Relations between graphs and integer-pair sequences

AbstractIn this paper a relation between graphs and finite integer-pair sequences is established. Necessary and sufficient conditions under which a given finite sequence of pairs of integers can realize a graph are found. Results are extended to diagraphs

Complexity results for scheduling tasks in fixed intervals on two types of machines

Suppose that $n$ independent tasks are to be scheduled without preemption on an unlimited number of parallel machines of two types: inexpensive slow machines and expensive fast machines. Each task requires a given processing time on a slow machine or a given smaller processing time on a fast machine. We make two different feasibility assumptions: (a) each task has a specified processing interval, the length of which is equal to the processing time on a slow machine; (b) each task has a specified starting time. For either problem type, we wish to find a feasible schedule of minimum total machine cost. It is shown that both problems are NP-hard in the strong sense. These results are complemented by polynomial algorithms for some special cases

Роль української інтелігенції польського походження у національному відродженні України другої половини XIX століття

У статті розглядається участь деяких українських інтелігентів польського походження в активізації процесу національного відродження Україн

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.

The Cancer Genome Atlas Comprehensive Molecular Characterization of Renal Cell Carcinoma

Renal cell carcinoma(RCC) is not a single disease, but several histologically defined cancers with different genetic drivers, clinical courses, and therapeutic responses. The current study evaluated 843 RCC from the three major histologic subtypes, including 488 clear cell RCC, 274 papillary RCC, and 81 chromophobe RCC. Comprehensive genomic and phenotypic analysis of the RCC subtypes reveals distinctive features of each subtype that provide the foundation for the development of subtype-specific therapeutic and management strategies for patients affected with these cancers. Somatic alteration of BAP1, PBRM1, and PTEN and altered metabolic pathways correlated with subtype-specific decreased survival, while CDKN2A alteration, increased DNA hypermethylation, and increases in the immune-related Th2 gene expression signature correlated with decreased survival within all major histologic subtypes. CIMP-RCC demonstrated an increased immune signature, and a uniform and distinct metabolic expression pattern identified a subset of metabolically divergent (MD) ChRCC that associated with extremely poor survival