Improved Approximation Algorithms for Box Contact Representations ⋆

Abstract. We study the following geometric representation problem: Given a graph whose vertices correspond to axis-aligned rectangles with fixed dimensions, arrange the rectangles without overlaps in the plane such that two rectangles touch if the graph contains an edge between them. This problem is called CONTACT REPRESENTATION OF WORD NETWORKS (CROWN) since it formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. CROWN is known to be NP-hard, and there are approximation algorithms for certain graph classes for the optimization version, MAX-CROWN, in which realizing each desired adjacency yields a certain profit. We present the first O(1)-approximation algorithm for the general case, when the input is a complete weighted graph, and for the bipartite case. Since the subgraph of realized adjacencies is necessarily planar, we also consider several planar graph classes (namely stars, trees, outerplanar, and planar graphs), improving upon the known results. For some graph classes, we also describe improvements in the unweighted case, where each adjacency yields the same profit. Finally, we show that the problem is APX-hard on bipartite graphs of bounded maximum degree.

Safety assessment and behavioral effects of Solanum guaraniticum leaf extract in rats

ABSTRACT Solanum guaraniticum is a medicinal plant traditionally used to treat gastric and liver diseases. However, there is no documented evidence corroborating its safety. The present study evaluated the potential toxicity of S. guaraniticum leaf extract after acute administration in rats. Single doses of the extract (1.250, 2.500, and 5.000 mg/kg) were administered by gavage, and the rats were then monitored for 48 h and/or 14 days. Mortality, acute signs of toxicity, and general activity in the open field test were assessed as well as hematological and biochemical parameters, enzymatic activity (δ-aminolevulinate dehydratase and acetylcholinesterase), and oxidative stress parameters (lipid peroxidation level, non-protein thiol content, tissue catalase activity, and serum ferrous reducing power). Phytochemical analysis was also performed by HPLC. The results showed that extract administration produced no deaths (LD50 > 5,000 mg/kg), and no significant adverse effects regarding food consumption, body weight gain, gross pathology, or other parameters. However, the open field tests showed a decrease in spontaneous activity (crossing and rearing) mainly at 48 h after treatment. The results suggest that S. guaraniticum extract is not acutely toxic, but causes alterations in central nervous system activity