Approximation Algorithms for the Capacitated Domination Problem

We consider the {\em Capacitated Domination} problem, which models a service-requirement assignment scenario and is also a generalization of the well-known {\em Dominating Set} problem. In this problem, given a graph with three parameters defined on each vertex, namely cost, capacity, and demand, we want to find an assignment of demands to vertices of least cost such that the demand of each vertex is satisfied subject to the capacity constraint of each vertex providing the service. In terms of polynomial time approximations, we present logarithmic approximation algorithms with respect to different demand assignment models for this problem on general graphs, which also establishes the corresponding approximation results to the well-known approximations of the traditional {\em Dominating Set} problem. Together with our previous work, this closes the problem of generally approximating the optimal solution. On the other hand, from the perspective of parameterization, we prove that this problem is {\it W[1]}-hard when parameterized by a structure of the graph called treewidth. Based on this hardness result, we present exact fixed-parameter tractable algorithms when parameterized by treewidth and maximum capacity of the vertices. This algorithm is further extended to obtain pseudo-polynomial time approximation schemes for planar graphs

Differential colonization with segmented filamentous bacteria and Lactobacillus murinus do not drive divergent development of diet-induced obesity in C57BL/6 mice

Alterations in the gut microbiota have been proposed to modify the development and maintenance of obesity and its sequelae. Definition of underlying mechanisms has lagged, although the ability of commensal gut microbes to drive pathways involved in inflammation and metabolism has generated compelling, testable hypotheses. We studied C57BL/6 mice from two vendors that differ in their obesogenic response and in their colonization by specific members of the gut microbiota having well-described roles in regulating gut immune responses. We confirmed the presence of robust differences in weight gain in mice from these different vendors during high fat diet stress. However, neither specific, highly divergent members of the gut microbiota (Lactobacillus murinus, segmented filamentous bacteria) nor the horizontally transmissible gut microbiota were found to be responsible. Constitutive differences in locomotor activity were observed, however. These data underscore the importance of selecting appropriate controls in this widely used model of human obesity

Exact (exponential) algorithms for the dominating set problem

We design fast exact algorithms for the problem of computing a minimum dominating set in undirected graphs. Since this problem is NP-hard, it comes with no big surprise that all our time complexities are exponential in the number n of vertices. The contribution of this paper are ‘nice’ exponential time complexities that are bounded by functions of the form c n with reasonably small constants

Domination when the stars are out

We algorithmize the recent structural characterization for claw-free graphs by Chudnovsky and Seymour. Building on this result, we show that Dominating Set on claw-free graphs is (i) fixed-parameter tractable and (ii) even possesses a polynomial kernel. To complement these results, we establish that Dominating Set is not fixed-parameter tractable on the slightly larger class of graphs that exclude K 1,4 as an induced subgraph. Our results provide a dichotomy for Dominating Set in K 1,l-free graphs and show that the problem is fixed-parameter tractable if and only if l¿=¿3. Finally, we show that our algorithmization can also be used to show that the related Connected Dominating Set problem is fixed-parameter tractable on claw-free graphs

A structural view on parameterizing problems: distance from triviality

Abstract. Based on a series of known and new examples, we propose the generalized setting of “distance from triviality ” measurement as a reasonable and prospective way of determining useful structural problem parameters in analyzing computationally hard problems. The underlying idea is to consider tractable special cases of generally hard problems and to introduce parameters that measure the distance from these special cases. In this paper we present several case studies of distance from triviality parameterizations (concerning Clique, Power Dominating Set, Set Cover, and Longest Common Subsequence) that exhibit the versatility of this approach to develop important new views for computational complexity analysis.