{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:39:49Z","timestamp":1787337589703,"version":"build-2736575974"},"reference-count":14,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2003,1]]},"abstract":"<jats:p>Convex and submodular functions play an important role in many applications, and in particular in combinatorial optimization. Here we study two special cases: convexity in one dimension and submodularity in two dimensions. The latter type of functions are equivalent to the well-known Monge matrices. A matrix $V = \\{v_{i,j}\\}_{i,j=0}^{i=n_1,j=n_2}$ is called a Monge matrix if for every $0 \\leq i &lt; i' \\leq n_1$ and $0 \\leq j &lt; j' \\leq n_2$ we have $v_{i,j}+v_{i',j'} \\le v_{i,j'}+v_{i',j}$. If inequality holds in the opposite direction, then V is an inverse Monge matrix (supermodular function). Many problems, such as the traveling salesperson problem and various transportation problems, can be solved more efficiently if the input is a Monge matrix.<\/jats:p>\n                  <jats:p>\n                    In this work we present testing algorithms for the above properties. A testing algorithm for a predetermined property $\\cal P$ is given query access to an unknown function f and a distance parameter $\\epsilon$. The algorithm should accept f with high probability if it has the property $\\cal P$ and reject it with high probability if more than an $\\epsilon$-fraction of the function values should be modified so that f obtains the property. Our algorithm for testing whether a 1-dimensional function $f:[n] \\rightarrow \\mathbb{R}$ is convex (concave) has query complexity and running time of $O\\left((\\log n) \/\\epsilon\\right)$. Our algorithm for testing whether an n\n                    <jats:sub>1<\/jats:sub>\n                    \u00d7 n\n                    <jats:sub>2<\/jats:sub>\n                    matrix V is a Monge (inverse Monge) matrix has query complexity and running time of $O\\left((\\log n_1\\cdot\\log n_2) \/\\epsilon\\right)$.\n                  <\/jats:p>","DOI":"10.1137\/s0097539702414026","type":"journal-article","created":{"date-parts":[[2003,11,7]],"date-time":"2003-11-07T11:19:22Z","timestamp":1068203962000},"page":"1158-1184","source":"Crossref","is-referenced-by-count":14,"title":["On Testing Convexity and Submodularity"],"prefix":"10.1137","volume":"32","author":[{"given":"Michal","family":"Parnas","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dana","family":"Ron","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ronitt","family":"Rubinfeld","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,2,17]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(95)00103-X"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"Tu\u011fkan Batu, Ronitt Rubinfeld, Patrick White, Fast approximate PCPs for multidimensional bin\u2010packing problems, Lecture Notes in Comput. 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Soc., Providence, R.I., 1963, 317\u201332728:1007","DOI":"10.1090\/pspum\/007\/0157778"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1145\/502090.502096"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"B. Lehmann, D. Lehmann, and N. Nisan,\n                      Combinatorial auctions with decreasing marginal utilities\n                      , in Proceedings of the ACM Conference on Electronic Commerce, Tampa, FL, 2001.","DOI":"10.1145\/501158.501161"},{"key":"R11","doi-asserted-by":"crossref","unstructured":"L. Lov\u00e1sz, Submodular functions and convexity, Springer, Berlin, 1983, 235\u201325785i:90109","DOI":"10.1007\/978-3-642-68874-4_10"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"Dana Ron, Property testing, Comb. Optim., Vol. 9, Kluwer Acad. Publ., Dordrecht, 2001, 597\u20136491966913","DOI":"10.1007\/978-1-4615-0013-1_15"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539793255151"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.2000.1989"}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0097539702414026","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:19:19Z","timestamp":1787336359000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0097539702414026"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1]]},"references-count":14,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2003,1]]}},"alternative-id":["10.1137\/S0097539702414026"],"URL":"https:\/\/doi.org\/10.1137\/s0097539702414026","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,1]]}}}