We study the complexity of fundamental distributed graph problems in the recently popular setting where information about the input graph is available to the nodes before the start of the computation. We focus on the most common such setting, known as the Supported LOCAL model, where the input graph (on which the studied graph problem has to be solved) is guaranteed to be a subgraph of the underlying communication network. Building on a successful lower bound technique for the LOCAL model called round elimination, we develop a framework for proving complexity lower bounds in the stronger Supported LOCAL model. Our framework reduces the task of proving a (deterministic or randomized) lower bound for a given problem $\Pi$ to the graph-theoretic task of proving non-existence of a solution to another problem $\Pi'$ (on a suitable graph) that can be derived from $\Pi$ in a mechanical manner. We use the developed framework to obtain substantial (and, in the majority of cases, asymptotically tight) Supported LOCAL lower bounds for a variety of fundamental graph problems, including maximal matching, maximal independent set, ruling sets, arbdefective colorings, and generalizations thereof. In a nutshell, for essentially any major lower bound proved in the LOCAL model in recent years, we prove a similar lower bound in the Supported LOCAL model. Our framework also gives rise to a new deterministic version of round elimination in the LOCAL model: while, previous to our work, the general round elimination technique required the use of randomness (even for obtaining deterministic lower bounds), our framework allows to obtain deterministic (and therefore via known lifting techniques also randomized) lower bounds in a purely deterministic manner. Previously, such a purely deterministic application of round elimination was only known for the specific problem of sinkless orientation [SOSA'23].

翻译：我们研究了近期流行的设定下基本分布式图问题的复杂度，即计算开始前节点已获取输入图信息。重点关注最常见的"支持LOCAL模型"（Supported LOCAL model）设定——待求解图问题的输入图必须是底层通信网络的子图。基于名为"轮消除"（round elimination）的LOCAL模型成功下界技术，我们构建了在更强的支持LOCAL模型中证明复杂度下界的框架。该框架将证明给定问题Π的（确定性或随机化）下界简化为图论任务：证明在另一问题Π'（定义于适当图上）的解不存在性，而Π'可通过机械方式从Π导出。利用所开发的框架，我们为多种基本图问题获得了重要（且大多数情况下渐近紧的）支持LOCAL下界，包括最大匹配、最大独立集、统治集、任意缺陷着色及其推广。简而言之，对于近年LOCAL模型中证明的几乎所有主要下界，我们在支持LOCAL模型中建立了类似下界。我们的框架还催生了LOCAL模型中轮消除技术的新确定性版本：此前通用轮消除技术需要随机性（即使证明确定性下界），而我们的框架能以纯确定性方式证明确定性（进而通过已知提升技术获得随机化）下界。此前，这类纯确定性轮消除应用仅已知于无源定向问题[SOSA'23]。