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Abstract — We present a new representation for individuals in problems that have cyclic permutations as solutions. To demonstrate its usefulness rigorously, we construct from it a simple randomized local search and a (1+1) evolutionary algorithm for the Eulerian cycle problem. Both have an expected run-time of Θ(m 2 log(m)), where m denotes the number of edges of the input graph. This clearly beats previous solutions, which all have an expected optimization time of Θ(m 3) or worse (PPSN ’06, CEC ’04). We are optimistic that our representation allows superior solutions also for other cyclic permutation problems. For NP-complete ones like the TSP, however, other means than theoretical run-time analyses are necessary. I.

We propose and analyze a novel genotype to represent walk and cycle covers in graphs, namely matchings in the adjacency lists. This representation admits the natural mutation operator of adding a random match and possibly also matching the former partners. To demonstrate the strength of this set-up, we use it to build a simple (1+1) evolutionary algorithm for the problem of finding an Eulerian cycle in a graph. We analyze several natural variants that stem from different ways to randomly choose the new match. Among other insight, we exhibit a (1+1) evolutionary algorithm that computes an Euler tour in a graph with m edges in expected optimization time Θ(m log m). This significantly improves the previous best evolutionary solution having expected optimization time Θ(m2 log m) in the worst-case, but also compares nicely with the runtime of an optimal classical algorithm which is of order Θ(m). A simple coupon collector argument indicates that our optimization time is asymptotically optimal for any randomized search heuristic.

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Let G = (V; E) be an undirected graph. Given an odd number k = 2l + 1, a matching M is said to be k-optimal if it does not admit an augmenting path of length less than or equal to k. We prove jM j jM 3 j(l+1)=(l+2), where M 3 is a maximum cardinality matching. If M is not already (k + 2)-optimal, using M , in O(jEj) time we compute a (k + 2)-optimal matching. We show that starting with any matching, repeated k-optimizations result in an optimal matching in O( p jM 3 jjEj) time. This approximation scheme extends to the minimum weight perfect matching, and the minimum weight uncapacitated perfect 2-matching problems over complete graphs, and complete bipartite graphs with edge weights of one and two. In particular we obtain a fast approximation algorithm for the traveling salesman problem over complete graphs with edge weights of one and two.

This dissertation presents a systematic design methodology for directed product evolution that uses both rigid body and compliant mechanisms to facilitate component combination in the domain of mechanical products. The methodology, known as effort flow analysis, is based on fundamental tenets from the Theory of Mechanics, and Graph Theory. Effort flow analysis uses a semantic network known as an effort flow diagram to model a product as a connected set of nodes and links. The nodes represent the components of the product and the links represent the interfaces between the components. The effort flow diagram is a quasi-static model of the flow of effort (force or torque in the mechanical domain) as it transits the body from input to output. In order to capture the effect of the relative motion that occurs at the interfaces, a basis set for relative motion is developed for effort flow analysis. The basis consists of 4 possible link type cases, (1) No relative motion at the interface or away from the interface, (2) Component relative motion that occurs away from the interface, (3) general Relative motion between components, and (4) Interface relative motion that occurs only at the interface. These are known as N- Links, C-Links, R-Links, and I-Links respectively. Rigid body combinations are sought for components connected by N-Links and compliant mechanism combinations are sought for components connected by the other link types. Component combination opportunities are sought based on the connection structure of the effort flow diagram. Guidance for component combination is captured in a set of 29 product-evolution guidelines that are derived from the results of an empirical study. In addition, an example solution is cataloged for each of the guidelines to foster design-by-analogy efforts by the designer. Finally, the work presents a novel design and prototype for a compliant one-piece umbrella frame, which serves as a proof of concept for the methodology. Possible future pursuits are presented in the areas of knowledge capture using guidelines and solution examples. In addition, the possibility of a function-to-component matrix for automated concept generation is considered.

BDDs (Binary Decision Diagrams) are often used to represent Boolean expressions in hardware synthesis, hardware and software verification and numerous other applica-tions. BDD computation, implemented using tree data structures with binary nodes, is inherently memory intensive, and therefore suffers from the von Neumann memory bottleneck. This thesis examines an approach which can speed up BDD computation through compression of the graphical structure, reducing the overhead of handling memory pointers, and thereby reducing memory access latency. The novel aspect of this work is that a compression technique is used which allows BDD computation directly on the compressed data without the overheads of compression and decompression. The need for such an approach is driven by technology in two ways: the increasing discrepancy between CPU and memory speeds, and the move towards larger (64 bit) memory architectures. The work applies graph enumeration techniques to classify graph topology using a structural identifier (SID) coding table, providing the basis for algorithms which generate compressed representations. The work uses these new representations to develop algorithms which can pre-calculate results for specific graph-traversal operations in Combination lookup tables (CLT). The work distinguishes itself from other methods of reducing BDD access latency, such as index based systems, by improving spatial locality and simplifying computation at the same time.

this paper. The general form is not needed, therefore only a special case is given here. In general the A--cycle problem is to decide whether there exists a cycle containing all elements of A, where A is a collection of vertices and edges from the graph. But let us restrict ourselves to the case when A ` V (G). For sake of simplicity we will suppose that the elements of A are pairwise non-adjacent. Otherwise a 2-degree vertex is put into the middle of the edge connecting the two vertices. (This new vertex is not in A in this case.) We say that a pair (X; Y ), where X ` V (G \Gamma A) and Y ` E(G \Gamma A \Gamma X), disconnects A,