2016/10/25 by Anna Adamaszek, Tomasz Kociumaka, Adamaszek, Anna +5
Computer Science · Engineering · #Computational Complexity (cs.CC) #Computational Geometry and Mesh Generation #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #Optimization and Packing Problems #Optimization and Search Problems
paper · pdf · doi:10.48550/arxiv.1610.07766
openalex publication_date 2016/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 stock-cutting, 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 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 in the input data are allowed. Their result has subsequently been improved to a (4/3 + ε) 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 quasi-polynomial time approximation scheme, as is the case for related two-dimensional packing problems like maximum independent set of rectangles or two-dimensional knapsack. In this paper 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 admitting only 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)).