Tsukiyama, S., Ide, M., Ariyoshi, H., Shirakawa, I.: A new algorithm for generating all the maximal independent sets. Tardos, E.: A strongly polynomial algorithm to solve combinatorial linear programs. 1 PDF View 2 excerpts, cites background and methods A PTAS for the minimum weight connected vertex cover P3 problem on unit disk graphs Limin Wang, Xiaoyan Zhang, Zhao Zhang, H. Marathe, M.V., Breu, H., Hunt III, H.B., Ravi, S.S., Rosenkrantz, D.S.: Simple heuristics for unit disk graphs. In this paper, constant-factor approximation algorithms for the problem with unit disk graphs and with graphs excluding a fixed minor are presented. Holt, Rinehart and Winston, New York (1976) The message complexity is linear for both the algorithms. Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Our results are distributed constant factor approximation algorithms for the MCDS problem in unit square graphs that run in 18 rounds and in unit disk graphs that run in 44 rounds. Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Hochbaum, D.S.: Efficient bounds for the stable set, vertex cover and set packing problems. 33rd IEEE Symposium on Foundations of Computer Science, pp. This process is experimental and the keywords may be updated as the learning algorithm improves.Īrora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and intractability of approximation problems. These keywords were added by machine and not by the authors. We also propose a strongly polynomial time 2-approximation algorithm for fractional coloring problem on a (general) unit disk graph. Our approach for the independent set problem implies a strongly polynomial time algorithm for the fractional coloring problem on unit disk graphs defined on a fixed width slab. The fractional coloring problem is a continuous version of the ordinary (vertex) coloring problem. We also propose an algorithm for fractional coloring problems on unit disk graphs. When the given unit disk graph is defined on a slab whose width is k, we propose an algorithm for finding a maximum independent set in \(\mathrm)\). In this paper, we consider the maximum independent set problems on unit disk graphs. Now we have solved an NP-hard problem in polynomial time, so we have a contradiction if we assume $P \neq NP$.Unit disk graphs are the intersection graphs of equal sized circles in the plane. If $M$ terminates, test if the output of $M$ is an unit disk configuration of $G$. If $M$ does not terminate in $p(n)$ steps, output NO. Now, we can solve unit disk recognition problem in polynomial time as follows: Given a graph $G$ with $n$ vertices, run $M$ with $G$ as input for $p(n)$ steps. Is there a Turing machine $M$ and a polynomial $p(n)$, such that if the input of $M$ is a unit disk graph $G$ with $n$ vertices encoded as an adjacency matrix, then $M$ terminates in at most $p(n)$ steps, and outputs an unit disk configuration of $G$? In particular, the machine $M$ is allowed to have undefined behavior if the input is not an unit disk graph encoded as adjacency matrix.Īnswer: Suppose that there is such Turing machine $M$ and polynomial $p(n)$. Approximation algorithms Dominating set Unit disk graph 1. Maybe there could be some other encodings in which only unit disk graphs could be represented, but formulating them would be another topic.)Įdit: I'll try to formalize the question and the answer more: (Here we assume that the input is some well-known encoding of a graph, and therefore the restriction that the input must be an unit disk graph doesn't really make the problem at all easier. Therefore the problem you pose is NP-hard in the sense that if it admits polynomial time algorithm, then P=NP. If there would be a polynomial time algorithm for your problem, it could be used to solve the NP-hard recognition problem in polynomial time by just giving the input to it and checking if its output is correct.
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