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)

Manuscripts (2)

An FPRAS for Common Integer Polymatroid Bases with Binary Capacities

October 5, 2026 · source · cite as \cite{OAI:An-FPRAS-for-Common-Integer-Polymatroid-Bases-with-Binary-Capacities-October-5-2026}

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.

Approximate counting of common bases of two matroids

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:Approximate-counting-of-common-bases-of-two-matroids-September-23-2026}

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.

Result 114 · Theoretical computer science. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.