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Comput."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>We consider the maximum independent set problem on sparse graphs with maximum degree $d$. We show that the Lov\u00e1sz $\\vartheta$-function based semidefinite program (SDP) has an integrality gap of $ \\widetilde{O}(d\/\\log^{3\/2} d)$, improving on the previous best result of $\\widetilde{O}(d\/\\log d)$. This improvement is based on a new Ramsey-theoretic bound on the independence number of $K_r$-free graphs for large values of $r$. We also show that for stronger SDPs, namely, those obtained using poly$\\log(d)$ levels of the ${SA}^{+}$ semidefinite hierarchy, the integrality gap reduces to $\\widetilde{O}(d\/\\log^2 d)$. This matches the best unique-games-based hardness result up to lower-order poly$(\\log\\log d)$ factors. Finally, we give an algorithmic version of this ${SA}^{+}$-based integrality gap result, albeit using $d$ levels of ${SA}^{+}$, via a coloring algorithm of Johansson.<\/jats:p>","DOI":"10.1137\/15m1051002","type":"journal-article","created":{"date-parts":[[2018,6,27]],"date-time":"2018-06-27T15:57:32Z","timestamp":1530115052000},"page":"1039-1055","source":"Crossref","is-referenced-by-count":6,"title":["On the Lov\u00e1sz Theta Function for Independent Sets in Sparse Graphs"],"prefix":"10.1137","volume":"47","author":[{"given":"Nikhil","family":"Bansal","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anupam","family":"Gupta","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guru","family":"Guruganesh","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,6,26]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579451"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01581168"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(80)90030-8"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1999.1910"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.4086\/toc.2011.v007a003"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2418(199610)9:3<271::AID-RSA1>3.0.CO;2-U"},{"key":"atypb7","unstructured":"N. Alon and J. Spencer,\n                      The Probabilistic Method\n                      , Wiley Interscience, New York, 1992."},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"N. Bansal, A. Gupta, and G. Guruganesh,\n                      On the Lov\u00e1sz Theta Function for Independent Sets in Sparse Graphs\n                      , CoRR,arXiv:1504.04767, 2015.","DOI":"10.1145\/2746539.2746607"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1007\/s00222-010-0247-x"},{"key":"atypb10","first-page":"447","author":"Chan S. O.","year":"2013","journal-title":"Proceedings of STOC"},{"key":"atypb11","first-page":"134","author":"Chung K.-M.","year":"2014","journal-title":"Proceedings of PODC"},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"E. Chlamtac and M. Tulsiani,\n                      Convex relaxations and integrality gaps\n                      , in Handbook on Semidefinite, Conic and Polynomial Optimization, M. F. Anjos and J. B. Lasserre, eds., Springer, New York, 2012.","DOI":"10.1007\/978-1-4614-0769-0_6"},{"key":"atypb13","first-page":"463","volume":"2","author":"Erd\u00f6s P.","year":"1935","journal-title":"Compos. Math."},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/S089548010240415X"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2013.09.003"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"M. Gr\u00f6tschel, L. Lov\u00e1sz, and A. Schrijver,\n                      Geometric Algorithms and Combinatorial Optimization\n                      , Springer-Verlag, Berlin, 1988.","DOI":"10.1007\/978-3-642-97881-4"},{"key":"atypb17","doi-asserted-by":"crossref","unstructured":"B. G\u00e4rtner and J. Matou\u0161ek,\n                      Approximation Algorithms and Semidefinite Programming\n                      , Springer, Heidelberg, 2012,https:\/\/doi.org\/10.1007\/978-3-642-22015-9.","DOI":"10.1007\/978-3-642-22015-9"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1006\/jagm.2000.1097"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.7155\/jgaa.00020"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539700381097"},{"key":"atypb21","first-page":"627","author":"J.","year":"1996","journal-title":"Proceedings of FOCS"},{"key":"atypb22","first-page":"475","volume":"1","author":"Halld\u00f3rsson M. M.","year":"1994","journal-title":"Nord. J. Comput."},{"key":"atypb23","unstructured":"A. Johansson,\n                      Asymptotic Choice Number for Triangle-Free Graphs\n                      , 1996."},{"key":"atypb24","unstructured":"A. Johansson,\n                      The Choice Number of Sparse Graphs\n                      . preprint, 1996."},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548300001528"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1145\/274787.274791"},{"key":"atypb27","first-page":"226","volume":"1","author":"Khot S.","year":"2006","journal-title":"Proceedings of ICALP"},{"key":"atypb28","unstructured":"M. Laurent,\n                      Networks and Semidefinite Programming\n                      ,http:\/\/homepages.cwi.nl\/~monique\/lnmb12\/Lecture_Notes_NSP_21_01_13.pdf(21 January 2013)."},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1287\/moor.28.3.470.16391"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.1979.1055985"},{"key":"atypb31","doi-asserted-by":"crossref","unstructured":"M. Molloy and B. 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