Algorithms with advice have received ample attention in the distributed and online settings, and they have recently proven useful also in dynamic settings. 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 is an important family of problems that includes various coloring and orientation problems on finite-degree graphs. These are constraint-satisfaction graph problems that can be defined with a finite set of valid input/output-labeled neighborhoods.

Our main results are:

1. Any locally checkable labeling problem can be solved with only $1$ bit of advice per node in graphs with sub-exponential growth (the number of nodes within radius $r$ is sub-exponential in $r$; for example, grids are such graphs). Moreover, we can make the set of nodes that carry advice bits arbitrarily sparse. As a corollary, any locally checkable labeling problem admits a locally checkable proof with $1$ bit per node in graphs with sub-exponential growth.
2. 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.
3. In any graph we can find an almost-balanced orientation (indegrees and outdegrees differ by at most one) 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.
4. 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.
5. 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, while prior work shows that this is not possible with a proof labeling scheme (PLS), which is a more restricted setting where the verifier can only see up to distance $1$.

Our work shows that for many problems the key threshold is not whether we can achieve, say, $1$ bit of advice per node, but whether we can make the advice arbitrarily sparse. To formalize this idea, we develop a general framework of composable schemas that enables us to build algorithms for local computation with advice in a modular fashion: once we have (1) a schema for solving $\Pi_1$ and (2) a schema for solving $\Pi_2$ assuming an oracle for $\Pi_1$, we can also compose them and obtain (3) a schema that solves $\Pi_2$ without the oracle. It turns out that many natural problems admit composable schemas, all of them can be solved with only $1$ bit of advice, and we can make the advice arbitrarily sparse.