Strongly polynomial deterministic algorithm for DisProdTV

Determine whether there exists a strongly polynomial-time deterministic algorithm for the DisProdTV problem that, given two product distributions P = P1 × … × Pn and Q = Q1 × … × Qn over a finite domain [M] and an accuracy parameter ε ∈ (0,1), outputs a (1±ε)-relative approximation to dTV(P,Q) using a number of arithmetic operations bounded by a polynomial only in n, M, and 1/ε, independent of the bit-length of the input probabilities.

Background

The paper reviews existing deterministic and randomized algorithms for approximating the total variation distance between products of discrete distributions (DisProdTV). While these are polynomial-time in the standard bit complexity model, the deterministic algorithm’s count of arithmetic operations depends on input representation length, hence it is not strongly polynomial.

The authors highlight that a strongly polynomial deterministic algorithm would require the arithmetic operation count to depend only on structural parameters (n, M, 1/ε), not on the size of the numerical encodings. They explicitly note that it is unknown whether such an algorithm exists.