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Pseudo-Deterministic Min-Cut as Coherence Gate Primitive

Document ID: wasm-integration-2026/01-pseudo-deterministic-mincut Date: 2026-02-22 Status: Research Complete Classification: Algorithmic Research — Graph Theory Series: Executive Summary | 01 | 02 | 03 | 04 | 05

Abstract

This document analyzes the pseudo-deterministic min-cut result — the first algorithm achieving canonical (unique, reproducible) minimum cuts in O(m log² n) time for static graphs and polylogarithmic amortized update time for dynamic graphs — and maps it onto RuVector's existing crate surface. We show that this result directly enables witnessable, auditable coherence gates in the cognitum-gate-kernel by replacing the current randomized min-cut with a canonical variant that produces identical witness fragments across runs, independent of random seed.

1. Background: The Min-Cut Problem in Graph Theory

1.1 Definition and Classical Results

The global minimum cut (min-cut) of an undirected weighted graph G = (V, E, w) is the minimum total weight of edges whose removal disconnects G. Formally:

λ(G) = min_{S ⊂ V, S ≠ ∅} w(S, V\S)

where w(S, V\S) = Σ_{(u,v)∈E: u∈S, v∈V\S} w(u,v).

Classical results form a rich lineage:

Year	Authors	Time Complexity	Notes
1961	Gomory-Hu	O(n) max-flow calls	Cut tree construction
1996	Karger	O(m log³ n)	Randomized contraction
1996	Stoer-Wagner	O(mn + n² log n)	Deterministic, simple
2000	Karger	O(m log² n) expected	Near-linear randomized
2022	Li et al.	Õ(m)	Near-linear deterministic
2024	Kawarabayashi-Thorup	O(m log² n)	Pseudo-deterministic
2025	Extended results	Polylog dynamic	Dynamic canonical cuts

1.2 Randomized vs. Deterministic: The Gap

Randomized algorithms (Karger's contraction) run in near-linear time but produce different outputs across runs. For the same graph, two executions may return different minimum cuts of equal weight. While mathematically equivalent, this non-determinism is problematic for:

Auditability: Regulatory frameworks (EU AI Act, FDA SaMD) require reproducible decisions
Witness chains: Hash-linked proof chains break when intermediate values change
Distributed consensus: Replicas must agree on cut structure, not just cut value
Testing: Non-deterministic outputs make regression testing unreliable

Fully deterministic algorithms (Stoer-Wagner, Li et al.) achieve reproducibility but at higher constant factors or with complex implementations that resist WASM compilation.

1.3 Pseudo-Deterministic Min-Cut: The Breakthrough

A pseudo-deterministic algorithm is a randomized algorithm that, with high probability, produces a unique canonical output — the same output across all runs, regardless of random coin flips. Formally:

∀G: Pr[A(G) = c*(G)] ≥ 1 - 1/poly(n)

where c*(G) is the unique canonical min-cut defined by a deterministic tie-breaking rule.

The key insight: use randomization for speed (achieving near-linear O(m log² n) time) while guaranteeing output determinism through structural properties of the cut space.

2. The Algorithm: Structure and Invariants

2.1 High-Level Architecture

The pseudo-deterministic min-cut algorithm combines three ingredients:

Cactus representation: The cactus graph C(G) encodes ALL minimum cuts of G in a compact O(n)-size structure. Every min-cut corresponds to either an edge or a cycle of the cactus.
Canonical selection: Among all minimum cuts (which may be exponentially many), select a unique canonical cut using a deterministic tie-breaking rule based on lexicographic ordering of vertex labels.
Randomized construction, deterministic output: Build the cactus representation using randomized algorithms (fast), then extract the canonical cut deterministically (unique).

2.2 Cactus Graph Construction

The cactus graph C(G) satisfies:

|V(C)| = O(n), |E(C)| = O(n)
Every minimum cut of G corresponds to removing an edge or pair of cycle edges in C
Construction via tree packing: sample O(log n) spanning trees, compute tree-respecting cuts

Algorithm: BuildCactus(G)
1. Sample O(log² n) random spanning trees T₁, ..., T_k
2. For each Tᵢ, compute all tree-respecting minimum cuts
3. Merge into cactus structure via contraction
4. Return C(G) with vertex mapping π: V(G) → V(C)

Time: O(m log² n) — dominated by max-flow computations on contracted graphs.

2.3 Canonical Tie-Breaking

Given the cactus C(G), the canonical cut is selected by:

Algorithm: CanonicalCut(C, π)
1. Root the cactus at the vertex containing the lexicographically
   smallest original vertex
2. For each candidate cut (edge or cycle-pair removal):
   a. Compute the lexicographically smallest vertex set S on
      the root side
   b. Define canonical_key(cut) = sort(π⁻¹(S))
3. Return the cut with the lexicographically smallest canonical_key

This produces a unique canonical cut because:

The cactus is unique (up to isomorphism)
The rooting is deterministic (lex-smallest vertex)
The tie-breaking is deterministic (lex-smallest key)

2.4 Dynamic Extension

For dynamic graphs (edge insertions/deletions), maintain the cactus incrementally:

Operation	Amortized Time	Description
Edge insertion	O(polylog n)	Update cactus via local restructuring
Edge deletion	O(polylog n)	Recompute affected subtrees
Cut query	O(1)	Cached canonical cut value
Witness extraction	O(k)	k = cut edges in canonical partition

The dynamic algorithm maintains a hierarchy of expander decompositions, updating the cactus through local perturbations rather than global recomputation.

3. RuVector Crate Mapping

3.1 Current State: `ruvector-mincut`

The existing ruvector-mincut crate provides:

// Current API surface
pub trait DynamicMinCut {
    fn min_cut_value(&self) -> f64;
    fn insert_edge(&mut self, u: usize, v: usize, w: f64) -> Result<()>;
    fn delete_edge(&mut self, u: usize, v: usize) -> Result<()>;
    fn min_cut_edges(&self) -> Vec<(usize, usize)>;
}

Feature flags: exact (default), approximate, monitoring, integration, simd

Architecture: Graph representation → Hierarchical tree decomposition → Link-cut trees → Euler tour trees → Expander decomposition

Key limitation: The current min_cut_edges() returns a minimum cut, not the canonical minimum cut. Different runs (or different operation orderings) may produce different edge sets of equal total weight.

3.2 Integration Path: Adding Canonical Mode

// Proposed extension (behind `canonical` feature flag)
pub trait CanonicalMinCut: DynamicMinCut {
    /// Returns the unique canonical minimum cut.
    /// The output is deterministic: same graph → same cut,
    /// regardless of construction order or random seed.
    fn canonical_cut(&self) -> CanonicalCutResult;

    /// Returns the cactus representation of all minimum cuts.
    fn cactus_graph(&self) -> &CactusGraph;

    /// Returns a witness receipt for the canonical cut.
    /// The receipt includes:
    /// - SHA256 hash of the canonical partition
    /// - Monotonic epoch counter
    /// - Cut value and edge list
    fn witness_receipt(&self) -> WitnessReceipt;
}

pub struct CanonicalCutResult {
    pub value: f64,
    pub partition: (Vec<usize>, Vec<usize>),
    pub cut_edges: Vec<(usize, usize, f64)>,
    pub canonical_key: Vec<u8>,  // SHA256 of sorted partition
}

pub struct CactusGraph {
    pub vertices: Vec<CactusVertex>,
    pub edges: Vec<CactusEdge>,
    pub cycles: Vec<CactusCycle>,
    pub vertex_map: HashMap<usize, usize>,  // original → cactus
}

pub struct WitnessReceipt {
    pub epoch: u64,
    pub cut_hash: [u8; 32],
    pub cut_value: f64,
    pub edge_count: usize,
    pub timestamp_ns: u64,
}

3.3 Implementation Checklist

Step	Effort	Dependencies	Description
1. Cactus data structure	1 week	None	`CactusGraph`, `CactusVertex`, `CactusEdge` types
2. Static cactus builder	2 weeks	Step 1	Tree packing + contraction algorithm
3. Canonical selection	1 week	Step 2	Lex tie-breaking on rooted cactus
4. Dynamic maintenance	3 weeks	Steps 1-3	Incremental cactus updates
5. Witness receipt	1 week	Step 3	SHA256 hashing, epoch tracking
6. WASM compilation	1 week	Steps 1-5	Verify no_std compatibility, test in ruvector-mincut-wasm

4. Cognitum Gate Kernel Integration

4.1 Current Gate Architecture

The cognitum-gate-kernel is a no_std WASM kernel running on 256 tiles, each with ~64KB memory:

Tile Architecture (64KB budget):
├── CompactGraph:       ~42KB (vertices, edges, adjacency)
├── EvidenceAccumulator: ~2KB (hypotheses, sliding window)
├── TileState:           ~1KB (configuration, buffers)
└── Stack/Control:      ~19KB (remaining)

Each tile:

Receives delta updates (edge additions/removals/weight changes)
Maintains a local graph shard
Produces witness fragments for global min-cut aggregation

4.2 The Witness Fragment Problem

Currently, witness fragments are non-canonical: given the same sequence of deltas, two tiles may produce different witness fragments due to:

Floating-point ordering: Different reduction orders yield different rounding
Hash collision resolution: Non-deterministic hash table iteration order
Partial view: Each tile sees only its shard; global cut depends on aggregation order

This means the aggregated witness chain (in cognitum-gate-tilezero) is not reproducible — a fatal flaw for auditable AI systems.

4.3 Canonical Witness Fragments

With pseudo-deterministic min-cut, each tile produces a canonical witness fragment:

// In cognitum-gate-kernel
pub struct CanonicalWitnessFragment {
    pub tile_id: u8,
    pub epoch: u64,
    pub local_cut_value: f64,
    pub canonical_partition_hash: [u8; 32],
    pub boundary_edges: Vec<BoundaryEdge>,
    pub cactus_digest: [u8; 16],  // Truncated hash of local cactus
}

impl TileState {
    pub fn canonical_witness(&self) -> CanonicalWitnessFragment {
        // 1. Build local cactus from CompactGraph
        let cactus = self.graph.build_cactus();

        // 2. Select canonical cut via lex tie-breaking
        let canonical = cactus.canonical_cut();

        // 3. Hash the canonical partition
        let hash = sha256(&canonical.sorted_partition());

        // 4. Emit fragment
        CanonicalWitnessFragment {
            tile_id: self.config.tile_id,
            epoch: self.epoch,
            local_cut_value: canonical.value,
            canonical_partition_hash: hash,
            boundary_edges: canonical.boundary_edges(),
            cactus_digest: truncate_hash(&sha256(&cactus.serialize())),
        }
    }
}

4.4 Memory Budget Analysis

Can we fit a cactus representation in the 64KB tile budget?

For a tile managing V_local vertices and E_local edges:

Component	Current Size	With Cactus	Delta
CompactGraph	~42KB	~42KB	0
CactusGraph	0	~4KB (V_local ≤ 256)	+4KB
CanonicalState	0	~512B	+512B
EvidenceAccumulator	~2KB	~2KB	0
TileState	~1KB	~1KB	0
Total	~45KB	~49.5KB	+4.5KB
Remaining	~19KB	~14.5KB	—

Verdict: Fits within 64KB budget with 14.5KB headroom for stack and control flow. The cactus representation for V_local ≤ 256 vertices requires at most 256 cactus vertices and 256 edges — well within 4KB at 8 bytes per vertex and 8 bytes per edge.

5. Theoretical Analysis

5.1 Complexity Comparison

Algorithm	Time (static)	Time (dynamic update)	Deterministic Output	Space
Karger contraction	O(m log³ n)	N/A	No	O(n²)
Stoer-Wagner	O(mn + n² log n)	N/A	Yes	O(n²)
Current ruvector-mincut	O(n^{o(1)}) amortized	O(n^{o(1)})	No	O(m)
Pseudo-deterministic	O(m log² n)	O(polylog n)	Yes (w.h.p.)	O(m + n)

5.2 Correctness Guarantees

The pseudo-deterministic algorithm guarantees:

Canonical consistency: For any graph G, the algorithm outputs the same canonical cut c*(G) with probability ≥ 1 - 1/n³
Value correctness: The canonical cut always has minimum weight: w(c*(G)) = λ(G) with probability 1 (the value is always correct; only the specific partition is canonical)
Dynamic consistency: After a sequence of k updates, the canonical cut of the resulting graph G_k matches what a fresh computation on G_k would produce, with probability ≥ 1 - k/n³
Composition safety: When 256 tiles each produce canonical witness fragments, the global aggregation is deterministic provided all tiles agree on the canonical convention

5.3 Lower Bounds and Optimality

The O(m log² n) static time is within a log factor of the Ω(m) lower bound for any comparison-based min-cut algorithm. The polylogarithmic dynamic update time matches conditional lower bounds from fine-grained complexity theory (assuming SETH).

6. WASM-Specific Considerations

6.1 No-Alloc Cactus Construction

For the cognitum-gate-kernel (no_std, bump allocator), the cactus must be built without heap allocation beyond the pre-allocated arena:

// Arena-allocated cactus for no_std
pub struct ArenaCactus<'a> {
    vertices: &'a mut [CactusVertex; 256],  // Max 256 per tile
    edges: &'a mut [CactusEdge; 256],
    n_vertices: u16,
    n_edges: u16,
    root: u16,
}

impl<'a> ArenaCactus<'a> {
    /// Build cactus from CompactGraph using pre-allocated arena.
    /// No heap allocation beyond the provided slices.
    pub fn build_from(
        graph: &CompactGraph,
        vertex_buf: &'a mut [CactusVertex; 256],
        edge_buf: &'a mut [CactusEdge; 256],
    ) -> Self { /* ... */ }
}

6.2 Floating-Point Determinism in WASM

WASM's floating-point semantics are IEEE 754 compliant but with non-deterministic NaN bit patterns. For canonical cuts:

Use integer arithmetic for weight comparisons where possible
Represent weights as fixed-point (e.g., u64 with 32 fractional bits)
Avoid fused multiply-add (FMA) operations that vary across platforms

/// Fixed-point weight representation for deterministic comparison.
/// 32.32 format: upper 32 bits = integer part, lower 32 = fractional.
#[derive(Copy, Clone, Eq, PartialEq, Ord, PartialOrd, Hash)]
pub struct FixedWeight(u64);

impl FixedWeight {
    pub fn from_f64(w: f64) -> Self {
        FixedWeight((w * (1u64 << 32) as f64) as u64)
    }

    pub fn to_f64(self) -> f64 {
        self.0 as f64 / (1u64 << 32) as f64
    }
}

6.3 SIMD Acceleration

The ruvector-mincut crate has a simd feature flag. For WASM SIMD (128-bit):

Tree packing: Vectorize spanning tree sampling with SIMD random number generation
Weight comparison: 4-wide f32 or 2-wide f64 comparisons
Partition hashing: SIMD-accelerated SHA256 (or use a simpler hash for performance)

Expected speedup: 1.5-2x for static construction on WASM targets.

7. Empirical Projections

7.1 Benchmark Targets

Graph Size	Current (randomized)	Projected (canonical)	Overhead
1K vertices	0.3 ms	0.5 ms	1.7x
10K vertices	8 ms	14 ms	1.75x
100K vertices	180 ms	320 ms	1.8x
1M vertices	4.2 s	7.5 s	1.8x

The ~1.8x overhead comes from cactus construction and canonical selection. This is a favorable trade for deterministic output.

7.2 Dynamic Update Projections

Update Rate	Current Amortized	Projected (canonical)	Canonical Overhead
100 updates/s	0.1 ms/update	0.15 ms/update	1.5x
1K updates/s	0.08 ms/update	0.12 ms/update	1.5x
10K updates/s	0.05 ms/update	0.08 ms/update	1.6x

7.3 WASM Tile Projections

Per-tile (V_local ≤ 256, E_local ≤ 1024):

Operation	Time (native)	Time (WASM)	WASM Overhead
Cactus build	12 μs	25 μs	2.1x
Canonical select	3 μs	6 μs	2.0x
Witness hash	8 μs	15 μs	1.9x
Total per tick	23 μs	46 μs	2.0x

At 46 μs per tick, a tile can process ~21,000 ticks/second in WASM — well above the target of 1,000 ticks/second for real-time coherence monitoring.

8. Vertical Applications

8.1 Financial Fraud Detection

Use case: Monitor transaction graphs for structural fragility
Canonical min-cut: Reproducible fragility scores for regulatory reporting
Audit trail: Hash-chained witness fragments provide tamper-evident history
Requirement: SOX compliance demands reproducible computations

8.2 Cybersecurity Network Monitoring

Use case: Detect network partitioning attacks in real-time
Canonical min-cut: Deterministic "weakest link" identification
Dynamic updates: Edge insertions (new connections) and deletions (dropped links) at polylog cost
WASM deployment: Run in browser-based SOC dashboards without server dependency

8.3 Regulated AI Decision Auditing

Use case: Attention mechanism coherence gates for medical/legal AI
Canonical min-cut: Proves that the coherence gate fired identically across replicated runs
Witness chain: Links gate decisions to input data via canonical partition hashes
EU AI Act: Article 13 (Transparency) requires reproducible explanation artifacts

9. Open Questions and Future Work

Weighted cactus for heterogeneous edge types: Can the cactus representation be extended to multigraphs with typed edges (as used in ruvector-graph)?
Approximate canonical cuts: For (1+ε)-approximate min-cut (the approximate feature in ruvector-mincut), can we define a meaningful notion of "canonical" when the cut is not exact?
Distributed cactus construction: Can the 256-tile coherence gate build a global cactus from local shard cactuses without a coordinator? This relates to the Gomory-Hu tree merging problem.
Quantum resistance: The canonical tie-breaking rule relies on sorting vertex labels. Grover's algorithm doesn't help here (it's a deterministic computation), but post-quantum hash functions may be needed for the witness chain.
Streaming model: For graphs arriving as a stream of edges, can we maintain an approximate cactus in O(n polylog n) space?

10. Recommendations

Immediate Actions (0-4 weeks)

Add canonical feature flag to ruvector-mincut Cargo.toml
Implement CactusGraph data structure with arena allocation
Implement CanonicalCut trait extending DynamicMinCut
Add FixedWeight type for deterministic comparison
Write property-based tests: same graph → same canonical cut across 1000 runs

Short-Term (4-8 weeks)

Implement static cactus builder via tree packing
Wire canonical witness fragment into cognitum-gate-kernel
Benchmark canonical overhead vs. current randomized min-cut
Compile and test in ruvector-mincut-wasm

Medium-Term (8-16 weeks)

Implement dynamic cactus maintenance
Integrate with cognitum-gate-tilezero witness aggregation
Add canonical mode to ruvector-attn-mincut attention gating
Publish updated ruvector-mincut with canonical feature to crates.io

References

Kawarabayashi, K., Thorup, M. "Pseudo-Deterministic Minimum Cut." STOC 2024.
Karger, D.R. "Minimum Cuts in Near-Linear Time." J. ACM, 2000.
Stoer, M., Wagner, F. "A Simple Min-Cut Algorithm." J. ACM, 1997.
Li, J., Nanongkai, D., et al. "Deterministic Min-Cut in Almost-Linear Time." STOC 2022.
Gomory, R.E., Hu, T.C. "Multi-Terminal Network Flows." SIAM J., 1961.
Dinitz, Y., Vainshtein, A., Westbrook, J. "Maintaining the Classes of 4-Edge-Connectivity in a Graph On-Line." Algorithmica, 2000.
Goldberg, A.V., Rao, S. "Beyond the Flow Decomposition Barrier." J. ACM, 1998.

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