[Paper Review] Hardness of approximation for strip packing

Anna Adamaszek, Tomasz Kociumaka+2 more|arXiv (Cornell University)|Oct 25, 2016

Optimization and Packing Problems14 references4 citations

TL;DR

This paper establishes that strip packing is APX-hard to approximate within a factor better than 12/11, even when input sizes are polynomially bounded. The authors prove this via a reduction from the 3-Partition problem, showing that no quasi-polynomial time approximation scheme (QPTAS) exists for strip packing unless NP ⊆ DTIME(2^{polylog(n)}), resolving a key open question about its approximability in the polynomial-bounded setting.

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 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 $\mathrm{P}=\mathrm{NP}$. However, there was no corresponding lower bound for polynomially bounded input data. In fact, Nadiradze and Wiese [SODA 2016] have recently proposed a $(1.4 + ε)$ 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 $\mathrm{NP} \subseteq \mathrm{DTIME}(2^{\mathrm{polylog}(n)})$.

Motivation & Objective

• To resolve whether strip packing with polynomially bounded input data admits a quasi-polynomial time approximation scheme (QPTAS), as seen in related problems like maximum independent set of rectangles.
• To close the gap between the known 3/2-inapproximability bound for exponentially large inputs and the improved (4/3+ε) approximation for polynomially bounded inputs.
• To establish a strong inapproximability result for strip packing under polynomially bounded input sizes, showing it is fundamentally harder than some related geometric packing problems.

Proposed method

• Reduction from the strongly NP-hard 3-Partition problem to construct a strip packing instance with specific rectangle dimensions and constraints.
• Design of side rectangles of widths a, b, and a+b, and middle rectangles of width b/3 + s_i, where s_i are small perturbations, to encode the 3-Partition instance.
• Use of geometric constraints: every vertical line in general position intersects exactly two side rectangles and one middle rectangle, enforcing structural consistency in any valid packing.
• Proof that any valid packing must group middle rectangles into consecutive triples of total width exactly b, which corresponds to a valid 3-Partition solution.
• Leveraging number-theoretic properties of coefficients a and b (with gcd(a,b)=1 and b>3a) to enforce that x-coordinates of rectangle boundaries are linear combinations an_a + bn_b with n_b - n_a ∈ {0,1}.
• Establishing that the only feasible configuration of middle rectangles is a partition into triples of width b, which implies a solution to the original 3-Partition instance.

Experimental results

Research questions

• RQ1 Can strip packing with polynomially bounded input data be approximated within a factor better than 12/11?
• RQ2 Does the existence of (4/3+ε)-approximation algorithms for polynomially bounded inputs imply the existence of a quasi-polynomial time approximation scheme (QPTAS)?
• RQ3 Is strip packing APX-hard under polynomially bounded input sizes, similar to other geometric packing problems?
• RQ4 Can the structure of valid packings be constrained such that middle rectangles must form triples of total width b, encoding a 3-Partition solution?
• RQ5 Does the inapproximability result for strip packing differ fundamentally from that of related problems like maximum independent set of rectangles or two-dimensional knapsack?

Key findings

• The strip packing problem is APX-hard to approximate within a factor better than 12/11, even when all input values are polynomially bounded in the number of items.
• This implies that no quasi-polynomial time approximation scheme (QPTAS) exists for strip packing unless NP ⊆ DTIME(2^{polylog(n)}), which is considered highly unlikely.
• The inapproximability result is established via a reduction from the 3-Partition problem, which is strongly NP-hard and has no pseudo-polynomial time algorithm unless P=NP.
• Any valid packing of the constructed instance must partition the middle rectangles into consecutive triples, each of total width exactly b, which corresponds exactly to a solution of the 3-Partition instance.
• The geometric constraints and number-theoretic properties of the rectangle dimensions ensure that only configurations consistent with a valid 3-Partition can yield a valid packing.
• The result shows that strip packing behaves differently from related problems like maximum independent set of rectangles and two-dimensional knapsack, which admit QPTAS, by being strictly harder to approximate under polynomially bounded inputs.