# On the Probe Complexity of Local Computation Algorithms

## TL;DR

This paper investigates the probe complexity of local computation algorithms (LCAs) for graph problems, establishing bounds and separations for different coloring and matching problems, and designing efficient algorithms for approximating maximum matching.

## Contribution

It provides new bounds on probe complexity for weak coloring and matching problems, including a separation between 2- and 3-coloring, and introduces an efficient randomized LCA for approximate maximum matching.

## Key findings

- Log-star probes suffice for weak 3-coloring.
- Omega(log n / log log n) probes are needed for weak 2-coloring.
- A new randomized LCA approximates maximum matching with degree-independent probes.

## Abstract

The Local Computation Algorithms (LCA) model is a computational model aimed at problem instances with huge inputs and output. For graph problems, the input graph is accessed using probes: strong probes (SP) specify a vertex $v$ and receive as a reply a list of $v$'s neighbors, and weak probes (WP) specify a vertex $v$ and a port number $i$ and receive as a reply $v$'s $i^{th}$ neighbor. Given a local query (e.g., "is a certain vertex in the vertex cover of the input graph?"), an LCA should compute the corresponding local output (e.g., "yes" or "no") while making only a small number of probes, with the requirement that all local outputs form a single global solution (e.g., a legal vertex cover). We study the probe complexity of LCAs that are required to work on graphs that may have arbitrarily large degrees. In particular, such LCAs are expected to probe the graph a number of times that is significantly smaller than the maximum, average, or even minimum degree. For weak probes, we focus on the weak coloring problem. Among our results we show a separation between weak 3-coloring and weak 2-coloring for deterministic LCAs: $\log^* n + O(1)$ weak probes suffice for weak 3-coloring, but $\Omega\left(\frac{\log n}{\log\log n}\right)$ weak probes are required for weak 2-coloring. For strong probes, we consider randomized LCAs for vertex cover and maximal/maximum matching. Our negative results include showing that there are graphs for which finding a \emph{maximal} matching requires $\Omega(\sqrt{n})$ strong probes. On the positive side, we design a randomized LCA that finds a $(1-\epsilon)$ approximation to \emph{maximum} matching in regular graphs, and uses $\frac{1}{\epsilon }^{O\left( \frac{1}{\epsilon ^2}\right)}$ probes, independently of the number of vertices and of their degrees.