[Rule] BinPacking (3-Partition) to MinimumGraphBandwidth

[QingyunQian]

Source: BinPacking (3-Partition special case: capacity = A, constraint A/4 < aᵢ < A/2)
Target: MinimumGraphBandwidth
Motivation: Landmark reduction establishing NP-completeness of graph bandwidth even for trees with maximum degree 3. This is one of the strongest hardness results in graph layout theory — the problem remains hard on the simplest possible tree structure. The reduction also demonstrates strong NP-completeness: hardness persists even when all input numbers are polynomially bounded.
Reference:

• Garey, Graham, Johnson & Knuth (1978). "Complexity results for bandwidth minimization." SIAM J. Appl. Math., 34(3), 477–495. — Section 8.
• Garey & Johnson (1979). Computers and Intractability. Problem GT40.

Reduction Algorithm

Source problem

A 3-Partition instance: integers a₁, ..., a₃ₙ with A/4 < aᵢ < A/2 and Σaᵢ = nA. The question: can these be partitioned into n triples, each summing to A? (The size constraint forces exactly 3 items per triple.)

In this codebase, 3-Partition is a special case of BinPacking with capacity = A.

Parameters

Given the source instance, define:

• k = smallest power of 2 ≥ 6nA + 9n − 5 (the bandwidth bound)
• d = k/2
• mⱼ = 6dn · aⱼ (tentacle size encoding integer aⱼ)
• c = 12dnA (channel capacity constant)

The "Siphonophore" construction

The constructed graph G is a tree with max degree 3, built from five components. The name comes from Garey et al.: "The free trees we shall construct bear more resemblance to pelagic hydrozoa of the order Siphonophora than to actual trees."

1. Stem — A path of 4dn + 6d + 1 vertices. Every d-th vertex is a boundary vertex. The 2n + 4 boundary vertices are labeled:

b₀, h₁, b₁, p₁, b₂, f₁, b₃, ..., pₙ, b₂ₙ, fₙ, b₂ₙ₊₁, h₂, b₂ₙ₊₂, h₃, b₂ₙ₊₃

2. Heads (H₁, H₂, H₃) — Three subtrees, each with 2dk − 1 vertices, attached at h₁, h₂, h₃. Each head is a recursively constructed binary-tree-like structure where every vertex is within distance d of its center. Heads act as rigid anchors: in any bandwidth-≤-k layout, H₁ occupies the leftmost positions and H₃ the rightmost, pinning all other components to the interior.

3. Polyps (Pᵢ, Fᵢ) — Between the heads, n P-polyps and n F-polyps fill the interior. Each starts as a full head, then has vertices removed to create channels — parallel slots through which tentacles pass. Polyp sizes:

• Fᵢ: 2dk − 1 − 6di vertices (channels admit filaments only)
• Pᵢ: 2dk − 1 − c − 18di + 12d vertices (channels admit nematocysts)

4. Tentacles — Each integer aⱼ becomes a tentacle attached at H₂:

• Filament: a path of 4dn vertices
• Nematocysts: 2mⱼ = 12dn · aⱼ vertices at the end

Tentacle length is proportional to aⱼ, encoding the integer as a geometric "ruler."

5. Calibration — Total vertices = (2n + 3)(2dk) + 1. This count is exact, making the layout maximally tight: every position must be used, forcing the stem to stretch to its maximum (adjacent stem vertices exactly k positions apart).

Correctness

The key lemma (proved by induction in Section 8 of the paper): each P-polyp region absorbs the nematocysts of exactly 3 tentacles, and their sizes must sum to c = 12dnA. This forces a valid 3-Partition.

Equivalence: 3-Partition has a solution ⟺ optimal bandwidth of G = k.

Solution extraction: The three tentacles whose nematocysts land in region P'ᵢ identify triple Tᵢ.

Size Overhead

Target metric	Formula
num_vertices	(2n + 3) · 2dk + 1, where n = num_items/3, A = capacity, k = next power of 2 ≥ 6nA + 9n − 5, d = k/2
num_edges	num_vertices − 1 (tree)

Polynomial in n and A. Concrete sizes:

n	A	k	num_vertices
1	6	64	20,481
1	10	128	131,073
2	10	256	458,753

Validation Method

The construction produces instances too large for brute-force solving (20K+ vertices at minimum). Use property-based tests:

1. Vertex count: num_vertices == (2n + 3) * 2 * d * k + 1
2. Tree: num_edges == num_vertices − 1 and graph is connected
3. Max degree ≤ 3 for all vertices
4. Head sizes: each head has exactly 2dk − 1 vertices
5. Tentacle sizes: tentacle j has 4dn + 12dn · aⱼ vertices
6. Consistency: sum of all component sizes equals total vertex count

Bandwidth equivalence relies on the published proof (Garey et al. 1978, Section 8, pp. 489–493).

Example

Source: n = 1, A = 6, integers (2, 2, 2). Partition: T₁ = {1, 2, 3}, sum = 6 ✓

Parameters: k = 64, d = 32, m₁ = m₂ = m₃ = 384

Constructed tree:

Component	Count	Size each	Total
Stem	1	321	321
Heads	3	4,095	12,285
P-polyps	1	(per formula)	—
F-polyps	1	(per formula)	—
Tentacles	3	896	2,688
Total			20,481

Optimal bandwidth = 64 ⟺ 3-Partition solvable ⟺ YES ✓

Property checks: vertices = 20,481 ✓, edges = 20,480 ✓, max degree = 3 ✓

[zazabap]

Issue Quality Check — Rule

Check	Result	Details
Usefulness	⚠️ Warn	Novel reduction, but target problem GraphBandwidth does not exist in the codebase
Non-trivial	✅ Pass	Sophisticated gadget construction (siphonophore: heads, polyps, tentacles)
Correctness	✅ Pass	All three references verified; claims consistent with literature
Well-written	❌ Fail	Multiple template sections incomplete or not implementable (see details)

Overall: 1 passed, 1 warning, 1 failed (Usefulness is a pass on novelty but warns on dependency)

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Usefulness

Novel reduction. No path exists from BinPacking to GraphBandwidth (or any proxy) in the current reduction graph. BinPacking currently has zero outgoing reductions, so this would be its first. The reduction establishes a landmark NP-completeness result (bandwidth hard even on degree-3 trees).

Warning: The target problem GraphBandwidth does not exist in the codebase. A [Model] issue and implementation for GraphBandwidth would need to be completed first before this rule can be implemented. Additionally, 3-Partition is treated as a special case of BinPacking, but the reduction requires the specific A/4 < a_i < A/2 constraint — the current BinPacking model does not encode this constraint, so the reduce_to() implementation would need to validate (or assume) this precondition.

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Non-trivial

Pass. The siphonophore construction is a highly non-trivial gadget reduction involving:

• Recursive branching "head" subtrees that act as rigid anchors
• "Polyp" subtrees with carefully calibrated channel gaps
• "Tentacle" paths whose lengths encode the integer values
• A tight vertex-count calibration that forces the layout to be maximally stretched

This is far from a variable substitution or subtype coercion — it is a genuine structural transformation with an intricate correctness argument (channel-counting induction).

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Correctness

All three references verified:

1. Garey, Graham, Johnson & Knuth (1978) — "Complexity results for bandwidth minimization," SIAM J. Appl. Math., 34(3), 477–495. Confirmed to exist. The paper proves NP-completeness of bandwidth for trees with max degree 3 and does use the "siphonophore" terminology for the tree construction. The paper is available at the UCSD math department URL cited in the issue. Already in the project bibliography as garey1979 (the book, not this specific paper — the 1978 journal article itself is not yet in references.bib).

2. Garey & Johnson (1979) — Computers and Intractability. Already in references.bib as garey1979. Problem GT40 is confirmed to be the bandwidth problem. Consistent with claims.

3. Papadimitriou (1976) — "The NP-Completeness of the bandwidth minimization problem," Computing, 16, 263–270. Confirmed via Springer (doi: 10.1007/BF02280884). This is correctly cited as the first NP-completeness proof for general graphs, distinct from the Garey et al. degree-3 tree result.

No contradictions found between the issue's claims and the literature.

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Well-written

Fail. Several issues prevent this from being directly implementable:

4a: Information Completeness

• Size Overhead table: The expressions use informal mathematical notation ((2n + 3)(2dk) + 1, where d = ½k and k = O(nA)) rather than concrete formulas referencing source problem getter methods. The #[reduction(overhead = {...})] macro requires expressions like num_vertices = "some_formula_of_source_fields". Since GraphBandwidth doesn't exist yet, the target metric names (num_vertices, max_degree, bandwidth_bound) are speculative.
• Validation Method: States "this is a theoretical NP-completeness reduction; validation means verifying the proof." This does not meet the template requirement. A valid validation method should describe how to computationally verify the reduction on small instances (e.g., construct a tiny 3-Partition instance, run the reduction, solve the target with brute force, and check equivalence). Even if the full construction is large, a simplified or scaled-down test case should be provided.

4b: Algorithm Completeness

The algorithm is described at a high level with references to the paper's construction, but several steps are not fully specified:

• Step 2 (heads): Says "constructed via a recursive branching pattern (binary-tree-like)" but does not give the exact recursion.
• Step 3 (polyps): Says "constructed by removing vertices from full head polyps" but does not specify which vertices to remove or how channels are created.
• The algorithm as written is a summary of the paper's approach, not a step-by-step procedure a programmer could implement without reading the original paper.

4c: Symbol Consistency

• The variable c is introduced in Step 3 (c = 12dnA) and reused in the logical equivalence section, but its relationship to other parameters could be clearer.
• The overhead table uses n and A but these are source-problem parameters that would need to be extracted from the BinPacking instance — the mapping is not explicit.

4d: Example Quality

• Not brute-force solvable: The example produces a tree with 458,753 vertices and bandwidth bound k = 256. This is far too large for hand verification or brute-force solving. The issue acknowledges the parameters but does not provide a smaller worked example.
• Not fully worked: The example shows the parameter calculations but does not trace through the construction step by step (e.g., listing the actual stem vertices, showing which tentacles go where).
• A minimal useful example would need much smaller parameters — but the construction inherently produces large instances even for n=2, which is a fundamental limitation of this reduction.

Recommendations

• Add a [Model] issue for GraphBandwidth first, as this rule depends on it.
• Consider providing pseudo-code (or Rust-like pseudo-code) for each construction step rather than prose descriptions.
• If a small brute-force-verifiable example is impossible due to the construction's inherent size, state this explicitly and describe an alternative validation strategy (e.g., property-based testing: verify the constructed tree has max degree 3, verify vertex count matches formula, verify stem structure).
• Add the 1978 Garey et al. journal article as a separate BibTeX entry (distinct from the 1979 book).

[GiggleLiu]

Fix-issue changelog

• Renamed target from GraphBandwidth to MinimumGraphBandwidth — matches model issue [Model] MinimumGraphBandwidth #133
• Restructured algorithm section: clear subsections (Parameters, Construction 1-5, Correctness)
• Replaced verbose example prose with component size table
• Updated validation: property-based testing (vertex count, tree structure, max degree, component sizes) instead of brute-force — construction produces 20K+ vertex trees at minimum
• Smallest valid example: n=1, A=6, producing 20,481 vertices
• Clarified that 3-Partition is modeled as BinPacking special case in the codebase
• Improved overall writing quality and conciseness

Applied by /fix-issue.

[GiggleLiu]

Closing as part of batch cleanup.