A Tighter Approximation Guarantee for Greedy Minimum Entropy Coupling

Abstract

We examine the minimum entropy coupling problem, where one must find the minimum entropy variable that has a given set of distributions S = {p1, . . . , pm} as its marginals. Although this problem is NP-Hard, previous works have proposed algorithms with varying approximation guarantees. In this paper, we show that the greedy coupling algorithm of [Kocaoglu et al., AAAI'17] is always within log 2 (e) (≈ 1.44) bits of the minimum entropy coupling. In doing so, we show that the entropy of the greedy coupling is upper-bounded by H( S) + log 2 (e). This improves the previously best known approximation guarantee of 2 bits within the optimal [Li, IEEE Trans. Inf. Theory '21]. Moreover, we show our analysis is tight by proving there is no algorithm whose entropy is upper-bounded by H( S) + c for any constant c < log 2 (e). Additionally, we examine a special class of instances where the greedy coupling algorithm is exactly optimal.