We consider the polyhedral structure of the precedence constrained knapsack (PCK) problem, also known as the partially ordered knapsack problem. A set of items N is given, along with a partial order, or set of precedence relationships, on the items, denoted by S ⊆ N × N. A precedence relationship (i, j) ∈ S exists if item i can be placed in the knapsack only if item j is in the knapsack. Each i...

Developing and promoting self-directed learning, as an one of the goals in higher education is important because it expected from students to reach the high level of individual capabilities, and able to identify their training needs in order to fix it and evaluate their own learning, without being in a special education system. Conclusion: The literature review showed that self-directed learnin...

Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of non-trivial cycles in G, using different techniques depending on whether or not G is an undirected graph. If G is undirected, then we give an algorithm to compute a shortest non-separating cycle in 2O(g)n log log n time. Similar algorithms are given to compute a shortest...

It is proved that every strongly connected directed graph with n nodes and at least ⌊(n + 1)/4⌋ edges must contain an even cycle. This is best possible, and the structure of extremal graphs is discussed.

In this note, it is proved that if 0.24817   , then any digraph on n vertices with minimum outdegree at least n  contains a directed cycle of length at most 5.

We observe recent results on the applications of extremal graph theory to cryptography. Classical Extremal Graph Theory contains Erdős Even Circuite Theorem and other remarkable results on the maximal size of graphs without certain cycles. Finite automaton is roughly a directed graph with labels on directed arrows. The most important advantage of Turing machine in comparison with finite automat...

Let D(n, p) be the random directed graph on n vertices where each of the n(n− 1) possible arcs is present independently with probability p. It is known that if p ≥ (log n+ ω(1))/n then D(n, p) typically has a directed Hamilton cycle, and this is best possible. We show that under the same condition, the number of directed Hamilton cycles in D(n, p) is typically n!(p(1 + o(1))) . We also prove a ...

epigenetics alterations, especially dna methylation, play a critical role in control of gene expressions. abnormal patterns of methylation are observed in early-stages of many cancers. therefore, methylation analysis is useful in primary detection of tumors. advances knowledge about the functional role of aberrant epigenetic modifications as potential biomarkers for cancer, have attracted consi...