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Computer Science, Data Structures and Algorithms, cs.DS,Computer Science, Computational Geometry, cs.CG
Abstract:
Consider the Maximum Weight Independent Set problem for rectangles: given a
family of weighted axis-parallel rectangles in the plane, find a maximum-weight
subset of non-overlapping rectangles. The problem is notoriously hard both in
the approximation and in the parameterized setting. The best known
polynomial-time approximation algorithms achieve super-constant approximation
ratios [Chalermsook and Chuzhoy, SODA 2009; Chan and Har-Peled, Discrete &
Comp. Geometry 2012], even though there is a $(1+\epsilon)$-approximation
running in quasi-polynomial time [Adamaszek and Wiese, FOCS 2013; Chuzhoy and
Ene, FOCS 2016]. When parameterized by the target size of the solution, the
problem is $\mathsf{W}[1]$-hard even in the unweighted setting [Marx, FOCS
2007].
To achieve tractability, we study the following shrinking model: one is
allowed to shrink each input rectangle by a multiplicative factor $1-\delta$
for some fixed $\delta>0$, but the performance is still compared against the
optimal solution for the original, non-shrunk instance. We prove that in this
regime, the problem admits an EPTAS with running time $f(\epsilon,\delta)\cdot
n^{\mathcal{O}(1)}$, and an FPT algorithm with running time $f(k,\delta)\cdot
n^{\mathcal{O}(1)}$, in the setting where a maximum-weight solution of size at
most $k$ is to be computed. This improves and significantly simplifies a PTAS
given earlier for this problem [Adamaszek et al., APPROX 2015], and provides
the first parameterized results for the shrinking model. Furthermore, we
explore kernelization in the shrinking model, by giving efficient kernelization
procedures for several variants of the problem when the input rectangles are
squares.