Testing Hamiltonicity (And Other Problems) in Minor-Free Graphs.

In this paper we provide sub-linear algorithms for several fundamental problems in the setting in which the input graph excludes a fixed minor, i.e., is a minor-free graph. In particular, we provide the following algorithms for minor-free unbounded degree graphs. A tester for Hamiltonicity with two-sided error with $poly(1/\epsilon)$-query complexity, where $\epsilon$ is the proximity parameter. A local algorithm, as defined by Rubinfeld et al. (ICS 2011), for constructing a spanning subgraph with almost minimum weight, specifically, at most a factor $(1+\epsilon)$ of the optimum, with $poly(1/\epsilon)$-query complexity. \end{enumerate} Both our algorithms use partition-oracles, a tool introduced by Hassidim et al. (FOCS 2009), which are oracles that provide access to a partition of the graph such that the number of cut-edges is small and each part of the partition is small. The polynomial dependence in $1/\epsilon$ of our algorithms is achieved by combining the recent $poly(d/\epsilon)$-query partition oracle of Kumar-Seshadhri-Stolman (ECCC 2021) for minor-free graphs with degree bounded by $d$. For bounded degree minor-free graphs we introduce the notion of {\em covering partition oracles} which is a relaxed version of partition oracles and design a $poly(d/\epsilon)$-time covering partition oracle for this family of graphs. Using our covering partition oracle we provide the same results as above (except that the tester for Hamiltonicity has one sided error) for minor free bounded degree graphs, as well as showing that any property which is monotone and additive (e.g. bipartiteness) can be tested in minor-free graphs by making $poly(d/\epsilon)$-queries. The benefit of using the covering partition oracle rather than the partition oracle in our algorithms is its simplicity and an improved polynomial dependence in $1/\epsilon$ in the obtained query complexity.