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.

中文翻译：

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在次要图中测试汉密尔顿性（和其他问题）

在本文中，我们提供了针对输入图中不包括固定次要图（即无次要图）的设置中的几个基本问​​题的次线性算法。特别是，我们为次要自由无界度图提供以下算法。具有双向误差的汉密尔顿性测试器，具有$ poly（1 / \ epsilon）$查询复杂度，其中$ \ epsilon $是接近性参数。Rubinfeld等人定义的局部算法。（ICS 2011），以构造具有几乎最小权重的跨度子图，具体而言，最多为最优因子$（1+ \ epsilon）$，查询复杂度为$ poly（1 / \ epsilon）$。\ end {enumerate}我们的两种算法都使用了由Hassidim等人引入的工具oracle-partition-oracles。（FOCS 2009），这些是提供对图的分区的访问的oracle，这样尖端的数量很小，分区的每个部分都很小。我们算法的$ 1 / \ epsilon $中的多项式相关性是通过结合最近的Kumar-Seshadhri-Stolman（ECCC 2021）的$ poly（d / \ epsilon）$-查询分区oracle来实现的，其次方图的度为$ d $。对于有界度次要无图，我们引入{\ em coverage partition oracles}的概念，它是分区oracle的宽松版本，并为此图系列设计了一个$ poly（d / \ epsilon）$时间覆盖的oracle。使用覆盖分区的oracle，我们为次要的自由界度图提供了与上述相同的结果（除了汉密尔顿性的测试器具有单侧误差），并显示了任何具有单调和加性的性质（例如，可以通过进行$ poly（d / \ epsilon）$查询来在次要自由图中进行测试。在我们的算法中使用覆盖分区的oracle而不是分区oracle的好处是它的简单性和在获得的查询复杂度中$ 1 / \ epsilon $中改进的多项式依赖性。