Abstract:In the paper, we write a linear algorithm for calculating the weighted domination number of a vertexweighted cactus. The algorithm is based on the well known depth first search (DFS) structure. Our algorithm needs less than 12n + 5b additions and 9n + 2b min-operations where n is the number of vertices and b is the number of blocks in the cactus.
The modified Hosoya polynomial of double weighted graphs, i.e. edge and vertex weighted graphs, is introduced that enables derivation of closed expressions for Hosoya polynomial of some special graphs including unicyclic graphs. Furthermore, the Hosoya polynomial is given as a sum of edge contributions generalizing well known analogous results for the Wiener number. A linear algorithm for computing the Hosoya polynomial on cactus graphs is provided. Hosoya polynomial is extensively studied in chemical graph theory, and in particular its weighted versions have interesting applications in theory of communication networks.
The k-assignment problem (or, the k-matching problem) on k-partite graphs is an NP-hard problem for k≥3. In this paper we introduce five new heuristics. Two algorithms, Bm and Cm, arise as natural improvements of Algorithm Am from (He et al., in: Graph Algorithms And Applications 2, World Scientific, 2004). The other three algorithms, Dm, Em, and Fm, incorporate randomization. Algorithm Dm can be considered as a greedy version of Bm, whereas Em and Fm are versions of local search algorithm, specialized for the k-matching problem. The algorithms are implemented in Python and are run on three datasets. On the datasets available, all the algorithms clearly outperform Algorithm Am in terms of solution quality. On the first dataset with known optimal values the average relative error ranges from 1.47% over optimum (algorithm Am) to 0.08% over optimum (algorithm Em). On the second dataset with known optimal values the average relative error ranges from 4.41% over optimum (algorithm Am) to 0.45% over optimum (algorithm Fm). Better quality of solutions demands higher computation times, thus the new algorithms provide a good compromise between quality of solutions and computation time.
The rapid worldwide evolution of LEDs as light sources has brought new challenges, which means that new methods are needed and new algorithms have to be developed. Since the majority of LED luminaries are of the multi-source type, established methods for the design of light engines cannot be used in the design of LED light engines. This is because in the latter case what is involved is not just the design of a good reflector or projector lens, but the design of several lenses which have to work together in order to achieve satisfactory results. Since lenses can also be bought off the shelf from several manufacturers, it should be possible to combine together different off the shelf lenses in order to design a good light engine. However, with so many different lenses to choose from, it is almost impossible to find an optimal combination by hand, which means that some optimization algorithms need to be applied. In order for them to work properly, it is first necessary to describe the input data (i.e. spatial light distribution) in a functional form using as few as possible parameters. In this paper the focus is on the approximation of the input data, and the implementation of the well-known mathematical procedure for the separation of linear and nonlinear parameters, which can provide a substantial increase in performance.
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