TY - JOUR
T1 - Hardness of approximation for strip packing
AU - Adamaszek, Anna Maria
AU - Kociumaka, Tomasz
AU - Pilipczuk, Marcin
AU - Pilipczuk, Michał
PY - 2017/9
Y1 - 2017/9
N2 - 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, for example, in scheduling and stock-cutting, and has been studied extensively. When the dimensions of the 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 P = 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 + ϵ)-approximation algorithm for this variant, thus showing that strip packing with polynomially bounded data can be approximated better than when exponentially large values are allowed in the input. Their result has subsequently been improved to a (4/3 + ϵ)-approximation by two independent research groups [FSTTCS 2016,WALCOM 2017]. This raises a questionwhether strip packing with polynomially bounded input data admits a quasi-polynomial time approximation scheme, as is the case for related twodimensional packing problems like maximum independent set of rectangles or two-dimensional knapsack. In this article, we answer this question in negative by proving that it is NP-hard to approximate strip packing within a factor better than 12/11, even when restricted to polynomially bounded input data. In particular, this shows that the strip packing problem admits no quasi-polynomial time approximation scheme, unless NP ⊆ DTIME(2polylog(n)).
AB - 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, for example, in scheduling and stock-cutting, and has been studied extensively. When the dimensions of the 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 P = 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 + ϵ)-approximation algorithm for this variant, thus showing that strip packing with polynomially bounded data can be approximated better than when exponentially large values are allowed in the input. Their result has subsequently been improved to a (4/3 + ϵ)-approximation by two independent research groups [FSTTCS 2016,WALCOM 2017]. This raises a questionwhether strip packing with polynomially bounded input data admits a quasi-polynomial time approximation scheme, as is the case for related twodimensional packing problems like maximum independent set of rectangles or two-dimensional knapsack. In this article, we answer this question in negative by proving that it is NP-hard to approximate strip packing within a factor better than 12/11, even when restricted to polynomially bounded input data. In particular, this shows that the strip packing problem admits no quasi-polynomial time approximation scheme, unless NP ⊆ DTIME(2polylog(n)).
KW - Approximation hardness
KW - Packing problems
KW - Strip packing
UR - http://www.scopus.com/inward/record.url?scp=85029784272&partnerID=8YFLogxK
U2 - 10.1145/3092026
DO - 10.1145/3092026
M3 - Journal article
AN - SCOPUS:85029784272
SN - 1942-3454
VL - 9
JO - ACM Transactions on Computation Theory
JF - ACM Transactions on Computation Theory
IS - 3
M1 - 14
ER -