Exponential Time Algorithms for the Minimum Dominating Set problem on Some Graph Classes

The Minimum Dominating Set problem remains NP-hard when restricted to chordal graphs, circle graphs and c-dense graphs (i.e. |E | ≥ cn 2 for a constant c, 0 < c < 1/2). For each of these three graph classes we present an exponential time algorithm solving the Minimum Dominating Set problem. The running times of those algorithms are O(1.4173 n) for chordal graphs, O(1.4956 n) for circle graphs, and O(1.2303 (1+ √ 1−2c)n)forc-dense graphs

Exponential time algorithms for the minimum dominating set problem on some graph classes

International audienceThe Minimum Dominating Set problem remains NP-hard when restricted to any of the following graph classes: $c$-dense graphs, chordal graphs, $4$-chordal graphs, weakly chordal graphs and circle graphs. Developing and using a general approach, for each of these graph classes we present an exponential time algorithm solving the Minimum Dominating Set problem faster than the best known algorithm for general graphs. Our algorithms have the following running time: $O(1.4124^n)$ for chordal graphs, $O(1.4776^n)$ for weakly chordal graphs, $O(1.4845^n)$ for $4$-chordal graphs, $O(1.4887^n)$ for circle graphs, and $O(1.2273^{(1+\sqrt{1-2c})n})$ for $c$-dense graphs

Generalized Domination in Graphs with Applications in Wireless Networks

The objective of this research is to study practical generalization of domination in graphs and explore the theoretical and computational aspects of models arising in the design of wireless networks. For the construction of a virtual backbone of a wireless ad-hoc network, two different models are proposed concerning reliability and robustness. This dissertation also considers wireless sensor placement problems with various additional constraints that reflect different real-life scenarios. In wireless ad-hoc network, a connected dominating set (CDS) can be used to serve as a virtual backbone, facilitating communication among the members in the network. Most literature focuses on creating the smallest virtual backbone without considering the distance that a message has to travel from a source until it reaches its desired destination. However, recent research shows that the chance of loss of a message in transmission increases as the distance that the message has to travel increases. We propose CDS with bounded diameter, called dominating s-club (DsC) for s ≥ 1, to model a reliable virtual backbone. An ideal virtual backbone should retain its structure after the failure of a certain number of vertices. The issue of robustness under vertex failure is considered by studying k-connected m-dominating set. We describe several structural properties that hold form ≥ k, but fail when m < k. Three different formulations based on vertex-cut inequalities are shown depending on the value of k and m. The computational results show that the proposed lazy-constraint approach compares favorably with existing methods for the minimum connected dominating set problem (for k = m = 1). The experimental results for k = m = 2, 3, 4 are presented as well. In the wireless sensor placement problem, the objective is often to place a minimum number of sensors while monitoring all sites of interest, and this can be modeled by dominating set. In some practical situations, however, there could be a location where a sensor cannot be placed because of environmental restrictions. Motivated by these practical scenarios, we introduce varieties of dominating set and the corresponding optimization problems. For these new problems, we consider the computational complexity, mathematical programming formulation, and analytical bounds on the size of structures of interest. We solve these problems using a general commercial solver and compare its performance with that of simulated annealing