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تسجيل مستخدم جديدBalanced Independent and Dominating Sets on Colored Interval Graphs
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In this paper, we study independent domination in directed graphs, which was recently introduced by Cary, Cary, and Prabhu. We provide a short, algorithmic proof that all directed acyclic graphs contain an independent dominating set. Using linear alg
ebraic tools, we prove that any strongly connected graph with even period has at least two independent dominating sets, generalizing several of the results of Cary, Cary, and Prabhu. We prove that determining the period of the graph is not sufficient to determine the existence of an independent dominating set by constructing a few examples of infinite families of graphs. We show that the direct analogue of Vizings Conjecture does not hold for independent domination number in directed graphs by providing two infinite families of graphs. We initialize the study of time complexity for independent domination in directed graphs, proving that the existence of an independent dominating set in directed acyclic graphs and strongly connected graphs with even period are in the time complexity class $P$. We also provide an algorithm for determining existence of an independent dominating set for digraphs with period greater than $1$.
We study the {em Budgeted Dominating Set} (BDS) problem on uncertain graphs, namely, graphs with a probability distribution $p$ associated with the edges, such that an edge $e$ exists in the graph with probability $p(e)$. The input to the problem con
sists of a vertex-weighted uncertain graph $G=(V, E, p, omega)$ and an integer {em budget} (or {em solution size}) $k$, and the objective is to compute a vertex set $S$ of size $k$ that maximizes the expected total domination (or total weight) of vertices in the closed neighborhood of $S$. We refer to the problem as the {em Probabilistic Budgeted Dominating Set}~(PBDS) problem and present the following results. begin{enumerate} dnsitem We show that the PBDS problem is NP-complete even when restricted to uncertain {em trees} of diameter at most four. This is in sharp contrast with the well-known fact that the BDS problem is solvable in polynomial time in trees. We further show that PBDS is wone-hard for the budget parameter $k$, and under the {em Exponential time hypothesis} it cannot be solved in $n^{o(k)}$ time. item We show that if one is willing to settle for $(1-epsilon)$ approximation, then there exists a PTAS for PBDS on trees. Moreover, for the scenario of uniform edge-probabilities, the problem can be solved optimally in polynomial time. item We consider the parameterized complexity of the PBDS problem, and show that Uni-PBDS (where all edge probabilities are identical) is wone-hard for the parameter pathwidth. On the other hand, we show that it is FPT in the combined parameters of the budget $k$ and the treewidth. item Finally, we extend some of our parameterized results to planar and apex-minor-free graphs. end{enumerate}
In this article, we study a generalized version of the maximum independent set and minimum dominating set problems, namely, the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem on unit disk graphs for a
positive integer $d>0$. We first show that the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem belongs to NP-hard class. Next, we propose a simple polynomial-time constant-factor approximation algorithms and PTAS for both the problems.
We investigate the computational complexity of the following problem. We are given a graph in which each vertex has an initial and a target color. Each pair of adjacent vertices can swap their current colors. Our goal is to perform the minimum number
of swaps so that the current and target colors agree at each vertex. When the colors are chosen from {1,2,...,c}, we call this problem c-Colored Token Swapping since the current color of a vertex can be seen as a colored token placed on the vertex. We show that c-Colored Token Swapping is NP-complete for c = 3 even if input graphs are restricted to connected planar bipartite graphs of maximum degree 3. We then show that 2-Colored Token Swapping can be solved in polynomial time for general graphs and in linear time for trees. Besides, we show that, the problem for complete graphs is fixed-parameter tractable when parameterized by the number of colors, while it is known to be NP-complete when the number of colors is unbounded.
A unit disk graph is the intersection graph of n congruent disks in the plane. Dominating sets in unit disk graphs are widely studied due to their application in wireless ad-hoc networks. Because the minimum dominating set problem for unit disk graph
s is NP-hard, numerous approximation algorithms have been proposed in the literature, including some PTAS. However, since the proposal of a linear-time 5-approximation algorithm in 1995, the lack of efficient algorithms attaining better approximation factors has aroused attention. We introduce a linear-time O(n+m) approximation algorithm that takes the usual adjacency representation of the graph as input and outputs a 44/9-approximation. This approximation factor is also attained by a second algorithm, which takes the geometric representation of the graph as input and runs in O(n log n) time regardless of the number of edges. Additionally, we propose a 43/9-approximation which can be obtained in O(n^2 m) time given only the graphs adjacency representation. It is noteworthy that the dominating sets obtained by our algorithms are also independent sets.
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