{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:09:38Z","timestamp":1758823778116},"reference-count":31,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2012,8,10]],"date-time":"2012-08-10T00:00:00Z","timestamp":1344556800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2012,11]]},"abstract":"<jats:p>A function<jats:italic>f<\/jats:italic>:<jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548312000363_char1\" \/><\/jats:private-char><jats:sub>2<\/jats:sub><jats:sup><jats:italic>n<\/jats:italic><\/jats:sup>\u2192 {0,1} is<jats:italic>odd-cycle-free<\/jats:italic>if there are no<jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>,.\u00a0.\u00a0.,<jats:italic>x<jats:sub>k<\/jats:sub><\/jats:italic>\u2208<jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548312000363_char1\" \/><\/jats:private-char><jats:sub>2<\/jats:sub><jats:sup><jats:italic>n<\/jats:italic><\/jats:sup>with<jats:italic>k<\/jats:italic>an odd integer such that<jats:italic>f<\/jats:italic>(<jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>) = \u00b7\u00b7\u00b7 =<jats:italic>f(x<jats:sub>k<\/jats:sub>)<\/jats:italic>= 1 and<jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>+ \u00b7\u00b7\u00b7 +<jats:italic>x<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>= 0. We show that one can distinguish odd-cycle-free functions from those \u03b5-far from being odd-cycle-free by making poly(1\/\u03b5) queries to an evaluation oracle. We give two proofs of this result, each shedding light on a different connection between testability of properties of Boolean functions and of dense graphs.<\/jats:p><jats:p>The first issue we study is directly reducing testing of linear-invariant properties of Boolean functions to testing associated graph properties. We show a black-box reduction from testing odd-cycle-freeness to testing bipartiteness of graphs. Such reductions have already been shown (Kr\u00e1l\u2019, Serra and Vena, and Shapira) for monotone linear-invariant properties defined by forbidding solutions to a<jats:italic>finite<\/jats:italic>number of equations. But for odd-cycle-freeness whose description involves an infinite number of forbidden equations, a reduction to graph property testing was not previously known. If one could show such a reduction more generally for any linear-invariant property closed under restrictions to subspaces, then it would likely lead to a characterization of the one-sided testable linear-invariant properties, an open problem raised by Sudan.<\/jats:p><jats:p>The second issue we study is whether there is an efficient<jats:italic>canonical<\/jats:italic>tester for linear-invariant properties of Boolean functions. A canonical tester for linear-invariant properties operates by picking a random linear subspace and then checking whether the restriction of the input function to the subspace satisfies a fixed property. The question is if, for every linear-invariant property, there is a canonical tester for which there is only a polynomial blow-up from the optimal query complexity. We answer the question affirmatively for odd-cycle-freeness. The general question remains open.<\/jats:p>","DOI":"10.1017\/s0963548312000363","type":"journal-article","created":{"date-parts":[[2012,8,10]],"date-time":"2012-08-10T13:21:56Z","timestamp":1344604916000},"page":"835-855","source":"Crossref","is-referenced-by-count":4,"title":["Testing Odd-Cycle-Freeness in Boolean Functions"],"prefix":"10.1017","volume":"21","author":[{"given":"ARNAB","family":"BHATTACHARYYA","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ELENA","family":"GRIGORESCU","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"PRASAD","family":"RAGHAVENDRA","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ASAF","family":"SHAPIRA","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2012,8,10]]},"reference":[{"key":"S0963548312000363_ref17","doi-asserted-by":"publisher","DOI":"10.1137\/090749621"},{"key":"S0963548312000363_ref2","doi-asserted-by":"publisher","DOI":"10.1006\/jagm.1994.1005"},{"key":"S0963548312000363_ref27","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20017"},{"key":"S0963548312000363_ref15","unstructured":"Chen V. , Sudan M. and Xie N. 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