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Approximation Algorithms for Connected Dominating Sets
 Algorithmica
, 1996
"... The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to either be in the dominating set, or adjacent to some node in the dominating set. We focus on the question of finding a connected dominating set of minimum size, whe ..."
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Cited by 277 (9 self)
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The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to either be in the dominating set, or adjacent to some node in the dominating set. We focus on the question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of O(H (\Delta)) are presented, where \Delta is the maximum degree, and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited, or has at least one of its neighbors visited. We study a generalization of the problem when the vertices have weights, and give an algorithm which achieves a performance ratio of 3 ln n. We also consider the ...
In Some Curved Spaces, One Can Solve NPHard Problems in Polynomial Time
"... In the late 1970s and the early 1980s, Yuri Matiyasevich actively used his knowledge of engineering and physical phenomena to come up with parallelized schemes for solving NPhard problems in polynomial time. In this paper, we describe one such scheme in which we use parallel computation in curved s ..."
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Cited by 6 (6 self)
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In the late 1970s and the early 1980s, Yuri Matiyasevich actively used his knowledge of engineering and physical phenomena to come up with parallelized schemes for solving NPhard problems in polynomial time. In this paper, we describe one such scheme in which we use parallel computation in curved spaces. 1 Introduction and Formulation of the Problem Many practical problems are NPhard. It is well known that many important practical problems are NPhard; see, e.g., [11, 14, 27]. Under the usual hypothesis that P̸=NP, NPhardness has the following intuitive meaning: every algorithm which solves all instances of the corresponding problem requires, for
On the Complexity of Optimal Reconfiguration Planning for Modular Reconfigurable Robots
"... Abstract — This paper presents a thorough analysis of the computational complexity of optimal reconfiguration planning problem for chaintype modular robots, i.e. finding the least number of reconfiguration steps to transform from the initial configuration into the goal configuration. It establishes ..."
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Cited by 1 (0 self)
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Abstract — This paper presents a thorough analysis of the computational complexity of optimal reconfiguration planning problem for chaintype modular robots, i.e. finding the least number of reconfiguration steps to transform from the initial configuration into the goal configuration. It establishes a formal proof that this problem is NPcomplete, even if the configurations are acyclic. This result gives a compelling reason that a polynomial algorithm for optimal reconfiguration plan is unlikely to exist. To facilitate future evaluation of reconfiguration algorithms, the paper also provides the lower and the upper bounds for the minimum number of reconfiguration steps for any given reconfiguration problem. I.