Finding a Bounded-Degree Expander Inside a Dense One

It follows from the Marcus-Spielman-Srivastava proof of the Kadison-Singer conjecture that if $G=(V,E)$ is a $Δ$-regular dense expander then there is an edge-induced subgraph $H=(V,E_H)$ of $G$ of constant maximum degree which is also an expander. As with other consequences of the MSS theorem, it is not clear how one would explicitly construct such a subgraph. We show that such a subgraph (although with quantitatively weaker expansion and near-regularity properties than those predicted by MSS) can be constructed with high probability in linear time, via a simple algorithm. Our algorithm allows a distributed implementation that runs in $\mathcal O(\log n)$ rounds and does $\mathcal O(n)$ total work with high probability. The analysis of the algorithm is complicated by the complex dependencies that arise between edges and between choices made in different rounds. We sidestep these difficulties by following the combinatorial approach of counting the number of possible random choices of the algorithm which lead to failure. We do so by a compression argument showing that such random choices can be encoded with a non-trivial compression. Our algorithm bears some similarity to the way agents construct a communication graph in a peer-to-peer network, and, in the bipartite case, to the way agents select servers in blockchain protocols.