Brief Announcement: Local Advice and Local Decompression

In this work we study local computation with advice: the goal is to solve a graph problem Π with a distributed algorithm in f (Δ) communication rounds, for some function f that only depends on the maximum degree Δ of the graph, and the key question is how many bits of advice per node are needed. Our main results are: (1) Any locally checkable labeling problem (LCL) can be solved in graphs with sub-exponential growth with only 1 bit of advice per node. Moreover, we can make the set of nodes that carry advice bits arbitrarily sparse. (2) The assumption of sub-exponential growth is necessary: assuming the Exponential-Time Hypothesis (ETH), there are LCLs 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. (4) 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 f(Δ) rounds. (5) In any graph of maximum degree Δ, we can find a Δ-coloring (if it exists) with 1 bit of advice per node, and again, we can make the advice arbitrarily sparse. (6) In any 3-colorable graph, we can find a 3-coloring with 1 bit of advice per node. Here, it remains open whether we can make the advice arbitrarily sparse.