Simple, Faster (min, +) Matrix Multiplication and APSP

Abstract: In this note, we provide a deterministic algorithm that computes the $(\min, +)$-Matrix Product $C = A * B$ of two $N \times N$ matrices $A$ and $B$ with polynomially bounded integer entries in $\tilde{O}(N<sup>{2.9964})$ time, improving upon the exponent of $O(N<sup>{2.9995})$ time obtained in the recent breakthrough by Anthropic (Alman and Vassilevska Williams, 2026). Our algorithm uses a phased approach first proposed by Bringmann, Grandoni, Saha, and Vassilevska Williams in 2016. The phased approach produces an approximation $\tilde{C}$ of $C$ in truly subcubic time such that the number of remaining triples $(i,k,j): A_{i,k}+B_{k,j} &lt; \tilde{C}(i,j)$ is truly subcubic. This provided strong evidence that a truly subcubic $(\min, +)$-Matrix Product and therefore, a truly subcubic algorithm for All Pairs Shortest Paths (APSP) may exist. However, to find the remaining triples in subcubic time, a sparse certificate is needed whose construction was not known. We use the thin matrix product theorem of the recent breakthrough to provide such a certificate. This gives a simple modular subcubic algorithm for APSP.

• The paper proposes a new deterministic algorithm for matrix multiplication and APSP, leveraging the $(\min,+)$ matrix product framework, demonstrating an improved running time of $O(N^{2.9963432})$ for polynomially bounded integer entries.
• The computed matrix leverage sparse-certificate verification to isolate combinatorial difficulties by providing $O(N^{2.9963432})$ running times for selected $(\min,+)$ Matrix Products and related problems, enhancing computational efficiency.
• Achievement of this improved exponent relies on updating the phased approximation-and-verification algorithm, incorporating a thin matrix product theorem that calculates specific matrix entries.

Problem setting and contribution

The paper studies the $(\min,+)$ matrix product of two $N \times N$ matrices, defined by

$$(A * B)[i,j] = \min_{k \in [N]} \{A[i,k] + B[k,j]\}.$$

This operation is equivalent, up to polylogarithmic factors and standard reductions, to APSP and several other fine-grained problems, including Negative Triangle, Replacement Paths, Metricity, $(\min,+)$-Verification, Weighted Tree Edit Distance, and two-dimensional Maximum Subarray. The central objective is to improve the exponent below three for polynomially bounded integer entries.

The main claimed result is a deterministic algorithm with running time

$O\left(N^{2.9963432}\right),$

together with a Las Vegas algorithm achieving expected time of the same exponent, up to polylogarithmic factors. The improvement is obtained by combining the phased approximation-and-verification framework for $(\min,+)$ multiplication with a thin matrix product theorem that computes selected entries of a rectangular Boolean or integer matrix product faster than dense multiplication. The paper’s principal methodological claim is that the sparse-certificate component required by the phased framework can be supplied without monotonicity or bounded-difference assumptions.

The stated contribution is therefore not a new reduction from APSP, but a modular exploitation of a recently assumed thin matrix product primitive. The resulting algorithm is conceptually decomposed into: additive approximation, modular candidate generation, sparse-certificate verification, and, for derandomization, a hierarchical deterministic search over candidate blocks (2610.20297).

Relation to the phased framework

The algorithm follows the framework in which the exact $(\min,+)$ product is reduced to a witness-equality problem. Given matrices $A$, $B$, and a candidate output matrix $C$, the witness-equality problem asks whether every position $N \times N$0 has some $N \times N$1 satisfying

$N \times N$2

The paper uses a standard recursive scaling argument. If $N \times N$3 and $N \times N$4 are obtained by halving and rounding the entries of $N \times N$5 and $N \times N$6, then $N \times N$7 gives a constant-additive approximation to $N \times N$8. For each pair $N \times N$9, the exact optimum must lie among three candidate values derived from the approximate product. The witness-equality procedure is then invoked on these candidates, and the recursion contributes a factor logarithmic in the entry bound $$(A * B)[i,j] = \min_{k \in [N]} \{A[i,k] + B[k,j]\}.$$0.

This reduction is important because it isolates the combinatorial difficulty: instead of computing all minima directly, the algorithm must efficiently determine whether a candidate value has an exact witness. The paper’s sparse-certificate strategy addresses precisely this problem.

The inner dimension is partitioned into contiguous blocks of size $$(A * B)[i,j] = \min_{k \in [N]} \{A[i,k] + B[k,j]\}.$$1. A block $$(A * B)[i,j] = \min_{k \in [N]} \{A[i,k] + B[k,j]\}.$$2 is a genuine block witness for $$(A * B)[i,j] = \min_{k \in [N]} \{A[i,k] + B[k,j]\}.$$3 if it contains some $$(A * B)[i,j] = \min_{k \in [N]} \{A[i,k] + B[k,j]\}.$$4 satisfying the candidate equality. The algorithm attempts to discard most blocks while retaining every genuine block. The total number of blocks is $$(A * B)[i,j] = \min_{k \in [N]} \{A[i,k] + B[k,j]\}.$$5 per output pair, so an exhaustive search would still require cubic time. The target is consequently a certificate containing only a truly subcubic number of false candidates.

Modular candidate filtering

For a prime $$(A * B)[i,j] = \min_{k \in [N]} \{A[i,k] + B[k,j]\}.$$6 and a block $$(A * B)[i,j] = \min_{k \in [N]} \{A[i,k] + B[k,j]\}.$$7, the algorithm declares $$(A * B)[i,j] = \min_{k \in [N]} \{A[i,k] + B[k,j]\}.$$8 a candidate for $$(A * B)[i,j] = \min_{k \in [N]} \{A[i,k] + B[k,j]\}.$$9 if some $(\min,+)$0 satisfies

$(\min,+)$1

Every genuine block witness passes this test for every prime. A false block can pass only if at least one nonzero difference

$(\min,+)$2

is divisible by the sampled prime. Since the entries are polynomially bounded, each nonzero $(\min,+)$3 has only $(\min,+)$4 prime divisors in an interval $(\min,+)$5. Applying the prime number theorem and a union bound over the $(\min,+)$6 elements of the block gives a false-positive probability of order

$(\min,+)$7

Choosing $(\min,+)$8 sufficiently larger than $(\min,+)$9 makes this probability a constant below one. Repeating the test with $O\left(N^{2.9963432}\right),$0 independent primes reduces the expected number of surviving false candidates to approximately $O\left(N^{2.9963432}\right),$1.

The modular tests are implemented through ordinary matrix multiplication. For a fixed block and prime, the paper constructs matrices whose inner dimension is $O\left(N^{2.9963432}\right),$2, encoding the residues of $O\left(N^{2.9963432}\right),$3 and $O\left(N^{2.9963432}\right),$4. For an output pair whose residue class matches $O\left(N^{2.9963432}\right),$5, a positive entry of the resulting product certifies the existence of a modular witness in the block. The crucial point is that only selected output positions need to be computed: positions are grouped according to the residue of $O\left(N^{2.9963432}\right),$6, and the thin matrix product theorem is applied separately to these sparse sets.

This is the main interface between the earlier phased framework and the thin matrix product result. The construction does not require $O\left(N^{2.9963432}\right),$7 or $O\left(N^{2.9963432}\right),$8 to be monotone, bounded-difference, or otherwise structured. Its cost is controlled by the rectangular dimension $O\left(N^{2.9963432}\right),$9 and by the sparsity of the queried output positions.

Use of thin matrix multiplication

The paper invokes the following primitive: for $(\min,+)$0 and $(\min,+)$1 with $(\min,+)$2, and for a set $(\min,+)$3 of at most $(\min,+)$4 output positions, the entries $(\min,+)$5 for $(\min,+)$6 can be computed deterministically in time

$(\min,+)$7

for suitable $(\min,+)$8 depending on $(\min,+)$9 and $(\min,+)$0. The paper attributes this primitive to Theorem 25 of the cited breakthrough work and uses it as a black box (2610.20297).

When a residue class contains more than the permitted number of queried positions, the set is partitioned into smaller subsets. Summing over all residue classes yields a candidate-verification cost of the form

$(\min,+)$1

for one block and one prime, with logarithmic factors suppressed in the exposition. Repeating over all $(\min,+)$2 blocks and the sampled primes leads to the total randomized witness-equality cost

$(\min,+)$3

The terms have distinct origins: scanning blocks, constructing residue-encoding matrices, invoking the thin product theorem, checking surviving genuine blocks, and processing false candidates. The last term is the expected cost of false-candidate enumeration.

The parameter choice reported in the paper is

$(\min,+)$4

Taking $(\min,+)$5 proportional to $(\min,+)$6 and choosing $(\min,+)$7 near $(\min,+)$8 balances the dominant terms. The resulting exponent is

$(\min,+)$9

Thus the numerical improvement is extremely small but formally below three. The same exponent propagates through the logarithmic-depth scaling reduction, yielding the stated $A$0 product bound and, via standard reductions, the corresponding APSP bound.

Randomized algorithm

The randomized algorithm samples primes independently from $A$1. Genuine block witnesses survive all modular tests, whereas false blocks survive with exponentially decreasing probability in the number of samples. For every output pair, the algorithm enumerates the elements of surviving candidate blocks and stops as soon as it finds an exact witness.

The early stopping rule is essential to the analysis. For a fixed output pair, at most one genuine block witness needs to be fully processed: once the algorithm reaches one, it finds an exact witness and terminates. Consequently, genuine blocks contribute only $A$2 total work. False candidates contribute expected time proportional to their number times the block size, which is $A$3 after setting the sampling parameter to $A$4.

The algorithm is Las Vegas rather than Monte Carlo because modular filtering can create false candidates but cannot eliminate a genuine block witness. The final exact enumeration therefore preserves correctness for every random outcome; randomness affects only the running time.

The paper claims an expected running time of $A$5 for polynomially bounded integer entries. This claim depends directly on the stated thin matrix product theorem and on the parameter instantiation attributed to it. Any change in the admissible $A$6, $A$7, or $A$8 values would change the final exponent.

Deterministic derandomization

The deterministic algorithm replaces independent prime sampling with a hierarchical selection of good primes. Blocks are recursively subdivided, and the search proceeds depth-first. At level $A$9, blocks have size approximately $B$0. The algorithm maintains a potential function that upper-bounds the number of false candidate blocks at that level.

For a block and prime $B$1, let $B$2 count nonzero modular witnesses—elements satisfying the congruence but not the exact equality. The potential is the product of these counts over a constant-size set of selected primes, summed over all output pairs and blocks. If the selected primes are well chosen, the potential decreases by a factor related to the block size at each additional prime. This yields a bound of the form

$B$3

where $B$4 is the number of primes selected at each level.

The difficulty is that the exact potential depends on whether modular witnesses are genuine, which is itself the problem being solved. The paper addresses this by introducing an efficiently computable estimator $B$5 that approximates the potential within a constant factor. The estimator uses inclusion-exclusion over auxiliary primes and computes tuple counts through matrix multiplication over a multivariate polynomial ring. For constant $B$6, the resulting algebraic overhead is polynomial in $B$7 and does not alter the exponent.

At each level, the algorithm evaluates $B$8 over a family of candidate primes and greedily chooses primes minimizing the estimator. This produces a deterministic set of moduli with the required potential bound. The construction is technically more involved than the randomized version and introduces a dependence on rectangular matrix multiplication costs of the form

$B$9

The paper then uses a data-structure variant of the thin matrix product theorem. This permits faster preprocessing for the residue-encoded products while increasing the cost of individual queries. The depth-first traversal makes this tradeoff effective: higher levels have fewer candidate blocks, so more expensive queries can be tolerated.

The reported deterministic parameters are

$C$0

The block size, prime range, and recursion depth are chosen as

$C$1

The choice $C$2 ensures that the geometric decay in false candidates dominates the growth of query costs across levels. With these settings, the prime-selection cost, data-structure preprocessing, queries, and final scans are each bounded by $C$3, up to polylogarithmic factors. The paper therefore claims the same $C$4 exponent deterministically.

Technical assessment of the claims

The paper’s strongest claim is that a recent thin matrix product theorem can be inserted into an older sparse-certificate framework without requiring structural assumptions on the input matrices. If all cited black-box results and parameter ranges hold exactly as stated, the modular algorithm is a plausible and relatively clean route to the claimed exponent. Its main conceptual value is the separation between candidate generation and exact verification.

Several aspects nevertheless require careful scrutiny.

First, the exponent is conditional on the precise form of the thin matrix product theorem, including its allowable range of $C$5, its dependence of $C$6 on $C$7, and the availability of the data-structure version used in the deterministic construction. The proof treats these results as black boxes and does not independently establish them.

Second, the presentation contains numerous transcription and notation defects in the supplied manuscript: inequalities and set delimiters are missing in several places, matrix dimensions are malformed, and some displayed expressions are syntactically incomplete. These defects do not necessarily invalidate the underlying strategy, but they make the proof difficult to verify line by line. In particular, the deterministic section requires a careful audit of the potential estimates, the estimator computation, and the cumulative preprocessing costs.

Third, the reduction assumes polynomially bounded integer entries. The paper explicitly leaves extension to real-valued matrices open. This restriction is substantive: the modular-prime filtering argument relies on nonzero integer differences having few prime divisors. A real-valued analogue cannot use the same divisibility argument without an additional discretization or algebraic encoding step.

Fourth, the abstract describes a deterministic algorithm in $C$8, while the theorem statement gives $C$9 and the randomized theorem gives expected $N \times N$00. The distinction between suppressed logarithmic factors and the explicit exponent should be maintained consistently. Moreover, the paper’s claims about equivalent APSP problems inherit the entry-domain restrictions and any assumptions in the reductions.

Finally, the manuscript reports that ChatGPT contributed ideas to the deterministic algorithm and parameter optimization. This is a disclosure rather than a mathematical component of the proof. The validity of the deterministic result rests on the formal lemmas and their verification, not on the stated origin of the ideas.

Limitations and open questions

The paper leaves open whether the exponent can be improved by strengthening the thin matrix product theorem. Within the presented framework, the exponent improvement is exactly governed by the product $N \times N$01, so better admissible rectangular dimensions or a larger decay parameter would directly improve the result.

The extension from polynomially bounded integers to real-valued matrices is also unresolved. The randomized algorithm’s correctness and sparsity analysis use modular divisibility in an essential way, and the paper does not provide a replacement for this mechanism over arbitrary reals.

A further open issue is whether the deterministic construction can be simplified or made quantitatively sharper. Its hierarchical potential argument introduces substantial machinery—multivariate polynomial rings, greedy prime selection, and level-dependent data structures—despite achieving the same exponent as the randomized procedure. The proof also depends on several parameter inequalities that should be checked against the exact constants and hypotheses of the cited thin-product theorem.

Conclusion

The paper presents a modular $N \times N$02 matrix multiplication algorithm based on additive approximation, modular sparse-certificate generation, and thin rectangular matrix products. Its central claim is a deterministic and Las Vegas running time of $N \times N$03 for polynomially bounded integer inputs, improving the previously stated $N \times N$04 exponent. The randomized construction is comparatively direct; the deterministic version replaces random prime sampling with potential minimization and a hierarchical depth-first search. The claimed improvement is small, but the framework is technically significant because it removes structural assumptions from the sparse-certificate stage. Its conclusions remain contingent on the cited thin matrix product primitives and on filling the manuscript’s substantial presentation and verification gaps (2610.20297).