Result 114, Theoretical computer science

Approximate counting of common integer polymatroid bases

Gives a fully polynomial randomized approximation scheme for counting common integer bases of two integral polymatroids of equal total rank, supplied by exact rank-value oracles. Capacities are binary-encoded, each integer vector counts once, and oracle calls and bit operations outside the oracles are polynomial on every execution. For matroids presented by independence oracles, the results also cover common independent sets of prescribed, unrestricted, or maximum cardinality, even when the ranks differ.

Lean formalization Algorithm or complexity result

The bigger picture

Why it matters

The manuscript reports an efficient randomized way to estimate how many integer allocations simultaneously satisfy two systems of capacity constraints. It addresses counting possibilities, not merely finding one, even when capacities are large.

What changes?

An integral polymatroid describes nonnegative integer allocations constrained by subset-total bounds, called ranks, with a diminishing-returns structure. A base attains the total rank. For two such systems of equal total rank, the claimed scheme uses exact rank-value oracles, which return subset bounds. On every execution, oracle calls and external bit operations are polynomial in ground-set size, binary input length, inverse relative-error tolerance, and log inverse failure probability. Total rank and capacities are binary-encoded; each vector counts once.

What does that help mathematicians do?

The estimate comes with user-chosen relative-error and failure-probability guarantees. Handling binary capacities directly avoids replacing a large capacity by that many labelled copies, which can make the representation exponentially larger and distort counts by distinguishing copies. The summary also extends counting to common independent sets of two matroids, combinatorial systems specifying allowable subsets, with prescribed, unrestricted, or maximum size, even when their ranks differ.

Are there practical applications?

Its immediate value is foundational for counting under overlapping combinatorial constraints. Matroids need only independence oracles, procedures that test whether a subset is allowed, rather than explicit representations. The demonstrated efficiency concerns oracle calls and work outside those oracles; it does not establish that evaluating the oracles is inexpensive in every application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

An FPRAS for Common Integer Polymatroid Bases with Binary Capacities

October 5, 2026 42 pages

We give a fully polynomial randomized approximation scheme for counting common integer bases of two polymatroids with the same total rank, supplied by exact rank-value oracles. The total rank and capacities are encoded in binary, and each integer vector is counted once. On every execution, the number of oracle calls and the bit work outside the oracles are bounded by a fixed polynomial in the ground-set size, the binary input length, the inverse relative-error tolerance, and the logarithm of the inverse failure probability. The algorithm handles binary capacities directly, without expanding them into labelled copies.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:An-FPRAS-for-Common-Integer-Polymatroid-Bases-with-Binary-Capacities-October-5-2026,
  author = {{OpenAI}},
  title = {{An FPRAS for Common Integer Polymatroid Bases with Binary Capacities}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-FPRAS-for-Common-Integer-Polymatroid-Bases-with-Binary-Capacities-October-5-2026/polymatroid-fpras.pdf}{OAI:An-FPRAS-for-Common-Integer-Polymatroid-Bases-with-Binary-Capacities-October-5-2026}},
  year = {2026}
}

Approximate counting of common bases of two matroids

September 23, 2026 26 pages Main result formalized in Lean

We give a fully polynomial randomized approximation scheme for counting the common bases of two arbitrary matroids of the same rank, supplied by independence oracles. The algorithm requires no explicit representation of either matroid and has polynomial bounds on both oracle calls and bit operations on every execution.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Approximate-counting-of-common-bases-of-two-matroids-September-23-2026,
  author = {{OpenAI}},
  title = {{Approximate counting of common bases of two matroids}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Approximate-counting-of-common-bases-of-two-matroids-September-23-2026/main.pdf}{OAI:Approximate-counting-of-common-bases-of-two-matroids-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/114.md.

Approximate counting of common integer polymatroid bases

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Approximate counting of common bases of two matroids

Scope

The formalized result gives a fully polynomial randomized approximation scheme for the number of common bases of two equal-rank matroids on an enumerated finite ground set, using independence oracles. For rational $0<\varepsilon,\delta<1$ , the nonnegative rational output has relative error at most $\varepsilon$ with probability at least $1-\delta$ . A zero count produces zero on every execution. Oracle calls and bit operations have polynomial bounds on every random tape in the input size, $\varepsilon^{-1}$ , and $\log(\delta^{-1})$ .

Comparator links

Result	Comparator statement
FPRAS for common matroid bases	CommonBasesFPRAS.lean