Triangular Scheme Revisited in the Light of n-permutable Categories

The first diagrammatic scheme was developed by H.P. Gumm under the name Shifting Lemma in case to characterize congruence modularity. A diagrammatic scheme is developed for the generalized semi distributive law in Mal'tsev categories. In this paper we study this diagrammatic scheme in the context of $n$-permutable, and of Mal'tsev categories in particular. Several remarks concerning the Triangular scheme case are included.

Minimum Dominating Set Problem for Unit Disks Revisited

In this article, we study approximation algorithms for the problem of computing minimum dominating set for a given set [Formula: see text] of [Formula: see text] unit disks in [Formula: see text]. We first present a simple [Formula: see text] time 5-factor approximation algorithm for this problem, where [Formula: see text] is the size of the output. The best known 4-factor and 3-factor approximation algorithms for the same problem run in time [Formula: see text] and [Formula: see text] respectively [M. De, G. K. Das, P. Carmi and S. C. Nandy, Approximation algorithms for a variant of discrete piercing set problem for unit disks, Int. J. of Computational Geometry and Appl., 22(6):461–477, 2013]. We show that the time complexity of the in-place 4-factor approximation algorithm for this problem can be improved to [Formula: see text]. A minor modification of this algorithm produces a [Formula: see text]-factor approximation algorithm in [Formula: see text] time. The same techniques can be applied to have a 3-factor and a [Formula: see text]-factor approximation algorithms in time [Formula: see text] and [Formula: see text] respectively. Finally, we propose a very important shifting lemma, which is of independent interest, and it helps to present [Formula: see text]-factor approximation algorithm for the same problem. It also helps to improve the time complexity of the proposed PTAS for the problem.