountered during the reassembling of G (denoted as α measure). Another is to minimize the “sum” of all edge-boundary degrees encountered during this process (denoted by β measure). Finally, we study different variations of graph reassembling (with respect to minimizing α or β) and their relation with problems such as Linear Arrangement, Routing Tree Embedding, and Tree Layout

A decomposition of locally finite graphs

We prove that every infinite, connected, locally finite graph G can be expressed as an edge-disjoint union of a leafless tree T, rooted at an arbitrarily chosen vertex of G, and a collection of finite graphs H1, H2, H3,...such that, for all i less than j, the vertices common to Hi and Hj lie in T, and no vertex of Hj lies on T between a vertex of Hi∩T and the root. © 1993