Given a set of discrete probability distributions, the minimum entropy coupling is the minimum entropy joint distribution that has the input distributions as its marginals. This has immediate relevance to tasks such as entropic causal inference for causal graph discovery and bounding mutual information between variables that we observe separately. Since finding the minimum entropy coupling is NP-Hard, various works have studied approximation algorithms. The work of [Compton, ISIT 2022] shows that the greedy coupling algorithm of [Kocaoglu et al., AAAI 2017] is always within $log_2(e) \approx 1.44$ bits of the optimal coupling. Moreover, they show that it is impossible to obtain a better approximation guarantee using the majorization lower-bound that all prior works have used: thus establishing a majorization barrier. In this work, we break the majorization barrier by designing a stronger lower-bound that we call the profile method. Using this profile method, we are able to show that the greedy algorithm is always within $log_2(e)/e \approx 0.53$ bits of optimal for coupling two distributions (previous best-known bound is within 1 bit), and within $(1 + log_2(e))/2 \approx 1.22$ bits for coupling any number of distributions (previous best-known bound is within 1.44 bits). We also examine a generalization of the minimum entropy coupling problem: Concave Minimum-Cost Couplings. We are able to obtain similar guarantees for this generalization in terms of the concave cost function. Additionally, we make progress on the open problem of [Kova\v{c}evi\'c et al., Inf. Comput. 2015] regarding NP membership of the minimum entropy coupling problem by showing that any hardness of minimum entropy coupling beyond NP comes from the difficulty of computing arithmetic in the complexity class NP. Finally, we present exponential-time algorithms for computing the exactly optimal solution.

翻译：给定一组离散概率分布，最小熵耦合是指以输入分布为边缘分布且具有最小熵的联合分布。该问题直接关联因果图发现中的熵因果推理以及分别观测变量间互信息界定的任务。由于寻找最小熵耦合是NP-难问题，已有诸多工作研究了近似算法。[Compton, ISIT 2022] 的研究表明，[Kocaoglu et al., AAAI 2017] 的贪婪耦合算法始终在最优耦合的 $log_2(e) \approx 1.44$ 比特内。此外，他们指出，使用所有先前工作采用的主化下界无法获得更好的近似保证，由此确立了主化障碍。在本工作中，我们通过设计一种称为轮廓法的更强下界突破了主化障碍。利用该轮廓法，我们证明贪婪算法对两个分布耦合时始终在最优解的 $log_2(e)/e \approx 0.53$ 比特内（此前最佳已知界为1比特内），对任意数量分布耦合时在 $(1 + log_2(e))/2 \approx 1.22$ 比特内（此前最佳已知界为1.44比特内）。我们还研究了最小熵耦合问题的推广形式：凹最小代价耦合。针对该推广问题，我们基于凹代价函数获得了类似的保证。此外，我们通过证明最小熵耦合的困难性若超出NP则源于复杂度类NP中算术计算的难度，从而推进了[Kovačević et al., Inf. Comput. 2015] 关于最小熵耦合问题NP成员资格的开放问题。最后，我们提出了计算精确最优解的指数时间算法。