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arXiv · 2109.00579

Moving intervals for packing and covering

Rain Jiang·Kai Jiang·Minghui Jiang
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Abstract

We study several problems on geometric packing and covering with movement. Given a family $\mathcal{I}$ of $n$ intervals of $κ$ distinct lengths, and another interval $B$, can we pack the intervals in $\mathcal{I}$ inside $B$ (respectively, cover $B$ by the intervals in $\mathcal{I}$) by moving $τ$ intervals and keeping the other $σ= n - τ$ intervals unmoved? We show that both packing and covering are W[1]-hard with any one of $κ$, $τ$, and $σ$ as single parameter, but are FPT with combined parameters $κ$ and $τ$. We also obtain improved polynomial-time algorithms for packing and covering, including an $O(n\log^2 n)$ time algorithm for covering, when all intervals in $\mathcal{I}$ have the same length.

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Rain Jiang, Kai Jiang, Minghui Jiang. 2021-09-01. Moving intervals for packing and covering. https://arxiv.org/abs/2109.00579

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