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arXiv · 1707.03132

Coloring Chains for Compression with Uncertain Priors

Noah Golowich
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Abstract

Haramaty and Sudan considered the problem of transmitting a message between two people, Alice and Bob, when Alice's and Bob's priors on the message are allowed to differ by at most a given factor. To find a deterministic compression scheme for this problem, they showed that it is sufficient to obtain an upper bound on the chromatic number of a graph, denoted $U(N,s,k)$ for parameters $N,s,k$, whose vertices are nested sequences of subsets and whose edges are between vertices that have similar sequences of sets. In turn, there is a close relationship between the problem of determining the chromatic number of $U(N,s,k)$ and a local graph coloring problem considered by Erdős et al. We generalize the results of Erdős et al. by finding bounds on the chromatic numbers of graphs $H$ and $G$ when there is a homomorphism $ϕ:H\rightarrow G$ that satisfies a nice property. We then use these results to improve upper and lower bounds on $χ(U(N,s,k))$.

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Noah Golowich. 2018-10-21. Coloring Chains for Compression with Uncertain Priors. https://arxiv.org/abs/1707.03132

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