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Comput."],"published-print":{"date-parts":[[2023,4]]},"abstract":"<jats:p>Let $G$ be a graph with $n$ vertices and maximum degree $d$. Fix some minor-closed property $\\mathcal{P}$ (such as planarity). We say that $G$ is $\\varepsilon$-far from $\\mathcal{P}$ if one has to remove $\\varepsilon dn$ edges to make it have $\\mathcal{P}$. The problem of property testing $\\mathcal{P}$ was introduced in the seminal work of Benjamini, Schramm, and Shapira (Symposium on the Theory of Computing 2008) that gave a tester with query complexity triply exponential in $\\varepsilon^{-1}$. Levi and Ron [ ACM Trans. Algorithms, 11 (2005), 24 2015] have given the best tester to date, with a quasi-polynomial (in $\\varepsilon^{-1}$) query complexity. It remained an open problem to show whether there is a property tester whose query complexity is $\\mathrm{poly}(d\\varepsilon^{-1})$, even for planarity. In this paper, we resolve this open question. For any minor-closed property, we give a tester with query complexity $d\\cdot \\mathrm{poly}(\\varepsilon^{-1})$. The previous line of work on (independent of $n$, two-sided) testers is primarily combinatorial. Our work, on the other hand, employs techniques from spectral graph theory. This paper is a continuation of recent work of the authors (Foundations of Computer Science 2018) analyzing random walk algorithms that find forbidden minors.<\/jats:p>","DOI":"10.1137\/20m1312824","type":"journal-article","created":{"date-parts":[[2022,8,18]],"date-time":"2022-08-18T09:52:16Z","timestamp":1660816336000},"page":"STOC19-323-STOC19-338","source":"Crossref","is-referenced-by-count":0,"title":["Random Walks and Forbidden Minors II: A $\\mathrm{poly}(d\\varepsilon^{-1})$-Query Tester for Minor-Closed Properties of Bounded-Degree Graphs"],"prefix":"10.1137","volume":"52","author":[{"given":"Akash","family":"Kumar","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"C.","family":"Seshadhri","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andrew M.","family":"Stolman","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2022,8,16]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1990-1065053-0"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1145\/1374376.1374433"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20462"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-14279-6"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511581274"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-22935-0_45"},{"key":"atypb7","volume-title":"Proceedings of the 45th International Colloquium on Automata, Languages, and Programming (ICALP)","author":"Fichtenberger H.","year":"2018"},{"key":"atypb8","volume-title":"On Testing Minor-Freeness in Bounded Degree Graphs with One-Sided Error, preprint, arXiv:1707.06126 [cs.DS]","author":"Fichtenberger H.","year":"2017"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1017\/9781108135252"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/s004930050060"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/s00453-001-0078-7"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2009.77"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1145\/321850.321852"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/110837863"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2011.07.004"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2018.00055"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2018.00055"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.4064\/fm-15-1-271-283"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1145\/2629508"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1109\/FSCS.1990.89553"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/120890946"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1995.1034"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1995.1006"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2004.08.001"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1137\/080744888"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1007\/BF01594196"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1007\/s00453-014-9897-1"}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M1312824","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:06:56Z","timestamp":1787339216000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M1312824"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,16]]},"references-count":27,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,4,30]]}},"alternative-id":["10.1137\/20M1312824"],"URL":"https:\/\/doi.org\/10.1137\/20m1312824","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,8,16]]}}}