Efficient $\varepsilon$-approximate minimum-entropy couplings

Abstract: Given $m \ge 2$ discrete probability distributions over $n$ states each, the minimum-entropy coupling is the minimum-entropy joint distribution whose marginals are the same as the input distributions. Computing the minimum-entropy coupling is NP-hard, but there has been significant progress in designing approximation algorithms; prior to this work, the best known polynomial-time algorithms attain guarantees of the form $$H(\operatorname{ALG}) \le H(\operatorname{OPT}) + c$$, where $c \approx 0.53$ for $m=2$, and $c \approx 1.22$ for general $m$ [CKQGK '23]. A main open question is whether this task is APX-hard, or whether there exists a polynomial-time approximation scheme (PTAS). In this work, we design an algorithm that produces a coupling with entropy $$H(\operatorname{ALG}) \le H(\operatorname{OPT}) + \varepsilon$$ in running time $$n^{{O(\operatorname{poly}(1/\varepsilon)} \cdot \operatorname{exp}(m) )}$$: showing a PTAS exists for constant $m$.