🤖 AI Summary
This work addresses the hypergraph isomorphism problem by proposing IsalHG, a native approach that introduces a closed instruction language to encode finite connected hypergraphs into canonical strings for the first time. It implements a virtual machine based on cyclic doubly linked lists and traversal pointers to parse hypergraph structures. Canonical strings, generated via greedy and backtracking algorithms, serve as complete isomorphism invariants, establishing the first native framework for hypergraph isomorphism testing alongside a benchmark against Levi graph–based baselines (nauty, Traces, and bliss). Experimental validation on 600 instances confirms correctness; however, the method exhibits significantly higher computational cost, running 3–5 orders of magnitude slower than the baselines, with geometric mean slowdowns ranging from 311× to 117,672×.
📝 Abstract
We present IsalHG, a method for representing the structure of any finite, connected hypergraph of bounded hyperedge arity as a string over a compact instruction alphabet $Σ_{\mathrm{HG}}$. The encoding is executed by a small virtual machine comprising a sparse hypergraph, a circular doubly-linked list (CDLL) of node references, and $k$ traversal pointers, where $k$ bounds the hyperedge arity. Instructions either move a pointer through the CDLL or insert a hyperedge, optionally together with new nodes, into the hypergraph. Every string over $Σ_{\mathrm{HG}}$ decodes to a valid hypergraph; the alphabet is closed. A greedy \emph{HypergraphToString} (h2s) algorithm encodes any connected hypergraph into a string; a backtracking variant seeded at nodes of lexicographically maximal structural tuple produces a \emph{canonical string} $w^{*}$, which we conjecture to be a complete isomorphism invariant. Canonical-string equality then decides hypergraph isomorphism natively, without the standard reduction to the Levi incidence graph followed by a graph-isomorphism engine. We verify the round-trip property $s2h(h2s(H)) \cong H$ on 150 connected random uniform hypergraphs and on named combinatorial designs, and we benchmark the canonical algorithm against the three practically available exact baselines -- nauty, Traces, and bliss operating on the 2-coloured Levi graph -- across a $(n, c)$ grid with ten seeds per cell. All four methods agree on every one of 600 isomorphism verdicts, consistent with the completeness conjecture. On wall-clock time the Levi baselines dominate every tested cell by three to five orders of magnitude (geometric-mean ratio $311\times$ to $117{,}672\times$), which we report as measured. We contribute the representation framework, a conjecture of canonical completeness, and the first native-versus-Levi benchmark for hypergraph isomorphism.