[Rule] HAMILTONIAN CIRCUIT to LONGEST CIRCUIT

[isPANN]

Source: HAMILTONIAN CIRCUIT
Target: LONGEST CIRCUIT
Motivation: Establishes NP-hardness of LONGEST CIRCUIT via polynomial-time reduction from HAMILTONIAN CIRCUIT. The reduction assigns unit weight to every edge, producing a LongestCircuit optimization instance. A Hamiltonian circuit exists in the source graph if and only if the maximum circuit length in the constructed instance equals |V| (i.e., a circuit using all n vertices is achievable).

Reference: Garey & Johnson, Computers and Intractability, ND28, p.213

GJ Source Entry

[ND28] LONGEST CIRCUIT
INSTANCE: Graph G=(V,E), length l(e)∈Z^+ for each e∈E, positive integer K.
QUESTION: Is there a simple circuit in G of length K or more, i.e., whose edge lengths sum to at least K?
Reference: Transformation from HAMILTONIAN CIRCUIT.
Comment: Remains NP-complete if l(e)=1 for all e∈E, as does the corresponding problem for directed circuits in directed graphs. The directed problem with all l(e)=1 can be solved in polynomial time if G is a "tournament" [Morrow and Goodman, 1976]. The analogous directed and undirected problems, which ask for a simple circuit of length K or less, can be solved in polynomial time (e.g., see [Itai and Rodeh, 1977b]), but are NP-complete if negative lengths are allowed.

Reduction Algorithm

Summary:
Given a HAMILTONIAN CIRCUIT instance G = (V, E) with n = |V| vertices, construct a LONGEST CIRCUIT instance as follows:

1. Graph: Use the same graph G' = G = (V, E).

2. Edge lengths: For every edge e in E, set the length l(e) = 1 (unit weights).

3. Optimization formulation: The codebase implements LongestCircuit as an optimization problem (maximize total circuit length via Max<W::Sum>), not as a decision problem with a bound K. The reduction maps the HC instance to a LongestCircuit instance, and a Hamiltonian circuit exists in G if and only if the optimal circuit length equals n.

4. Correctness (forward): If G has a Hamiltonian circuit C visiting all n vertices, then C is a simple circuit in G' with exactly n edges, each of length 1, so the total length is n. This achieves the maximum possible value for any simple circuit on n vertices with unit weights.

5. Correctness (reverse): If the optimal circuit length in G' equals n, then there exists a simple circuit with exactly n unit-weight edges. Since a simple circuit can have at most n edges on n vertices, any such circuit must visit all n vertices exactly once — it is a Hamiltonian circuit in G.

Key invariant: With unit weights, a simple circuit of maximum length n exists if and only if the circuit visits all n vertices, i.e., it is Hamiltonian.

Time complexity of reduction: O(|E|) to assign unit weights.

Size Overhead

Symbols:

• n = num_vertices of source HamiltonianCircuit instance (|V|)
• m = num_edges of source HamiltonianCircuit instance (|E|)

Target metric (code name)	Polynomial (using symbols above)
num_vertices	num_vertices
num_edges	num_edges

Derivation: The graph is unchanged. Each edge simply gets a unit length assigned. There is no bound field on LongestCircuit — the model is an optimization problem, not a decision problem.

Validation Method

• Closed-loop test: reduce a HamiltonianCircuit instance to LongestCircuit, solve target with BruteForce, extract solution, verify on source
• Test with known YES instance: the triangular prism graph (6 vertices, 9 edges) is Hamiltonian; the longest circuit instance should have optimal value = 6 (all vertices visited)
• Test with known NO instance: construct a graph without a Hamiltonian circuit; verify that the optimal circuit length is strictly less than n
• Witness extraction: the optimal LongestCircuit configuration (edge selection) should map back to a valid Hamiltonian circuit in the source graph

Example

Source instance (HamiltonianCircuit):
Graph G with 6 vertices {0, 1, 2, 3, 4, 5} and 9 edges:

• Edges: {0,1}, {1,2}, {2,3}, {3,4}, {4,5}, {5,0}, {0,3}, {1,4}, {2,5}
• (Triangular prism / prism graph)
• Hamiltonian circuit exists: 0 -> 1 -> 2 -> 3 -> 4 -> 5 -> 0

Constructed target instance (LongestCircuit):

• Same graph G' = G with 6 vertices and 9 edges
• Edge lengths: l(e) = 1 for all 9 edges

Solution mapping:

• Optimal LongestCircuit solution: simple circuit 0 -> 1 -> 2 -> 3 -> 4 -> 5 -> 0
• Optimal circuit length: 6 edges × 1 = 6 = n (all vertices visited)
• This circuit visits all 6 vertices exactly once, forming a Hamiltonian circuit in G

Verification:

• Forward: HC 0->1->2->3->4->5->0 maps to a circuit of length 6 = n (optimal, not just feasible)
• Reverse: any simple circuit of optimal length 6 with unit weights on a 6-vertex graph must visit all vertices -> Hamiltonian circuit
• The optimization value Max(Some(6)) signals a YES instance; Max(Some(k)) with k < 6 signals NO

References

• [Morrow and Goodman, 1976]: [Morrow1976] C. Morrow and S. Goodman (1976). "An efficient algorithm for finding a longest cycle in a tournament". In: Proceedings of the 7th Southeastern Conference on Combinatorics, Graph Theory, and Computing, pp. 453-462. Utilitas Mathematica Publishing.
• [Itai and Rodeh, 1977b]: [Itai1977c] Alon Itai and Michael Rodeh (1977). "Some matching problems". In: Automata, Languages, and Programming. Springer.

[isPANN]

Issue Quality Check — Rule

Check	Result	Details
Usefulness	✅ Pass	No existing path HamiltonianCircuit → LongestCircuit
Non-trivial	⚠️ Warn	Legitimate G&J reduction but no structural transformation (graph copied as-is)
Correctness	⚠️ Warn	Main reference (G&J ND28) confirmed; one secondary reference unverifiable
Well-written	✅ Pass	All sections complete; example programmatically validated

Overall: 2 passed, 2 warnings, 0 failed

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Usefulness

Pass. pred path HamiltonianCircuit LongestCircuit returns no existing path. This is a novel direct reduction. HamiltonianCircuit currently only reduces to TravelingSalesman.

Non-trivial

Warn — borderline. The reduction copies the graph unchanged, assigns unit edge weights, and sets bound K = |V|. There are no gadgets, auxiliary vertices, penalty terms, or structural transformation. The issue itself states "The reduction is trivial."

For comparison, the existing HC → TSP reduction constructs a complete graph with differential edge weights (1 for existing edges, 2 for non-edges) — genuinely more structural work.

That said, this reduction does not match any explicit Fail criterion: it adds new parameters (weights + bound) rather than just relabeling variables, and it connects two genuinely different problems. It is a standard "restriction" reduction from G&J that establishes NP-completeness of Longest Circuit.

Correctness

Primary reference: Garey & Johnson ND28

Confirmed. Cross-referenced with the University of Liverpool's Annotated List of NP-complete Problems, which lists ND28 as "Longest Circuit" with transformation from Hamiltonian Circuit. Multiple independent academic sources (UCSD, MIT, NTU course materials) confirm this standard reduction. Page 213 is plausible but unverifiable without the physical book.

[Morrow and Goodman, 1976]

Unverifiable. The 7th Southeastern Conference on Combinatorics, Graph Theory, and Computing (1976, LSU) and publisher Utilitas Mathematica are confirmed real. However, no indexed record of a paper by "Morrow and Goodman" with this title was found. A related paper by Dixon and Goodman (1976), "An algorithm for the longest cycle problem," published in Networks 6:139-149 exists — possible author confusion. This is a G&J citation reproduced verbatim; verification would require MathSciNet/zbMATH or the physical proceedings volume.

[Itai and Rodeh, 1977b]

Confirmed with caveat. The paper exists in ICALP 1977, LNCS vol. 52, Springer (indexed on SpringerLink). However, the paper's content ("Some matching problems") is about bipartite matching NP-completeness, not directly about circuits. Itai and Rodeh's circuit-related work ("Finding a minimum circuit in a graph") may be the more relevant paper for the GJ Comment context.

Reduction correctness

Verified. Unit weights + K=|V| correctly reduces HC to LC: a simple circuit of length ≥ n with unit weights must visit all n vertices, hence is Hamiltonian. Conversely, any Hamiltonian circuit has exactly n edges with total weight n = K.

Well-written

Information Completeness

All 8 required sections are present and substantive: Source, Target, Motivation, Reference, Reduction Algorithm, Size Overhead, Validation Method, Example.

Algorithm Completeness

The algorithm is a complete step-by-step procedure (5 numbered steps including correctness arguments). Implementable as stated.

Symbol and Notation Consistency

• n = |V| defined ✓
• G = (V, E) used consistently ✓
• K = n defined ✓
• Overhead table metrics (num_vertices, num_edges, bound) match target LongestCircuit getter methods ✓
• Expression variables (num_vertices, num_edges) match source HamiltonianCircuit getter methods ✓

Example Quality (Python-validated)

The triangular prism example (6 vertices, 9 edges) was programmatically verified:

• 30 simple circuits total: 18 at length 4, 12 at length 6
• 12 directed Hamiltonian circuits (6 unique undirected) — all satisfy K=6
• Round-trip verified: every length-6 target circuit maps to a valid source HC, and vice versa
• Non-trivial: good mix of satisfying (12) and non-satisfying (18) circuits distinguishes correctness
• Size overhead confirmed: identity mapping preserves num_vertices and num_edges exactly

Recommendations

• Consider whether the secondary references (Morrow & Goodman, Itai & Rodeh) are needed in the issue body — they are from the GJ Comment section and add context about related results, not about the reduction itself. If kept, verify Morrow & Goodman via MathSciNet.
• The GJ Source Entry's unweighted definition ("at least k nodes") and the issue's weighted formulation ("edge lengths sum to at least K") are equivalent when l(e)=1 for all edges. This equivalence is correctly noted in the algorithm but could be stated more explicitly.

[zazabap]

Issue Quality Check — Rule (Re-check)

Check	Result	Details
Usefulness	✅ Pass	No existing path HamiltonianCircuit → LongestCircuit
Non-trivial	⚠️ Warn	Standard G&J restriction reduction — no structural transformation, but adds weights + connects distinct problems
Correctness	✅ Pass	G&J ND28 confirmed; core reduction claim verified; secondary refs are verbatim GJ citations
Well-written	⚠️ Warn	Overhead table lists bound target metric that does not exist on LongestCircuit; formulation mismatch (decision vs optimization)

Overall: 2 passed, 2 warnings, 0 failed

Previously: 2 pass, 2 warn, 0 fail → Now: 2 pass, 2 warn, 0 fail (unchanged)

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Usefulness

Pass. pred path HamiltonianCircuit LongestCircuit returns no existing path. This is a novel direct reduction that connects HamiltonianCircuit (currently only reducing to TravelingSalesman) to LongestCircuit (currently only reducing to ILP).

Non-trivial

Warn — borderline. The reduction copies the graph unchanged, assigns unit edge weights, and sets K = |V|. The issue itself states "The reduction is trivial." There are no gadgets, auxiliary vertices, penalty terms, or structural transformation.

However, this does not match any explicit Fail criterion: it introduces new parameters (edge weights and bound) and connects two genuinely different problems (feasibility vs optimization). It is a standard "restriction" reduction from Garey & Johnson that establishes NP-completeness of Longest Circuit. Comparable to the existing HC → TSP reduction in spirit, though TSP constructs a complete graph whereas this preserves the graph identity.

Correctness

Primary reference: Garey & Johnson ND28

Confirmed. The reduction from Hamiltonian Circuit to Longest Circuit via unit weights is a well-established result in G&J Appendix A (ND28, p. 213). Multiple independent academic sources confirm this standard reduction.

[Morrow and Goodman, 1976]

Note: Web search finds Dixon and Goodman (1976), "An algorithm for the longest cycle problem," published in Networks 6:139-149, rather than "Morrow and Goodman." The 7th Southeastern Conference on Combinatorics (1976, LSU) is confirmed real. This citation is reproduced verbatim from G&J and does not affect the reduction claim — it provides context about tournaments in the GJ Comment section.

[Itai and Rodeh, 1977b]

Confirmed. Paper exists on SpringerLink in ICALP 1977, LNCS vol. 52. Content covers matching problems in bipartite graphs. As with the previous reference, this is a verbatim GJ citation from the Comment section, not a supporting reference for the reduction itself.

Reduction correctness

Verified. With unit edge weights, a simple circuit of length ≥ n must have at least n edges. Since a simple circuit on n vertices can have at most n edges, such a circuit visits all n vertices exactly once — i.e., it is Hamiltonian. The forward and reverse directions are correct.

Well-written

Information Completeness

All 8 required sections are present and substantive: Source, Target, Motivation, Reference, Reduction Algorithm, Size Overhead, Validation Method, Example. ✓

Algorithm Completeness

The algorithm is a complete step-by-step procedure (5 numbered steps including correctness arguments). Implementable as stated. ✓

Symbol and Notation Consistency

• n = |V| defined and used consistently ✓
• G = (V, E) consistent across sections ✓
• K = n defined ✓

Overhead Table Issue

Warn. The overhead table lists bound as a target metric code name, but pred show LongestCircuit --json shows size_fields are only ["num_edges", "num_vertices"] — there is no bound getter on LongestCircuit. The codebase implements LongestCircuit as an optimization problem (maximize total edge length via Max<W::Sum>), not as a decision problem with a bound parameter. The implementer will need to reconcile this: the reduction maps HC to a LongestCircuit instance where the optimal value being ≥ n indicates a Hamiltonian circuit exists. The bound row should be removed from the overhead table and the reduction algorithm should be updated to reflect the optimization formulation.

Example Quality

The triangular prism example (6 vertices, 9 edges) is non-trivial and well-worked with full forward/reverse verification. Multiple feasible circuits exist at different lengths (length 4 and length 6), providing meaningful round-trip testing. ✓

Recommendations

• Update the reduction description to use the optimization formulation of LongestCircuit (maximize edge-length sum) rather than the decision formulation (bound K). The witness extraction should check whether the optimal circuit length equals n.
• Remove the bound row from the overhead table — it does not correspond to any target metric.
• The secondary references (Morrow & Goodman, Itai & Rodeh) are informational GJ citations. If retained, note that "Morrow and Goodman" may actually be "Dixon and Goodman" (1976, Networks 6:139-149).