3-dimensional matching¶

A matching in a tripartite 3-uniform hypergraph: a set of triples no two of which share any coordinate, with optimization and perfect-decision variants.

Version: v1 · 2026-09-08 · History
Domain-specific #: 3155
Origin domain: combinatorial optimization
Subdomain: combinatorial optimization

Core Idea¶

Maximum 3DM is NP-hard and exact cover by 3-sets is closely related; weights, balanced part sizes and whether the goal is maximum cardinality or a perfect matching must be explicit. Candidate triples connect one element from each of three sets, pairwise coordinate-disjointness enforces resource exclusivity and search selects the largest or a covering subfamily. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application¶

3-dimensional matching belongs to combinatorial optimization and is useful where the analyst can specify the typed combinatorial optimization carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the finite disjoint sets X, Y and Z, allowed triple set, matching subset, no-shared-coordinate constraint, maximum or perfect objective, weights if any, decision threshold and complexity claim are explicit. The scope is broad within that domain but bounded by the need for the finite disjoint sets X, Y and Z, allowed triple set, matching subset, no-shared-coordinate constraint, maximum or perfect objective, weights if any, decision threshold and complexity claim are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the finite disjoint sets X, Y and Z, allowed triple set, matching subset, no-shared-coordinate constraint, maximum or perfect objective, weights if any, decision threshold and complexity claim are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to 3-dimensional matching. 3-dimensional matching compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed combinatorial optimization carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the finite disjoint sets X, Y and Z, allowed triple set, matching subset, no-shared-coordinate constraint, maximum or perfect objective, weights if any, decision threshold and complexity claim are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of combinatorial optimization because they reuse the typed combinatorial optimization carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Candidate triples connect one element from each of three sets, pairwise coordinate-disjointness enforces resource exclusivity and search selects the largest or a covering subfamily., and type the carrier, state every parameter and convention in the definition, test that the finite disjoint sets X, Y and Z, allowed triple set, matching subset, no-shared-coordinate constraint, maximum or perfect objective, weights if any, decision threshold and complexity claim are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction 3-dimensional matching Domain-specific

Parents (1) — more general patterns this builds on

3-dimensional matching is a kind of Two-Sided Matching Prime

The proposed strict upward parent is prime:two_sided_matching.

Hierarchy paths (2) — routes to 2 parentless roots

3-dimensional matching → Two-Sided Matching → Allocation → Scarcity → Constraint

Show alternative path (1)

Neighborhood in Abstraction Space¶

3-dimensional matching sits in a crowded region of the domain-specific corpus (5^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 & Network Flows (24 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08