[Rule] 3-DIMENSIONAL MATCHING to 3-MATROID INTERSECTION

[isPANN]

Source: 3-DIMENSIONAL MATCHING (3DM)
Target: 3-MATROID INTERSECTION

Motivation: Establishes NP-completeness of 3-matroid intersection by exhibiting 3DM as a canonical special case in which each of the three matroids is a partition matroid over one coordinate of the triples. The reduction is the textbook witness that, while 2-matroid intersection is polynomial-time solvable (Edmonds), adding a third matroid jumps to NP-hardness — and it does so already for the simplest combinatorial matroids (partitions). This makes 3-matroid intersection the natural matroid-theoretic home for 3DM, hypergraph matching, and related "triple packing" problems.
Reference: Garey & Johnson, Computers and Intractability, SP11, p. 223; Lawler, Combinatorial Optimization: Networks and Matroids (1976), Ch. 9.

GJ Source Entry

[SP11] 3-MATROID INTERSECTION
INSTANCE: Three matroids M_1 = (E, I_1), M_2 = (E, I_2), M_3 = (E, I_3) on a common finite ground set E, each specified by an independence oracle (or equivalent compact description), positive integer K ≤ |E|.
QUESTION: Is there a subset X ⊆ E with |X| ≥ K such that X ∈ I_1 ∩ I_2 ∩ I_3?

Reference: [Lawler, 1976]. Transformation from 3-DIMENSIONAL MATCHING.
Comment: Polynomial for k = 2 matroids (Edmonds' matroid intersection theorem). NP-complete already when all three matroids are partition matroids.

Reduction Algorithm

Summary:
Given a 3DM instance with disjoint sets X, Y, Z of size q each and a collection T ⊆ X × Y × Z of triples, construct a 3-MATROID INTERSECTION instance (E, M_1, M_2, M_3, K) as follows:

1. Ground set: E := T. Each triple t = (x, y, z) becomes one ground-set element.

2. Partition matroid M_1 (X-coordinate): Partition E by first coordinate. For each x ∈ X let block B^X_x := { t ∈ T : t = (x, ·, ·) }. Declare S ⊆ E independent in M_1 iff |S ∩ B^X_x| ≤ 1 for every x ∈ X. (Equivalently: no two chosen triples share their X-element.)

3. Partition matroid M_2 (Y-coordinate): Same construction with blocks B^Y_y := { t ∈ T : t = (·, y, ·) }.

4. Partition matroid M_3 (Z-coordinate): Same construction with blocks B^Z_z := { t ∈ T : t = (·, ·, z) }.

5. Threshold: K := q.

All three matroids are partition matroids with per-block capacity 1, so each can be described in O(|T|) space and admits a constant-time independence oracle.

Correctness:

• (⇒) If M ⊆ T is a 3DM matching of size q, then |M| = q and M uses each X-, Y-, Z-element at most once. Hence M is independent in M_1, M_2, M_3, giving a common independent set of size K = q.
• (⇐) If X ⊆ E is common independent of size ≥ K = q in all three matroids, no two of its triples share any coordinate. So |X| triples cover q distinct X-elements, q distinct Y-elements, q distinct Z-elements; since |X| = |Y| = |Z| = q, they cover all of X ∪ Y ∪ Z. Therefore X is a perfect 3-dimensional matching.

The transformation is computable in O(|T|) time and is many-one (witness-preserving): the selected triples are literally the matching.

Size Overhead

Symbols:

• q = |X| = |Y| = |Z| (each side of the 3DM instance)
• m = |T| = num_triples (number of input triples)

Target metric (code name)	Polynomial (using symbols above)
ground_size (= |E|)	m
num_matroids	3
k (independence threshold)	q

Derivation:

• Ground set is exactly the set of triples: |E| = |T| = m.
• Each matroid is the partition matroid associated with one of the three coordinate sides; it has q blocks of total size m and uniform capacity 1, so its description is O(m + q) = O(m).
• Threshold K equals the side length q of the 3DM instance (a perfect matching has exactly q triples).
• Overall construction: O(m) time, O(m) space, 3 matroids — strictly linear overhead.

Validation Method

• Closed-loop test (YES instance): Take any 3DM instance with a known perfect matching, run the reduction, solve 3-MATROID INTERSECTION with brute force over subsets of E, confirm the optimum equals q and the recovered set is exactly the matching.
• Closed-loop test (NO instance): Build a 3DM instance that admits no perfect matching (e.g. force some Y-element to appear in no triple, or force two X-elements to share their only triple); confirm the maximum common independent size is < q.
• Oracle sanity check: For each matroid, randomly sample subsets of E, verify that the independence oracle accepts iff the projection onto its coordinate is injective.
• Boundary cases: q = 1 (one triple per side, trivial YES); q = 2 with two triples covering disjoint coordinates (YES) vs. two triples sharing a coordinate (NO).

Example

Source instance (3DM):
X = {x_1, x_2}, Y = {y_1, y_2}, Z = {z_1, z_2} (q = 2). Triples:

• t_1 = (x_1, y_1, z_1)
• t_2 = (x_2, y_2, z_2)
• t_3 = (x_1, y_2, z_2)
• t_4 = (x_2, y_1, z_1)

This 3DM instance has two perfect matchings: {t_1, t_2} and {t_3, t_4}.

Constructed target instance (3-MATROID INTERSECTION):
Ground set E = {t_1, t_2, t_3, t_4}.

M_1 (partition by X-coordinate):

• block B^X_{x_1} = {t_1, t_3}, capacity 1
• block B^X_{x_2} = {t_2, t_4}, capacity 1

M_2 (partition by Y-coordinate):

• block B^Y_{y_1} = {t_1, t_4}, capacity 1
• block B^Y_{y_2} = {t_2, t_3}, capacity 1

M_3 (partition by Z-coordinate):

• block B^Z_{z_1} = {t_1, t_4}, capacity 1
• block B^Z_{z_2} = {t_2, t_3}, capacity 1

Threshold K = 2.

Verification:

• X* = {t_1, t_2}: X-projection {x_1, x_2} (M_1 ✓), Y-projection {y_1, y_2} (M_2 ✓), Z-projection {z_1, z_2} (M_3 ✓), |X*| = 2 ≥ K. YES.
• X' = {t_1, t_3}: both have X-coordinate x_1, violates M_1. Rejected.
• X'' = {t_1, t_4}: share y_1 and z_1, violates M_2 and M_3. Rejected.

So the 3-matroid intersection optimum is 2, matching the 3DM perfect matching size.

References

• [Lawler, 1976] E. L. Lawler. Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, 1976. Chapter 9 (matroid intersection) — establishes NP-hardness of 3-matroid intersection via the partition-matroid encoding of 3DM.
• [Garey & Johnson, 1979] M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Problem SP11, p. 223.
• [Edmonds, 1970] J. Edmonds. "Submodular functions, matroids, and certain polyhedra." In Combinatorial Structures and Their Applications, pp. 69–87, 1970. Polynomial-time algorithm for 2-matroid intersection (context for why 3-matroid intersection is the natural hardness threshold).

Specialization Note

This rule's source problem (3-DIMENSIONAL MATCHING / 3DM) is a specialization of SET PACKING (MaximumSetPacking) restricted to 3-element sets from three disjoint universes. Implementation should wait until 3DM is available as a codebase model.

[isPANN]

auto-pipeline: parked on OnHold — issue is a placeholder/Specialization Note with no Reduction Algorithm section (Motivation skipped pending general-source implementation). Human triage needed.

[isPANN]

Body refresh

• Replaced placeholder Specialization Note with researched reduction algorithm, size overhead, validation method, example, and references for G&J SP11.
• Specialization-note footer retained: 3DM still blocked on its codebase model.
• Moved OnHold → Backlog.

Applied by /fix-issue-style refresh subagent.

[isPANN]

Round-3 Web Verification — VERIFIED

Conducted independent web literature research (not trusting the in-repo cache or round-1 self-rating). The reduction body matches the construction in primary and authoritative sources.

Construction verified against sources

The body's reduction (E = T; three partition matroids M_1, M_2, M_3, one per coordinate, with per-block capacity 1; K = q) is exactly the textbook construction:

• arXiv 2412.02217v2 (Doron-Arad, Hassidim, Kulik, 2024), "You (Almost) Can't Beat Brute Force for 3-Matroid Intersection" — explicitly describes the 3DM → 3-MI reduction: ground set is the triplet set T ⊆ X_1 × X_2 × X_3, three partition matroids defined by parts T^i_a = "all triplets with a in the i-th entry," each block capacity 1, and "a common independent set in the three matroids is a matching." The cardinality threshold for a perfect matching is |X_1|. Confirms our K = q and the partition-matroid encoding.
• Wikipedia, Matroid_intersection — confirms NP-hardness at three matroids and that 3DM is a canonical NP-complete special case; the page's worked detail uses the Hamiltonian-path encoding (two partition matroids + graphic matroid) rather than the 3DM encoding, but both are standard.
• Wikipedia, Partition_matroid — confirms the formal definition: blocks C_i with capacity d_i, independent iff |I ∩ C_i| ≤ d_i for every block. Matches the M_1/M_2/M_3 in the body.
• G&J SP11 (R77_3dm_threematroidintersection.json) — independence-oracle formulation; "transformation from 3DM"; comment notes 2-matroid intersection is polynomial via Edmonds/Lawler.

Cache-vs-body diff

• G&J phrases the question as |E'|=K; body uses |X| ≥ K. These are equivalent decision problems by hereditariness of matroid independence (any independent superset can shrink to exactly K). No correction needed.

Example sanity check

Ran the round-trip on the body's worked example (q=2, four triples): brute force over subsets of size 2 yields exactly the two common independent sets {t_1, t_2} and {t_3, t_4}, which are the two perfect 3-matchings. Sets that violate any coordinate constraint (e.g. {t_1, t_3} sharing x_1) are rejected. Correctness, threshold, and the partition encoding all verify.

Verified URLs

• https://arxiv.org/html/2412.02217v2 (3DM → 3-MI construction in preliminaries)
• https://arxiv.org/abs/2412.02217 (abstract: "3-Dimensional Matching … notable canonical NP-complete special cases")
• https://en.wikipedia.org/wiki/Matroid_intersection (NP-hardness at three matroids)
• https://en.wikipedia.org/wiki/Partition_matroid (formal block/capacity definition)
• https://kunigami.blog/2013/11/11/lawler-and-an-introduction-to-matroids/ (Lawler, Combinatorial Optimization: Networks and Matroids, 1976 attribution)

Residual uncertainty

• Lawler 1976 (the cited textbook) is not freely available online; could not verbatim verify the page where Lawler first published the partition-matroid 3DM construction. Attribution is consistent across all secondary sources (Schrijver Ch. 41, Wikipedia, Doron-Arad et al. 2024).
• Wikipedia's worked NP-hardness derivation uses Hamiltonian Path rather than 3DM, but the Doron-Arad arXiv paper presents the 3DM partition-matroid encoding explicitly, which is what G&J SP11 cites.

No body rewrite required.

[isPANN]

Issue Quality Check — Rule (Re-check, --force)

Check	Result	Details
Usefulness	❌ Fail	Direct edge ThreeDimensionalMatching -> ThreeMatroidIntersection already exists in the codebase (PR #857) with identical construction and identical overhead
Non-trivial	✅ Pass	Genuine structural transformation: builds three partition matroids from coordinate projections
Correctness	✅ Pass	Construction matches G&J SP11, Lawler 1976, and Doron-Arad et al. (arXiv:2412.02217); per-coordinate partition matroid with block-capacity 1
Completeness	✅ Pass	Literature explicitly covers all 3DM instances; hand-traced corner cases (empty T, q=1, duplicate triples) all behave correctly
Well-written	✅ Pass	All template sections present, algorithm is implementable step-by-step, overhead symbols are defined and match pred show size fields

Overall: 4 passed, 1 failed

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Usefulness — Fail

The proposed reduction already exists as a primitive direct edge in the reduction graph. pred path ThreeDimensionalMatching ThreeMatroidIntersection returns a 1-step path with overhead

ground_set_size = num_triples
num_groups      = 3 * universe_size
bound           = universe_size

which is byte-for-byte identical to the overhead the issue proposes (|E| = m, num_matroids = 3 (i.e. 3 blocks-of-blocks of total count 3q), K = q). The implementation lives in src/rules/threedimensionalmatching_threematroidintersection.rs and was merged via PR #857 (commit c2e60cb4). The construction in that file is the same partition-matroid encoding the issue describes (three group-lists keyed by each of the three coordinates), and extract_solution is the identity, matching the witness-preserving claim.

Because this is a duplicate of an existing direct edge with the same overhead, there is nothing to add to the reduction graph. The redundancy-style criterion in this skill ("new rule must reduce the overhead of the existing path") is not met — the proposed rule equals, rather than improves on, what is already merged.

Suggested resolution: close the issue as already-done and link to PR #857, or repurpose it (e.g. as a documentation/paper task if the rule is missing a reduction-rule entry in docs/paper/reductions.typ).

Non-trivial — Pass

The mapping is not a relabeling or subtype coercion: it constructs three new combinatorial objects (partition matroids) over the ground set E = T, each derived from a coordinate projection. Independence is defined by per-block capacity constraints, which is a genuine structural transformation.

Correctness — Pass

Literature verification:

• Garey & Johnson 1979, SP11 (p. 223) — lists 3-MATROID INTERSECTION as NP-complete with reference to Lawler (1976) and the transformation from 3-DIMENSIONAL MATCHING. The decision form (|X| ≥ K) is equivalent to G&J's |X| = K by matroid hereditariness.
• Lawler 1976, Combinatorial Optimization: Networks and Matroids, Ch. 9 — original publication of this partition-matroid encoding. The cited textbook is the standard attribution; not freely available online but consistent across all secondary sources.
• Doron-Arad, Hassidim, Kulik (arXiv:2412.02217v2, 2024) — explicitly writes the 3DM → 3-MI reduction using ground set = triples, three partition matroids one per coordinate, per-block capacity 1, threshold |X_1|. This matches the issue's construction exactly.
• Edmonds 1970 — cited only as context (2-matroid intersection is polynomial); claim is standard textbook fact.

No claim in the issue body contradicts the literature. The (⇒)/(⇐) proof sketch is correct: common independence in M_1 ∩ M_2 ∩ M_3 is exactly the constraint that no two chosen triples share any coordinate, which characterizes a (partial) 3DM matching.

Completeness — Pass

Literature evidence. Doron-Arad et al. (2024) state the construction for "any 3DM instance," with no precondition; G&J SP11 similarly. There is no hidden restriction in the source (e.g. distinct triples, perfect-matchability) — the reduction is many-one for every 3DM instance.

Codebase corner-case research. pred show ThreeDimensionalMatching shows the source has fields universe_size: usize and triples: Vec<(usize, usize, usize)>, so the source model admits: empty triple sets, repeated triples, triples that share coordinates, and universe_size = 0.

Traced corner cases against the issue's algorithm:

1. Empty triple set (T = ∅, q = 3). E = ∅; each matroid has 3 empty blocks with capacity 1; K = 3. Only common independent set is ∅, size 0 < K → target NO. Source has no perfect matching (no triples at all). Consistent.
2. q = 1 with one triple (x_1, y_1, z_1). E = {t}; each matroid has the singleton block {t} with capacity 1; K = 1. {t} is common independent → target YES. Source's perfect matching is {t}. Consistent.
3. Duplicate triples (e.g. T = [(0,0,0), (0,0,0)], q = 1). Each occurrence becomes a distinct ground-set element; both fall into the same block in every matroid, so the common independence cap is 1; K = 1 → YES. Source still has a perfect matching {(0,0,0)}. Consistent.

Existing src/rules/threedimensionalmatching_threematroidintersection.rs handles these corner cases the same way (iterates triples by index and pushes into per-coordinate blocks).

Well-written — Pass

All template sections are present and substantive: Source, Target, Motivation, Reference, Reduction Algorithm (with numbered steps and (⇒)/(⇐) proof), Size Overhead (with symbol legend and a per-metric table), Validation Method (YES/NO closed loop, oracle sanity, boundary cases), Example (q=2, four triples, two distinct perfect matchings — has two suboptimal feasible subsets that the verification block walks through), References.

Symbol consistency:

• q = |X| = |Y| = |Z|, m = |T|, and K are all defined in the Symbols block before first use.
• Target metric names (ground_set_size, num_groups, bound, num_matroids) match pred show ThreeMatroidIntersection --json size fields (ground_set_size, bound, num_groups).
• Note: the table uses num_matroids = 3 as a separate line — in the registered overhead this is folded into num_groups = 3 * universe_size (i.e. 3 matroids × q blocks each). This is presentational, not a bug.

Example complexity: the worked instance has two distinct perfect matchings ({t_1, t_2} and {t_3, t_4}) plus at least two infeasible subsets shown ({t_1, t_3}, {t_1, t_4}), satisfying the round-trip test rule of thumb (more than two suboptimal/infeasible feasible solutions in addition to the optimum).

Recommendations

• Close as duplicate of PR [Rule] 3-DIMENSIONAL MATCHING to 3-MATROID INTERSECTION #857, or convert into a follow-up task (e.g. ensure the existing rule has a paper reduction-rule entry in docs/paper/reductions.typ and a canonical example in src/example_db/rule_builders.rs — a quick check shows the canonical example is already in place at src/rules/threedimensionalmatching_threematroidintersection.rs lines 57–76).
• The "Specialization Note" footer about 3DM blocking on its codebase model is stale: ThreeDimensionalMatching is already in src/models/set/ and is fully reduceable via the existing rule. The note can be removed.