Moving intervals for packing and covering

TL;DR

This paper investigates the computational complexity of geometric packing and covering problems involving intervals with movement constraints, establishing hardness results and developing fixed-parameter tractable algorithms, including efficient solutions for uniform-length intervals.

Contribution

It proves W[1]-hardness for packing and covering with movement constraints and provides fixed-parameter algorithms based on combined parameters, along with improved polynomial algorithms for uniform intervals.

Findings

Packing and covering are W[1]-hard with respect to any single parameter among $ ext{kappa}$, $ au$, and $ ext{sigma}$.

The problems are fixed-parameter tractable when combining parameters $ ext{kappa}$ and $ au$.

An $O(n ext{log}^2 n)$ algorithm is developed for covering with uniform-length intervals.

Abstract

We study several problems on geometric packing and covering with movement. Given a family $\mathcal{I}$ of $n$ intervals of $\kappa$ 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 $\tau$ intervals and keeping the other $\sigma = n - \tau$ intervals unmoved? We show that both packing and covering are W[1]-hard with any one of $\kappa$, $\tau$, and $\sigma$ as single parameter, but are FPT with combined parameters $\kappa$ and $\tau$. 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.