Hardness of approximation for strip packing
Abstract
Strip packing is a classical packing problem, where the goal is to pack a set of rectangular objects into a strip of a given width, while minimizing the total height of the packing. The problem has multiple applications, e.g. in scheduling and stockcutting, and has been studied extensively. When the dimensions of objects are allowed to be exponential in the total input size, it is known that the problem cannot be approximated within a factor better than $3/2$, unless $\mathrm{P}=\mathrm{NP}$. However, there was no corresponding lower bound for polynomially bounded input data. In fact, Nadiradze and Wiese [SODA 2016] have recently proposed a $(1.4 + \epsilon)$ approximation algorithm for this variant, thus showing that strip packing with polynomially bounded data can be approximated better than when exponentially large values in the input data are allowed. Their result has subsequently been improved to a $(4/3 + \epsilon)$ approximation by two independent research groups [FSTTCS 2016, arXiv:1610.04430]. This raises a question whether strip packing with polynomially bounded input data admits a quasipolynomial time approximation scheme, as is the case for related twodimensional packing problems like maximum independent set of rectangles or twodimensional knapsack. In this paper we answer this question in negative by proving that it is NPhard to approximate strip packing within a factor better than $12/11$, even when admitting only polynomially bounded input data. In particular, this shows that the strip packing problem admits no quasipolynomial time approximation scheme, unless $\mathrm{NP} \subseteq \mathrm{DTIME}(2^{\mathrm{polylog}(n)})$.
 Publication:

arXiv eprints
 Pub Date:
 October 2016
 arXiv:
 arXiv:1610.07766
 Bibcode:
 2016arXiv161007766A
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Complexity