Strongly Polynomial Deterministic Algorithm for DisProdTV

Determine whether there exists a strongly polynomial-time deterministic algorithm for DisProdTV: given discrete distributions P_i and Q_i over a finite domain [M] for i = 1, …, n and ε in (0, 1), output a value Z satisfying (1−ε) d_TV(P, Q) ≤ Z ≤ (1+ε) d_TV(P, Q), while performing 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

DisProdTV is the problem of approximating the total variation distance between two product distributions P = Π_i P_i and Q = Π_i Q_i over [M]^n to a prescribed relative error. Prior work provides a deterministic polynomial-time (in bit complexity) algorithm whose number of arithmetic operations depends on input representation length, hence it is not strongly polynomial.

The paper highlights that, despite progress on polynomial-time algorithms, it remains unknown whether a deterministic algorithm exists whose arithmetic operation count depends only on structural parameters (n, M, 1/ε) and not on the bit-length of the inputs, i.e., a strongly polynomial algorithm in the sense of combinatorial optimization.