{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T04:23:12Z","timestamp":1750220592185,"version":"3.41.0"},"reference-count":23,"publisher":"Association for Computing Machinery (ACM)","issue":"1","license":[{"start":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T00:00:00Z","timestamp":1611187200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Comput. Theory"],"published-print":{"date-parts":[[2021,3,31]]},"abstract":"<jats:p>\n            We prove a simple, nearly tight lower bound on the approximate degree of the two-level AND-OR tree using symmetrization arguments. Specifically, we show that\n            <jats:sup>\u02dc<\/jats:sup>\n            deg(AND\n            <jats:sub>\n              <jats:italic>m<\/jats:italic>\n            <\/jats:sub>\n            \u02c6 OR\n            <jats:sub>\n              <jats:italic>n<\/jats:italic>\n            <\/jats:sub>\n            ) =\n            <jats:sup>\u02dc<\/jats:sup>\n            \u03a9(\u221a\n            <jats:italic>mn<\/jats:italic>\n            ). We prove this lower bound via reduction to the OR function through a series of symmetrization steps, in contrast to most other proofs that involve formulating approximate degree as a linear program [6, 10, 21]. Our proof also demonstrates the power of a symmetrization technique involving Laurent polynomials (polynomials with negative exponents) that was previously introduced by Aaronson et al. [2].\n          <\/jats:p>","DOI":"10.1145\/3434385","type":"journal-article","created":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T11:11:18Z","timestamp":1611227478000},"page":"1-11","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Lower Bounding the AND-OR Tree via Symmetrization"],"prefix":"10.1145","volume":"13","author":[{"given":"William","family":"Kretschmer","sequence":"first","affiliation":[{"name":"University of Texas at Austin"}]}],"member":"320","published-online":{"date-parts":[[2021,1,21]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2008.91"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.4230\/LIPIcs.CCC.2020.7"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/335305.335394"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.4086\/toc.2005.v001a003"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/502090.502097"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2009.18"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/276698.276713"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(01)00144-X"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1145\/3188745.3188784"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-39206-1_26"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.4086\/toc.2016.v012a016"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1145\/237814.237866"},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.5555\/1759210.1759241"},{"key":"e_1_2_1_14_1","volume-title":"Perceptrons: An Introduction to Computational Geometry","author":"Minsky Marvin","year":"1969","unstructured":"Marvin Minsky and Seymour Papert . 1969 . Perceptrons: An Introduction to Computational Geometry . MIT Press . Marvin Minsky and Seymour Papert. 1969. Perceptrons: An Introduction to Computational Geometry. MIT Press."},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1145\/129712.129757"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/129712.129758"},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1109\/SFCS.1986.44"},{"key":"e_1_2_1_18_1","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2009.18"},{"key":"e_1_2_1_19_1","doi-asserted-by":"publisher","DOI":"10.1145\/1993636.1993643"},{"key":"e_1_2_1_20_1","doi-asserted-by":"publisher","DOI":"10.1145\/2213977.2214044"},{"key":"e_1_2_1_21_1","doi-asserted-by":"publisher","DOI":"10.4086\/toc.2013.v009a020"},{"key":"e_1_2_1_22_1","doi-asserted-by":"publisher","DOI":"10.1145\/3313276.3316408"},{"key":"e_1_2_1_23_1","unstructured":"Yaoyun Shi. 2002. Approximating linear restrictions of Boolean functions. Retrieved from https:\/\/web.eecs.umich.edu\/ shiyy\/mypapers\/linear02-j.ps.  Yaoyun Shi. 2002. Approximating linear restrictions of Boolean functions. Retrieved from https:\/\/web.eecs.umich.edu\/ shiyy\/mypapers\/linear02-j.ps."}],"container-title":["ACM Transactions on Computation Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3434385","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3434385","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T21:31:47Z","timestamp":1750195907000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3434385"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,21]]},"references-count":23,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,3,31]]}},"alternative-id":["10.1145\/3434385"],"URL":"https:\/\/doi.org\/10.1145\/3434385","relation":{},"ISSN":["1942-3454","1942-3462"],"issn-type":[{"type":"print","value":"1942-3454"},{"type":"electronic","value":"1942-3462"}],"subject":[],"published":{"date-parts":[[2021,1,21]]},"assertion":[{"value":"2020-03-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2020-09-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2021-01-21","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}