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Register a new userOn Continuity Properties for Infinite Rectangle Packing
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Added by
Zhiheng Liu
Publication date
2017
fields
Informatics Engineering
and research's language is
English
Authors
Zhiheng Liu
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By rectangle packing we mean putting a set of rectangles into an enclosing rectangle, without any overlapping. We begin with perfect rectangle packing problems, then prove two continuity properties for parallel rectangle packing problems, and discuss how they might be used to obtain negative results for perfect rectangle packing problems.
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In the $d$-dimensional hypercube bin packing problem, a given list of $d$-dimensional hypercubes must be packed into the smallest number of hypercube bins. Epstein and van Stee [SIAM J. Comput. 35 (2005)] showed that the asymptotic performance ratio $rho$ of the online bounded space variant is $Omega(log d)$ and $O(d/log d)$, and conjectured that it is $Theta(log d)$. We show that $rho$ is in fact $Theta(d/log d)$, using probabilistic arguments.
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Let $G$ be a graph of order $n(G)$ and vertex set $V(G)$. Given a set $Ssubseteq V(G)$, we define the external neighbourhood of $S$ as the set $N_e(S)$ of all vertices in $V(G)setminus S$ having at least one neighbour in $S$. The differential of $S$ is defined to be $partial(S)=|N_e(S)|-|S|$. In this paper, we introduce the study of the $2$-packing differential of a graph, which we define as $partial_{2p}(G)=max{partial(S): Ssubseteq V(G) text{ is a }2text{-packing}}.$ We show that the $2$-packing differential is closely related to several graph parameters, including the packing number, the independent domination number, the total domination number, the perfect differential, and the unique response Roman domination number. In particular, we show that the theory of $2$-packing differentials is an appropriate framework to investigate the unique response Roman domination number of a graph without the use of functions. Among other results, we obtain a Gallai-type theorem, which states that $partial_{2p}(G)+mu_{_R}(G)=n(G)$, where $mu_{_R}(G)$ denotes the unique response Roman domination number of $G$. As a consequence of the study, we derive several combinatorial results on $mu_{_R}(G)$, and we show that the problem of finding this parameter is NP-hard. In addition, the particular case of lexicographic product graphs is discussed.
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