Strip packing problem¶

The strip packing problem is a 2-dimensional geometric minimization problem.

Version: v1 · 2026-09-28 · History
Domain-specific #: 12320
Domain group: Formal Sciences
Origin domain: Operations Research
Subdomains: Packing Problems, Combinatorial Optimization → Operations Research

Core Idea¶

Strip packing problem is treated here as the recurring mathematics_logic_statistics identity summarized by this source-grounded definition: The strip packing problem is a 2-dimensional geometric minimization problem.

The strip packing problem is a 2-dimensional geometric minimization problem. Given a set of axis-aligned rectangles and a strip of bounded width and infinite height, determine an overlapping-free packing of the rectangles into the strip, minimizing its height. This problem is a cutting and packing problem and is classified as an Open Dimension Problem according to Wäscher et al.

This problem arises in the area of scheduling, where it models jobs that require a contiguous portion of the memory over a given time period. Another example is the area of industrial manufacturing, where rectangular pieces need to be cut out of a sheet of material (e.g., cloth or paper) that has a fixed width but infinite length, and one wants to minimize the wasted material. This problem was first studied in 1980.

For Strip packing problem, the abstraction is narrower than the article's general subject matter: a positive case must preserve The strip packing problem is a 2-dimensional geometric minimization problem. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics_logic_statistics, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — Each following level (starting at level three) is defined by a horizontal line through the top of the largest item on the previous level.
Constitutive relation — An instance I = (\mathcal{I},W) of the strip packing problem consists of a strip with width W = 1 and infinite height, as well as a set \mathcal{I} of rectangular items.
Operating condition — Especially the width of the strip is given by an arbitrary integer number larger than 1.
Recognition evidence — Geometry: In the standard variant of this problem, the set of given items consists of rectangles.
Admissible variation — However, variants have been studied where rotating by 90 degrees or even an arbitrary angle is allowed.
Characteristic consequence — One of these requirements is to be able to cut the items from the strip by horizontal or vertical edge-to-edge cuts.
Failure boundary — Furthermore, unless P = NP , there cannot be a pseudo-polynomial time algorithm that has an approximation ratio smaller than 5/4 , which can be proven by a reduction from the strongly NP-complete 3-partition problem.

What It Is Not¶

Not the whole field of mathematics_logic_statistics. The node requires the specific identity stated by The strip packing problem is a 2-dimensional geometric minimization problem.
Not an over-broad reading. When not mentioned differently, the strip packing problem is a 2-dimensional problem.
Not an over-broad reading. However, it also has been studied in three or even more dimensions.
Not an over-broad reading. Rotation: In the classical strip packing problem, the items are not allowed to be rotated.
Not automatically Smallest-Circle Problem. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Strip packing problem applies literally inside mathematics_logic_statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

Variants. However, there are applications that have explicit requirements on the structure of the packing.
Definition. This definition is used for all polynomial time algorithms.
Definition. An instance I = (\mathcal{I},W) of the strip packing problem consists of a strip with width W = 1 and infinite height, as well as a set \mathcal{I} of rectangular items.
Definition. Each item i \in \mathcal{I} has a width w_i \in (0,1] \cap \mathbb{Q} and a height h_i \in (0,1] \cap \mathbb{Q} .
Definition. A packing of the items is a mapping that maps each lower-left corner of an item i \in \mathcal{I} to a position (x_i,y_i) \in ([0,1-w_i] \cap \mathbb{Q}) \times \mathbb{Q}_{\geq 0} inside the strip.
Definition. An inner point of a placed item i \in \mathcal{I} is a point from the set \mathrm{inn}(i) = {(x,y) \in \mathbb{Q} \times \mathbb{Q}| x_i .

Outside mathematics_logic_statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

Clarity¶

A clear use of Strip packing problem names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is The strip packing problem is a 2-dimensional geometric minimization problem. The strongest recognition evidence in the frozen account is: Geometry: In the standard variant of this problem, the set of given items consists of rectangles. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification When not mentioned differently, the strip packing problem is a 2-dimensional problem. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Strip packing problem compresses multiple mathematics_logic_statistics details into a stable diagnostic relation. The source shows both the central mechanism—an instance I = (\mathcal{I},W) of the strip packing problem consists of a strip with width W = 1 and infinite height, as well as a set \mathcal{I} of rectangular items.—and the practical consequence—one of these requirements is to be able to cut the items from the strip by horizontal or vertical edge-to-edge cuts. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics_logic_statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: The strip packing problem is a 2-dimensional geometric minimization problem.
Check operation and conditions. Especially the width of the strip is given by an arbitrary integer number larger than 1.
Demand recognition evidence. Geometry: In the standard variant of this problem, the set of given items consists of rectangles.
Test variation. Change an implementation or setting while preserving however, variants have been studied where rotating by 90 degrees or even an arbitrary angle is allowed.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

Knowledge Transfer¶

Within the home domain. Knowledge about Strip packing problem transfers literally when a new case preserves the same carrier type, relation, and recognition test. However, there are applications that have explicit requirements on the structure of the packing. This definition is used for all polynomial time algorithms.

Beyond the home domain. No canonical parent is asserted for Strip packing problem. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

In an often considered subcase, all the items have to be squares. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → The strip packing problem is a 2-dimensional geometric minimization problem; recognition evidence → Geometry: In the standard variant of this problem, the set of given items consists of rectangles

Applied / In Practice¶

In the latter case, it is referred to as irregular strip packing. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Variants; invariant → The strip packing problem is a 2-dimensional geometric minimization problem; boundary → the case exits the class when when not mentioned differently, the strip packing problem is a 2-dimensional problem

Structural Tensions¶

T1 — Stable identity versus admissible variation. When not mentioned differently, the strip packing problem is a 2-dimensional problem. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. However, it also has been studied in three or even more dimensions. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. Rotation: In the classical strip packing problem, the items are not allowed to be rotated. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. However, variants have been studied where rotating by 90 degrees or even an arbitrary angle is allowed. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. Each following level (starting at level three) is defined by a horizontal line through the top of the largest item on the previous level. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Strip packing problem literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. An instance I = (\mathcal{I},W) of the strip packing problem consists of a strip with width W = 1 and infinite height, as well as a set \mathcal{I} of rectangular items. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Strip packing problem distinguish that the broader parent Pattern leaves together?

Terminal boundary synthesis. For Strip packing problem, the terminal identity test begins with the definition The strip packing problem is a 2-dimensional geometric minimization problem.. A reviewer must then establish the carrier and operation described by Each following level (starting at level three) is defined by a horizontal line through the top of the largest item on the previous level. and An instance I = (\mathcal{I},W) of the strip packing problem consists of a strip with width W = 1 and infinite height, as well as a set \mathcal{I} of rectangular items.. Recognition is constrained by Especially the width of the strip is given by an arbitrary integer number larger than 1., while admissible variation is limited by Geometry: In the standard variant of this problem, the set of given items consists of rectangles. and the collapse boundary However, variants have been studied where rotating by 90 degrees or even an arbitrary angle is allowed.. The source-domain setting in mathematics logic statistics matters because However, there are applications that have explicit requirements on the structure of the packing. and This definition is used for all polynomial time algorithms. specify where those roles have literal occupants. The strongest negative controls are The node requires the specific identity stated by The strip packing problem is a 2-dimensional geometric minimization problem. and When not mentioned differently, the strip packing problem is a 2-dimensional problem.; a case satisfying either exclusion should not be rescued merely because its label or examples look familiar.

Terminal adjudication sequence. First, bind the claimed instance to a concrete carrier and state the criterion by which The strip packing problem is a 2-dimensional geometric minimization problem. is recognized. Second, vary implementation, scale, notation, and example while holding An instance I = (\mathcal{I},W) of the strip packing problem consists of a strip with width W = 1 and infinite height, as well as a set \mathcal{I} of rectangular items. fixed; persistence supports one identity rather than several topic fragments. Third, remove Especially the width of the strip is given by an arbitrary integer number larger than 1. or trigger However, variants have been studied where rotating by 90 degrees or even an arbitrary angle is allowed. and verify that the classification fails. Fourth, compare the result with the two negative controls instead of relying on name similarity. Fifth, check scope against However, there are applications that have explicit requirements on the structure of the packing. and record any qualification supplied by mathematics logic statistics. Finally, audit the graph claim. The approved unparented placement prevents a weak lexical resemblance from becoming a false ontological claim; a later edge must preserve every constitutive role stated here. This sequence makes the entry rejectable, keeps analogy separate from literal transfer, and exposes which fact would require revision.

Counterfactual boundary matrix. Evaluate Strip packing problem under four controlled substitutions. In the carrier substitution, replace the concrete entities while retaining Each following level (starting at level three) is defined by a horizontal line through the top of the largest item on the previous level.; the identity should persist only if the new carrier has the same operative type. In the operation substitution, replace An instance I = (\mathcal{I},W) of the strip packing problem consists of a strip with width W = 1 and infinite height, as well as a set \mathcal{I} of rectangular items. while preserving surface vocabulary; the identity should fail unless the replacement entails the same relation. In the evidence substitution, change the instrument, representation, or witness used for Especially the width of the strip is given by an arbitrary integer number larger than 1.; classification may persist when the new evidence warrants the same fact. In the scope substitution, move the case outside However, there are applications that have explicit requirements on the structure of the packing. and ask whether This definition is used for all polynomial time algorithms. still gives the roles literal occupants. These four tests separate constitutive structure from implementation, evidence, and familiar examples. They also identify the exact revision needed when a source expands or narrows the recognized class.

Neighbor and residual test. The negative controls The node requires the specific identity stated by The strip packing problem is a 2-dimensional geometric minimization problem. and When not mentioned differently, the strip packing problem is a 2-dimensional problem. define two directions of possible overreach. A reviewer should construct one case that satisfies the first control but not Strip packing problem, one that satisfies Strip packing problem but not the control, and the corresponding pair for the second control. If no such asymmetric pair can be stated, the candidate may duplicate a neighbor or the distinction may depend only on wording. When the specialist identity fails but a thinner relation remains, record that residual separately instead of stretching Strip packing problem. The approved unparented placement prevents a weak lexical resemblance from becoming a false ontological claim; a later edge must preserve every constitutive role stated here. The resulting decision trail makes later DAG densification possible without treating today's uncertainty as a hierarchy fact.

Structural–Framed Character¶

Strip packing problem is structural-leaning. Its structural side is the repeatable organization summarized by The strip packing problem is a 2-dimensional geometric minimization problem. Its framed side is the mathematics_logic_statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: Especially the width of the strip is given by an arbitrary integer number larger than 1. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. The strip packing problem is a 2-dimensional geometric minimization problem. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: Each following level (starting at level three) is defined by a horizontal line through the top of the largest item on the previous level. An instance I = (\mathcal{I},W) of the strip packing problem consists of a strip with width W = 1 and infinite height, as well as a set \mathcal{I} of rectangular items. It further constrains recognition and variation through: Especially the width of the strip is given by an arbitrary integer number larger than 1. Geometry: In the standard variant of this problem, the set of given items consists of rectangles.

What is domain-bound. mathematics logic statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Strip packing problem literal. Its documented scope includes the condition that However, there are applications that have explicit requirements on the structure of the packing. Another bounded application condition is that This definition is used for all polynomial time algorithms. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—However, variants have been studied where rotating by 90 degrees or even an arbitrary angle is allowed.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a kind of Packing Problem.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Strip packing problem. The reviewed identity is: The strip packing problem is a 2-dimensional geometric minimization problem. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Strip packing problem Domain-specific

Parents (1) — more general patterns this builds on

Strip packing problem is a kind of Packing Problem Domain-specific

It minimizes used strip height under geometric nonoverlap constraints.

Hierarchy path (1) — routes to 1 parentless root

Strip packing problem → Packing Problem → Optimization

Neighborhood in Abstraction Space¶

Strip packing problem sits in a crowded region of the domain-specific corpus (38^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Combinatorial Optimization & Discrete Structures (31 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish The strip packing problem is a 2-dimensional geometric minimization problem?
Smallest-Circle Problem. The smallest-circle problem asks for the unique minimum-radius Euclidean disk containing a finite set of planar points, equivalently minimizing the maximum point-to-center distance in the two-dimensional Euclidean 1-center case. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Sphere packing. Arrange nonoverlapping equal-radius balls in a specified ambient space to maximize a declared finite or asymptotic density under explicit boundary, periodicity, and congruence conventions. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Guillotine cutting. A rectangular stock-cutting constraint in which every cut must pass straight from one edge of the current rectangular piece to the opposite edge, recursively partitioning it into two rectangles. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Strip packing problem remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics_logic_statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Strip_packing_problem (revision 1354281542).
Preserved source candidate: https://repositorio-aberto.up.pt/bitstream/10216/109367/2/234741.pdf
Preserved source candidate: https://ris.utwente.nl/ws/files/5327458/ParallelJobStripPacking.pdf
Preserved source candidate: http://doc.rero.ch/record/318381/files/10951_2006_Article_8497.pdf
Preserved source candidate: https://ris.utwente.nl/ws/files/6547720/Online_scheduling_of_parallel_jobs_on_two_machines_is_2_competitive.pdf
Preserved source candidate: https://research.utwente.nl/en/publications/a-note-on-the-lower-bound-for-online-strip-packing

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.