Search Trajectory Networks Applied to the Cyclic Bandwidth Sum Problem

Search trajectory networks (STNs) were proposed as a tool to analyze the behavior of metaheuristics in relation to their exploration ability and the search space regions they traverse. The technique derives from the study of fitness landscapes using local optima networks (LONs). STNs are related to LONs in that both are built as graphs, modelling the transitions among solutions or group of solutions in the search space. The key difference is that STN nodes can represent solutions or groups of solutions that are not necessarily locally optimal. This work presents an STN-based study for a particular combinatorial optimization problem, the cyclic bandwidth sum minimization. STNs were employed to analyze the two leading algorithms for this problem: a memetic algorithm and a hyperheuristic memetic algorithm. We also propose a novel grouping method for STNs that can be generally applied to both continuous and combinatorial spaces

Treewidth and related graph parameters

For modeling some practical problems, graphs play very important roles. Since many modeled problems can be NP-hard in general, some restrictions for inputs are required. Bounding a graph parameter of the inputs is one of the successful approaches. We study this approach in this thesis. More precisely, we study two graph parameters, spanning tree congestion and security number, that are related to treewidth. Let G be a connected graph and T be a spanning tree of G. For e ∈ E(T), the congestion of e is the number of edges in G connecting two components of T − e. The edge congestion of G in T is the maximum congestion over all edges in T. The spanning tree congestion of G is the minimum congestion of G in its spanning trees. In this thesis, we show the spanning tree congestion for the complete k-partite graphs, the two-dimensional tori, and the twodimensional Hamming graphs. We also address lower bounds of spanning tree congestion for the multi-dimensional hypercubes, the multi-dimensional grids, and the multi-dimensional Hamming graphs. The security number of a graph is the cardinality of a smallest vertex subset of the graph such that any “attack” on the subset is “defendable.” In this thesis, we determine the security number of two-dimensional cylinders and tori. This result settles a conjecture of Brigham, Dutton and Hedetniemi [Discrete Appl. Math. 155 (2007) 1708–1714]. We also show that every outerplanar graph has security number at most three. Additionally, we present lower and upper bounds for some classes of graphs.学位記番号：工博甲39

Cyclic cutwidths of the two-dimensional ordinary and cylindrical meshes

The cutwidth problem is to find a linear layout of a network so that the maximal number of cuts of a line separating consecutive vertices is minimized (see e.g. [7]). A related and more natural problem is the cyclic cutwidth when a circular layout is considered. The main question is to compare both measures cw and ccw for specific networks, whether adding an edge to a path and forming a cycle reduces the cutwidth essentially. We prove exact values for the cyclic cutwidths of the two-dimensional ordinary and cylindrical meshes Pm×Pn and Pm×Cn, respectively. Especially, if m[greater-or-equal, slanted]n+3, then ccw(Pm×Pn)=cw(Pm×Pn)=n+1 and if n is even then ccw(Pn×Pn)=n-1 while cw(Pn×Pn)=n+1 and if m[greater-or-equal, slanted]2,n[greater-or-equal, slanted]3, then ccw(Pm×Cn)=min{m+1,n+2}