Metro-Line Crossing Minimization: Hardness, Approximations, and Tractable Cases

Crossing minimization is one of the central problems in graph drawing. Recently, there has been an increased interest in the problem of minimizing crossings between paths in drawings of graphs. This is the metro-line crossing minimization problem (MLCM): Given an embedded graph and a set L of simple paths, called lines, order the lines on each edge so that the total number of crossings is minimized. So far, the complexity of MLCM has been an open problem. In contrast, the problem variant in which line ends must be placed in outermost position on their edges (MLCM-P) is known to be NP-hard. Our main results answer two open questions: (i) We show that MLCM is NP-hard. (ii) We give an $O(\sqrt{\log |L|})$-approximation algorithm for MLCM-P

Desilication of highly siliceous zeolite ZSM-5 with NaOH and NaOH/tetrabutylamine hydroxide

The results of both chemical and XPS analysis pointed out that desilication of highly siliceous ZSM-5 of Si/Al = 164 was more effective in the surface zone than in the bulk, contrary to zeolite ZSM-5 of Si/Al = 31.6. According to the IR studies in parent zeolite the concentration of protonic sites was very close to the concentration of Al indicating that all Al atoms can form Si-OH-Al. The results of our quantitative IR studies strongly support the realumination thesis, i.e. some Al atoms extracted in basic solutions are subsequently reinserted forming new acidic hydroxyls. In desilicated zeolites all Al atoms were able to form protonic sites, however part of them dehydroxylated during the activation of zeolite producing Lewis acid sites according to the stoichiometry: one protonic site was transformed into one Lewis site. Low temperature nitrogen adsorption revealed that the alkaline treatment of highly siliceous zeolite with 0.2 M NaOH/TBAOH mixture produced mesopores of smaller diameter and narrower pore size distribution than in the case of zeolite of medium Si/Al ratio. This result can be explained by low concentration of Al which similarly as TBA(+) cations plays the role of pore directing agents (PDA). Contrary to TEA(+), the presence of Al in desilication mixture, led to the formation of larger pores. Therefore, in highly siliceous zeolite TBA(+) played dominant role as PDA producing narrower pores. Highly siliceous zeolite with uniform distribution of relatively narrow pores may be useful catalyst or catalyst support. The influence of desilication temperature on porosity development was also investigated. The increase of desilication temperature from 338 to 353 K resulted in both more extensive demetalation (more Si and Al is extracted) and the distinct increase of the volume and surface of mesopores. Both lower concentration of protonic sites and higher concentration of Lewis sites confirmed partial zeolite destruction upon desilication at elevated temperature. The experiments of pivalonitrile sorption followed by IR spectroscopy showed a significant increase of accessibility of acid sites to bulky pivalonitrile molecules. (C) 2012 Elsevier Inc. All rights reserved.The research was partially carried out with the equipment purchased thanks to the financial support of the European Regional Development Fund in the framework of the Polish Innovation Economy Operational Program (Contract No. POIG.02.01.00-12-023/08).Sadowska, K.; Góra-Marek, K.; Drozdek, M.; Kustrowski, P.; Datka, J.; Martínez-Triguero, J.; Rey Garcia, F. (2013). Desilication of highly siliceous zeolite ZSM-5 with NaOH and NaOH/tetrabutylamine hydroxide. Microporous and Mesoporous Materials. 168:195-205. https://doi.org/10.1016/j.micromeso.2012.09.033S19520516

Minimization of Fixed-Polarity AND/XOR Canonical Networks

A method to extract the cubes of the Generalized Reed-Muller form (GRM) of a Boolean function with a given polarity is presented. The method does not require exponential space and time complexity and it achieves the lower bound time complexity. The proof of the method's correctness constitutes the first half of this paper. Also, a separate heuristic algorithm to find the optimal polarity that requires the least number of cubes in the GRM representation is proposed. The algorithm is fast and derives the polarity for variables and extracts all cubes simultaneously. It is based on the concept of a Boolean center for vertices, which emulates the center of gravity concept in geometry. The experimental results for the heuristic algorithm agree strongly with our observations and analysis. 1 Introduction It has been known [1-2][4-8][10-11][13-17][20,21] that for arithmetic functions and errorcorrecting circuits the best realizations include XOR gates. High testability of the AND/XOR designs ..

Cube diagram bundles: a new representation of strongly unspecified multiple-valued functions and relations

Efficient function representation is very important for speed and memory requirements of multiple-valued decomposers. This paper presents a new representation of multiple-valued relations (functions in particular), called Multiple-Valued Cube Diagram Bundles (MVCDB). MVCDBs improve on Rough Partition representation by labeling their blocks with variable values and by representing blocks efficiently. The MVCDB representation is especially efficient for very strongly unspecified multiplevalued input, multiple-valued output functions and relations, typical for Machine Learning applications