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Testing Equivalence to the Hamiltonian Cycle Polynomial

Author Agrim Dewan

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LIPIcs.MFCS.2026.90.pdf
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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Equivalence Testing
  • Hamiltonian Cycle Polynomial
  • Symmetries
  • Lie Algebra
  • Circuit identities

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Abstract

The Hamiltonian Cycle polynomial, denoted as HC_n, is defined to be the sum of the weighted Hamiltonian Cycles in an n-vertex complete digraph, with vertices labeled 1 to n and edges weighted by formal variables x_{i,j}. The Permanent and HC, defined as the family {HC_n | n ≥ 1}, were studied by Valiant (STOC 1979), with the former shown to be VNP-complete over all fields of characteristic other than 2, and the latter to be VNP-complete over every field. Since its introduction, HC has been studied from the perspective of circuit lower bounds by Jerrum-Snir (JACM 1982), determinantal complexity by Huttenhain-Ikenmeyer (LAA 2016), and its connection with the Permanent and the Determinant polynomials by Goulden-Jackson (EJC 1981) and Grochow (ToC 2017). It has been the most prominent choice for generalising results to all fields, such as in Malod (CCC 2007) and Grochow-Mulmuley-Qiao (ICALP 2016), owing to its VNP-completeness over every field. Hrubes (ToCT, 2016) showed the VNP-completeness of many graph-based polynomial families over every field by using HC.
In Kayal (STOC 2012), a randomised polynomial time algorithm was given for the following problem: Given an n²-variate degree-n polynomial f(𝐱) ∈ 𝔽[𝐱] as a black box, decide if there exists A ∈ GL_{n²}(𝔽) such that f(𝐱) = Perm_n(A𝐱). Here, the Permanent polynomial Perm_n computes the permanent of the n × n symbolic matrix (x_{i,j}). This problem is known as testing equivalence to the Permanent, or alternatively, ET for Permanent. 
In this work, we study ET for HC. While both families are VNP-complete, the efficient ET algorithm for Permanent does not imply the same for HC. Besides, there are crucial differences between the two polynomials that make studying the complexity of ET for HC interesting: The underlying decision problem corresponding to the Permanent is in P (detecting perfect matchings in a bipartite graph), but that for HC (detecting Hamiltonian cycles in a digraph) is NP-complete. The Permanent polynomial is known to be characterised by its symmetries as shown by Mulmuley-Sohoni (SIAM J. Computing, 2001). This property yields an efficient algorithm for the circuit-testing problem for the Permanent, a special case of ET for the Permanent, in which we check whether a given circuit computes the Permanent. In contrast, we show HC_n is not characterised by its symmetries. 
In this work, we give a randomised polynomial time ET algorithm for HC with mild constraints on the underlying field. The algorithm is obtained by studying and completely characterising the Lie algebra and the symmetries of HC_n. We show that, like the Permanent polynomial, the symmetries of HC_n are generated by permutation and scaling matrices over large enough fields. However, we also show that, unlike the Permanent polynomial, HC_n is not characterised by its symmetries. Nevertheless, like the Permanent polynomial, HC_n is downward self-reducible, as shown in Zhang-Bai (TCS 2011), which implies HC_n is characterised by circuit identities and that we can efficiently test whether a given circuit C computes HC_n. We also get a Flip theorem for HC_n as a result of its circuit identities.

Cite As Get BibTex

Agrim Dewan. Testing Equivalence to the Hamiltonian Cycle Polynomial. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 90:1-90:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.90

Author Details

Agrim Dewan
  • Indian Institute of Science, Bengaluru, India

Acknowledgements

The author thanks Chandan Saha for suggesting the problem and Abhiram Aravind for going through the proofs, which helped improve the presentation of the paper. The author also thanks the anonymous reviewers of MFCS 2026 for their constructive and informative feedback, particularly one reviewer for pointing to the Pascal Determinant polynomial family, which is VNP-complete and for which an ET algorithm can be designed using knowledge of its continuous symmetries.

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