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: v2 · 2026-09-19 · History
Domain-specific #: 3155
Domain group: Formal Sciences
Origin domain: Operations Research
Subdomain: Combinatorial Optimization → Operations Research

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.

How would you explain it like I'm…

Trios With No Repeats

Imagine making teams of three: one kid, one hat and one ball. You have a list of which trios are allowed. Every kid, hat and ball can be in only one team. The puzzle is to make as many teams as you can — and that turns out to be really hard when the lists get big.

No-Sharing Triple Picking

In 3-dimensional matching you have three separate groups, like people, rooms and time slots, and a list of allowed triples, each using one from every group. You want to pick triples so that nothing is used twice — no person, room or time slot in two picked triples. Sometimes the goal is to pick as many triples as possible; sometimes it's to cover every item exactly once. For big problems, no one knows a fast way to always find the best answer.

Disjoint Triple Matching

In 3-dimensional matching (3DM) you're given three finite, disjoint sets X, Y and Z and a set of allowed triples, each containing one element from X, one from Y and one from Z. A matching is a collection of these triples in which no two share an element in any coordinate — each element is used at most once, like an exclusive resource. You can ask for a maximum matching (as many triples as possible) or a perfect matching (every element covered exactly once). Maximum 3DM is NP-hard, meaning no efficient algorithm is known for all cases, and it is closely related to 'exact cover by 3-sets'.

An instance of 3-dimensional matching consists of finite, pairwise disjoint sets X, Y, Z and an allowed set T ⊆ X × Y × Z of triples. A matching M ⊆ T is a set of triples that are pairwise disjoint in every coordinate, encoding exclusive use of resources. The objective must be declared: maximum cardinality, maximum weight if triples carry weights, or existence of a perfect matching (often with |X| = |Y| = |Z|), with the decision version asking whether a matching of at least a given size exists. Maximum 3DM is NP-hard, and the problem is closely related to exact cover by 3-sets. Claims about the problem, especially complexity claims, depend on these specifications, so weights, balance of part sizes, the objective and any threshold should be explicit; ordinary bipartite matching is a neighboring but different problem.

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.
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) — outgoing typed relations

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

It remains its own entry because its identity is fixed by 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.

Hierarchy paths (2) — routes to 2 parentless roots

3-dimensional matching → is a kind of Two-Sided Matching → is a kind of Allocation → presupposes Scarcity → is a decomposition of Constraint

Show alternative path (1)

Neighborhood in Abstraction Space¶

3-dimensional matching sits in a crowded region of the domain-specific corpus (3^rd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Combinatorial Optimization on Graphs (16 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25