Recent work on distributed graph algorithms [e.g. STOC 2022, ITCS 2022, PODC 2020] has drawn attention to the following open question: are round elimination fixed points a universal technique for proving lower bounds? That is, given a locally checkable problem $\Pi$ that requires at least $\Omega(\log n)$ rounds in the deterministic LOCAL model, can we always find a relaxation $\Pi'$ of $\Pi$ that is a nontrivial fixed point for the round elimination technique [see STOC 2016, PODC 2019]? If yes, then a key part of distributed computational complexity would be also decidable. The key obstacle so far has been a certain family of homomorphism problems [ITCS 2022], which require $\Omega(\log n)$ rounds, but the only known proof is based on Marks' technique [J.AMS 2016]. We develop a new technique for constructing round elimination lower bounds systematically. Using so-called tripotent inputs we show that the aforementioned homomorphism problems indeed admit a lower bound proof that is based on round elimination fixed points. Hence we eliminate the only known obstacle for the universality of round elimination. Yet we also present a new obstacle: we show that there are some problems with inputs that require $\Omega(\log n)$ rounds, yet there is no proof that is based on relaxations to nontrivial round elimination fixed points. Hence round elimination cannot be a universal technique for problems with inputs (but it might be universal for problems without inputs). We also prove the first fully general lower bound theorem that is applicable to any problem, with or without inputs, that is a fixed point in round elimination. Prior results of this form were only able to handle certain very restricted inputs.

翻译：近期关于分布式图算法的研究[例如STOC 2022, ITCS 2022, PODC 2020]引发了以下开放性问题：轮次消去不动点是否构成证明下界的普适性技术？即给定在确定性LOCAL模型中至少需要$\Omega(\log n)$轮求解的局部可验证问题$\Pi$，我们是否总能找到$\Pi$的一个松弛版本$\Pi'$，使其成为轮次消去技术[参见STOC 2016, PODC 2019]的非平凡不动点？若可证明，则分布式计算复杂性的关键部分也将具备可判定性。目前的主要障碍在于某类同态问题族[ITCS 2022]——这些问题需要$\Omega(\log n)$轮求解，但现有唯一证明依赖于Marks技术[J.AMS 2016]。本文开发了一种系统性构建轮次消去下界的新技术。通过使用所谓的三幂输入，我们证明前述同态问题确实存在基于轮次消去不动点的下界证明，从而消除了轮次消去普适性理论的唯一已知障碍。然而我们也提出了新的障碍：证明存在某些带输入的问题需要$\Omega(\log n)$轮求解，但无法通过松弛至非平凡轮次消去不动点的方式证明，这意味着轮次消去技术对带输入问题不具备普适性（但对无输入问题可能仍具普适性）。此外，我们首次证明了完全通用的下界定理，该定理适用于任意作为轮次消去不动点的问题（无论是否带输入），而此前同类结果仅能处理某些特定受限输入。