Indeed, this is my third posting for this trilogy, where prior works were recommended in my 248th and 263th postings. The current work resolves an open problem that was bypassed in the 2nd work.
Chapeau (for the entire trilogy).
Consider the family of bounded degree graphs in any minor-closed family (such as planar graphs). Let d be the degree bound and n be the number of vertices of such a graph. Graphs in these classes have hyperfinite decompositions, where, for a sufficiently small eps > 0, one removes $eps dn$ edges to get connected components of size independent of n.
An important tool for sublinear algorithms and property testing for such classes is the partition oracle, introduced by the seminal work of Hassidim-Kelner-Nguyen-Onak (FOCS 2009). A partition oracle is a local procedure that gives consistent access to a hyperfinite decomposition, without any preprocessing. Given a query vertex v, the partition oracle outputs the component containing v in time independent of n. All the answers are consistent with a single hyperfinite decomposition.
The partition oracle of Hassidim et al. runs in time d^poly(d/eps)-per query. They pose the open problem of whether poly(d/eps)-time partition oracles exist. Levi-Ron (ICALP 2013) give a refinement of the previous approach, to get a partition oracle that runs in time d^log(d/eps)-per query.
In this paper, we resolve this open problem and give poly(d/eps)-time partition oracles for bounded degree graphs in any minor-closed family. Unlike the previous line of work based on combinatorial methods, we employ techniques from spectral graph theory. We build on a recent spectral graph theoretical toolkit for minor-closed graph families, introduced by the authors to develop efficient property testers. A consequence of our result is a poly(d/eps)-query tester for any property of minor-closed families (such as bipartite planar graphs). Our result also gives poly(d/eps)-query algorithms for additive $eps n$-approximations for problems such as maximum matching, minimum vertex cover, maximum independent set, and minimum dominating set for these graph families.
See ECCC TR21-008