Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Online Dominating Set and Independent Set   Finding minimum dominating set and maximum independent set for graphs in the
classical online setup are notorious due to their disastrous $\Omega(n)$ lower
bound of the competitive ratio that even holds for interval graphs, where $n$
is the number of vertices. In this paper, inspired by Newton number, first, we
introduce the independent kissing number $\zeta$ of a graph. We prove that the
well known online greedy algorithm for dominating set achieves optimal
competitive ratio $\zeta$ for any graph. We show that the same greedy algorithm
achieves optimal competitive ratio $\zeta$ for online maximum independent set
of a class of graphs with independent kissing number $\zeta$. For minimum
connected dominating set problem, we prove that online greedy algorithm
achieves an asymptotic competitive ratio of $2(\zeta-1)$, whereas for a family
of translated convex objects the lower bound is $\frac{2\zeta-1}{3}$. Finally,
we study the value of $\zeta$ for some specific families of geometric objects:
fixed and arbitrary oriented unit hyper-cubes in $I\!\!R^d$, congruent balls in
$I\!\!R^3$, fixed oriented unit triangles, fixed and arbitrary oriented regular
polygons in $I\!\!R^2$. For each of these families, we also present lower
bounds of the minimum connected dominating set problem.

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The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: quant-ph
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The categories of this paper are: q-alg
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ph
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The categories of this paper are: math
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The categories of this paper are: supr-con
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The categories of this paper are: hep-th