Tight Lower Bounds in the Supported LOCAL Model

TL;DR

This paper develops a deterministic round-elimination framework for the Supported LOCAL model, where nodes know the input graph G' as a subgraph of the network G. Central to the approach is a lift operation, $ ext{lift}$, that links 0-round solvability of a problem $\Pi$ to a purely graph-theoretic solvability question for $\text{lift}(\Pi)$ on the support graph, enabling reductions to graph-theoretic existence questions and eliminating the need for randomness in the core argument. The authors obtain tight or near-tight lower bounds for a broad set of fundamental problems, including maximal matching, maximal independent set, ruling sets, arbdefective colorings, and arbdefective colored ruling sets, extending the scope of LOCAL lower bounds to the Supported LOCAL setting. They also derive a lifting theorem that translates deterministic lower bounds into randomized ones, and show that deterministic bounds in Supported LOCAL immediately imply corresponding bounds in LOCAL via standard lifting. Overall, the framework unifies and extends major lower-bound results from LOCAL to the Supported LOCAL model, offering a practical route to deterministic lower bounds in related locality models and hypergraphs.

Abstract

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 $Π$ to the graph-theoretic task of proving non-existence of a solution to another problem $Π'$ (on a suitable graph) that can be derived from $Π$ 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].

Figures (3)

• Figure 1: Black diagram of $\Pi$.
• Figure 2: Black diagram of $\Pi$, in the case where $\mathcal{C}$ contains $3$ colors denoted with $\mathsf{A}$, $\mathsf{B}$, and $\mathsf{C}$, and $\beta = 2$.
• Figure 3: An example of a solution to the maximal matching problem in the black-white formalism.

Theorems & Definitions (56)

• Theorem 1.1: Simplified version of \ref{['lem:re-works']}
• Theorem 1.2: Simplified version of \ref{['th:lift-bipartite']}
• Theorem 1.3: Simplified version of \ref{['lem:derand-supported']}
• Theorem 1.4: Simplified version of \ref{['thm:approach-bipartite']}
• Theorem 1.5: Simplified version of \ref{['thm:lb-xy-matching']}
• Theorem 1.6: Simplified version of \ref{['thm:arb-coloring']}
• Theorem 1.7: Simplified version of \ref{['thm:acbrs']}
• Lemma 2.1: Alon10
• Definition 3.1
• Theorem 3.2