Distributed Computation with Local Advice

TL;DR

This paper investigates the amount of advice needed for distributed graph algorithms, showing that in graphs with sub-exponential growth, many problems can be solved with just 1 bit of advice per node, but some problems require more under certain complexity assumptions.

Contribution

The paper demonstrates that all LCL problems can be solved with 1 bit of advice in specific graph classes and establishes lower bounds under ETH, also providing methods for graph orientation, edge compression, and coloring with minimal advice.

Findings

Any LCL can be solved with 1 bit of advice in sub-exponential growth graphs.

Some problems require more than constant advice bits assuming ETH.

Almost-balanced orientation and coloring can be achieved with 1 bit of advice.

Abstract

In this work we study local computation with advice: the goal is to solve a graph problem $\Pi$ with a distributed algorithm in $T(\Delta)$ communication rounds, for some function $T$ that only depends on the maximum degree $\Delta$ of the graph, and the key question is how many bits of advice per node are needed. Some of our results regard Locally Checkable Labeling problems (LCLs), which are constraint-satisfaction graph problems that can be defined with a finite set of valid input/output-labeled neighborhoods. Our main results are: - Any LCL can be solved with only $1$ bit of advice per node in graphs with sub-exponential growth. Moreover, we can make the set of nodes that carry advice bits arbitrarily sparse. As a corollary, any LCL admits a locally checkable proof with $1$ bit per node in graphs with sub-exponential growth. - The assumption of sub-exponential growth is complemented by a conditional lower bound: assuming the Exponential-Time Hypothesis, there are locally checkable labeling problems that cannot be solved in general with any constant number of bits per node. - In any graph we can find an almost-balanced orientation with $1$ bit of advice per node, and again we can make the advice arbitrarily sparse. As a corollary, we can also compress an arbitrary subset of edges so that a node of degree $d$ stores only $d/2 + 2$ bits, and we can decompress it locally, in $T(\Delta)$ rounds. - In any graph of maximum degree $\Delta$, we can find a $\Delta$-coloring (if it exists) with $1$ bit of advice per node, and again, we can make the advice arbitrarily sparse. - In any $3$-colorable graph, we can find a $3$-coloring with $1$ bit of advice per node. As a corollary, in bounded-degree graphs there is a locally checkable proof that certifies $3$-colorability with $1$ bit of advice per node.