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

Authors: Reut Levi and Nadav Shoshan

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

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. 1) A tester for Hamiltonicity with two-sided error with poly(1/ε)-query complexity, where ε is the proximity parameter. 2) 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+ε) of the optimum, with poly(1/ε)-query complexity. 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/ε of our algorithms is achieved by combining the recent poly(d/ε)-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 covering partition oracles which is a relaxed version of partition oracles and design a poly(d/ε)-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/ε)-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/ε in the obtained query complexity.

Cite as

Reut Levi and Nadav Shoshan. Testing Hamiltonicity (And Other Problems) in Minor-Free Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 61:1-61:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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  author =	{Levi, Reut and Shoshan, Nadav},
  title =	{{Testing Hamiltonicity (And Other Problems) in Minor-Free Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{61:1--61:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.61},
  URN =		{urn:nbn:de:0030-drops-147540},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.61},
  annote =	{Keywords: Property Testing, Hamiltonian path, minor free graphs, sparse spanning sub-graphs}
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