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The concept of rainbow connection number of a graph was introduced by Chartrand et al. in 2008. Inspired by this concept, other concepts on colored version of connectivity in graphs were introduced, such as the monochromatic connection number by Caro and Yuster in 2011, the proper connection number by Borozan et al. in 2012, and the conflict-free connection number by Czap et al. in 2018, as well as some other variants of connection numbers later on. Chakraborty et al. proved that to compute the rainbow connection number of a graph is NP-hard. For a long time, it has been tried to fix the computational complexity for the monochromatic connection number, the proper connection number and the conflict-free connection number of a graph. However, it has not been solved yet. Only the complexity results for the strong version, i.e., the strong proper connection number and the strong conflict-free connection number, of these connection numbers were determined to be NP-hard. In this paper, we prove that to compute each of the monochromatic connection number, the proper connection number and the conflict free connection number for a graph is NP-hard. This solves a long standing problem in this field, asked in many talks of workshops and papers. [ABSTRACT FROM AUTHOR]

In order to run a dataflow with as low cost as possible, it is often faced with deciding which data-sets in a data-set sequence should be stored, with the rest regenerated. The Intermediate Data-set Storage problem arises from this situation. The current best algorithm for this problem takes O ( n 4 ) time. In this paper, we present two improved algorithms for this problem, the first of which can achieve a time complexity O ( n 2 ) , the second of which O ( r n ) , where n is the number of data-sets in a dataflow, r is a numerical number which indicates how large it is for the maximum storage cost to be divided by the minimum computation cost in the dataflow. [ABSTRACT FROM AUTHOR]

In this paper we prove that the Minimum Latency Aggregation Scheduling (MLAS) problem in the Signal-to-Interference-Noise-Ratio (SINR) model is APX-hard in the uniform power model. [ABSTRACT FROM AUTHOR]

In this paper, at first we describe a digraph representing all the weak-order extensions of a partially ordered set and algorithms for generating them. Then we present a digraph representing all of the minimal weak-order extensions of a partially ordered set. This digraph also implies generation algorithms. Finally, we prove that the number of weak-order extensions of a partially ordered set is a comparability invariant, whereas the number of minimal weak-order extensions of a partially ordered set is not a comparability invariant. [Copyright &y& Elsevier]