Ternary-Restricted Flip-Graph Exploration

• The paper introduces an algorithmic framework that uses local ternary-preserving flips and plus-operators to achieve a rank-23, 58-addition scheme for 3×3 matrix multiplication.
• The methodology employs stochastic local search and greedy common subexpression elimination to navigate the vast reconfiguration space while maintaining integer arithmetic.
• The approach extends to triangulation and discrete geometry, ensuring robust connectivity and efficient enumeration of combinatorial structures under strict ternary constraints.

Ternary-restricted flip-graph exploration refers to the algorithmic and combinatorial study of flip-graphs where states (vertices) correspond to mathematical objects—typically matrix multiplication schemes, triangulations, or geometric arrangements—subject to the strict constraint that all relevant coefficients, entries, or combinatorial features lie in the discrete set $\{-1,0,1\}$. This framework is motivated by the search for efficient, portable algorithms with low additive and multiplicative complexity in computational mathematics and discrete geometry, notably in matrix multiplication and triangulation enumeration. The flip-graph encodes local transformations (flips) as edges; the ternary restriction constrains all moves to remain within the set of admissible objects.

1. Formal Definitions and Flip-Graph Structure

A flip-graph is an undirected graph where each vertex corresponds to a valid combinatorial or algebraic structure, and edges represent admissible local modifications, typically “flips.” In the context of matrix multiplication schemes, each vertex $S = (U, V, W)$ represents a set of rank-$r$ decompositions for $n \times n$ matrix multiplication, with $U, V, W$ being collections of $r \times n \times n$ tensors. $S$ is valid iff it solves the Brent equations for bilinear matrix multiplication. The ternary restriction enforces that all coefficients in $U, V, W$ must be elements of $\{-1,0,1\}$, i.e. for all tensor entries $u^{(t)}_{ij}, v^{(t)}_{ij}, w^{(t)}_{ij}$, $|x| \le 1$.

A flip is a local transformation satisfying:

• Preservation of the Brent equations (algorithm correctness).
• No increase in rank.
• All coefficients after the flip remain in $\{-1,0,1\}$.

Operationally, this often involves applying a $2 \times 2$ unimodular transformation to pairs of rank-1 summands, with the accompanying transformation of the remaining structure so that functional correctness is preserved (Perminov, 26 Dec 2025).

2. The Role of the Ternary Restriction

The ternary restriction ($\{-1,0,1\}$) is motivated by both algebraic generality (validity over arbitrary rings and fields) and complexity considerations. Schemes with larger or fractional coefficients may achieve lower rank or additive complexity but lose universality and become less efficient or hardware-friendly. The ternary domain guarantees:

• All arithmetic is integer and bounded.
• Portability across fields with characteristic $\neq 2$.
• Simple bitwise representations, enabling hardware acceleration (Perminov, 25 Nov 2025).
• Reduction in additive cost, as subexpressions are more amenable to common subexpression elimination (CSE).

In triangulation flip-graphs or geometric arrangements, "ternary-restricted" may analogously refer to forbidding certain edge sets, or restricting attention to triangle-specific local moves, which greatly influences both the structure and connectivity of the resulting graph (Bigdeli et al., 2022, Radtke et al., 2023).

3. Exploration Algorithms and Pipeline

The exploration of ternary-restricted flip-graphs is typically performed via stochastic local search, metaheuristic walks, or parallel population-based algorithms. For $3 \times 3$ matrix multiplication, the pipeline consists of:

1. Initialization: Start from the naive scheme (e.g., with 27 multiplications).
2. Phase 1: Use random ternary-preserving flips and occasional rank-increasing "plus-operator" steps to reduce rank to the target (e.g., 23).
   • The plus-operator injects new rank-1 summands, allowing escape from local optima, always maintaining ternary coefficients.
3. Phase 2: Apply a greedy intersection-reduction (a CSE heuristic) which identifies common subexpressions among the linear forms and systematically introduces auxiliary variables to eliminate additive redundancy, minimizing the total number of additions.
4. Phase 3: Perturb the scheme randomly (e.g., single flip, chance of plus) to escape new local minima and restart.
5. Iteration: Repeat the cycle until a minimum-addition, minimum-rank scheme is found.

The entire process operates efficiently due to the ternary restriction, as invalid flips (which would lead outside $\{-1,0,1\}$) are instantly rejected. The search space is vast but navigable with the local flip plus restart/escape mechanisms. Empirical results for $3 \times 3$ multiplication find a new best scheme of rank 23 and 58 additions in 30 minutes on commodity hardware (Perminov, 26 Dec 2025).

4. Applications: Matrix Multiplication and Triangulation

The ternary-restricted flip-graph methodology has produced significant results in fast matrix multiplication:

• For $3 \times 3$ multiplication, the method yields a rank-23, 58-addition scheme, improving the best-known additive complexity in the ternary case (Perminov, 26 Dec 2025).
• GPU-accelerated algorithms discover optimal or near-optimal schemes in formats $4 \times 5 \times 12$, $5 \times 6 \times 10$, and $6 \times 7 \times 9$, matching or surpassing rational-coefficient algorithms while remaining within the ternary domain (Perminov, 25 Nov 2025).
• In triangulation flip-graphs, algorithms efficiently enumerate and sample triangulations subject to forbidden edges, with $O(n \log n)$ or $O(n)$ algorithms for connectivity and component membership queries (Bigdeli et al., 2022).

The approach extends to discrete geometry and combinatorics, where triangle-restricted flips in arrangements of pseudolines and pseudocircles generate highly structured reconfiguration graphs, enabling quantitative analysis of their connectivity, diameter, and combinatorial properties (Radtke et al., 2023).

5. Structural Properties and Complexity

Key structural aspects of ternary-restricted flip-graphs include:

• Connectivity: For combinatorial reconfiguration (e.g., pseudoline arrangements), triangle-restricted flip-graphs are $(n-2)$-connected and of tight diameter $\Theta(n^3)$, underlining robustness and navigability (Radtke et al., 2023).
• Cut characterization: In triangulations, forbidden (ternary) edge sets affect the connectivity of the restricted flip-graph. The flip cut number for a convex $n$-gon is $n-3$: removing any fewer than $n-3$ edges preserves connectivity (Bigdeli et al., 2022).
• Exponential growth: The space of valid ternary schemes grows exponentially with rank and dimension; flip-graph-based search scales polynomially in the number of moves attempted but overall exploration remains limited by this growth (Perminov, 26 Dec 2025, Perminov, 25 Nov 2025).

Local flips, plus-operators, and CSE heuristics provide a feasible mechanism for systematic exploration within nontrivial computational boundaries.

6. Limitations and Extensions

Current limitations of ternary-restricted flip-graph exploration stem from:

• Exponential scaling beyond moderate ranks/dimensions.
• Absence of known lower bounds for additive complexity in the ternary domain; optimality remains conjectural (Perminov, 26 Dec 2025).
• Non-universality: some minimal-rank schemes with non-ternary coefficients are unreachable; the ternary framework cannot realize them regardless of exploration depth (Perminov, 25 Nov 2025).

Potential extensions include:

• Heuristics leveraging look-ahead flips, memory structures (e.g., tabu lists), or machine learning-guided operator selection.
• Hybridization with rational/integer coefficient search, constraint programming, or basis transformation methods.
• Automating advanced CSE or integrating reinforcement learning approaches for prioritized flip selection (Perminov, 25 Nov 2025).

The methodology generalizes to other formats and reconfiguration problems, though practical scalability and success depend on problem-specific structure and search space topology.

7. Connections and Theoretical Impact

Ternary-restricted flip-graph exploration unifies computational algebra and discrete geometry via structurally similar local-move graphs:

• In matrix multiplication, it enables systematic, hardware-friendly algorithm discovery with strong universal validity (Perminov, 26 Dec 2025, Perminov, 25 Nov 2025).
• In triangulation and arrangement theory, it underpins the analysis of connectivity, enumeration, and reconfiguration diameter for local-move ensembles (Bigdeli et al., 2022, Radtke et al., 2023).
• The study reveals deep combinatorial principles governing algorithmic complexity, universality, and reconfiguration phase transitions under local constraints.

Ongoing research decomposes both theoretical and practical frontiers, offering toolkits for further exploration and application in computational mathematics and discrete structures.