Moving intervals for packing and covering

09/01/2021

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by   Rain Jiang, et al.

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We study several problems on geometric packing and covering with movement. Given a family ℐ of n intervals of κ distinct lengths, and another interval B, can we pack the intervals in ℐ inside B (respectively, cover B by the intervals in 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(nlog^2 n) time algorithm for covering, when all intervals in ℐ have the same length.