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Given two discrete random variables $X$ and $Y$, with probabilitydistributions ${\bf p} =(p_1, \ldots , p_n)$ and ${\bf q}=(q_1, \ldots , q_m)$,respectively, denote by ${\cal C}({\bf p}, {\bf q})$ the set of all couplingsof ${\bf p}$ and ${\bf q}$, that is, the set of all bivariate probabilitydistributions that have ${\bf p}$ and ${\bf q}$ as marginals. In this paper, westudy the problem of finding the joint probability distribution in ${\calC}({\bf p}, {\bf q})$ of minimum entropy (equivalently, the joint probabilitydistribution that maximizes the mutual information between $X$ and $Y$), and wediscuss several situations where the need for this kind of optimizationnaturally arises. Since the optimization problem is known to be NP-hard, wegive an efficient algorithm to find a joint probability distribution in ${\calC}({\bf p}, {\bf q})$ with entropy exceeding the minimum possible by at most 1,thus providing an approximation algorithm with additive approximation factor of1. Leveraging on this algorithm, we extend our result to the problem of findinga minimum--entropy joint distribution of arbitrary $k\geq 2$ discrete randomvariables $X_1, \ldots , X_k$, consistent with the known $k$ marginaldistributions of $X_1, \ldots , X_k$. In this case, our approximation algorithmhas an additive approximation factor of $\log k$. We also discuss some relatedapplications of our findings.

How to Find a Joint Probability Distribution of Minimum Entropy (almost) given the Marginals