The minrank of random graphs

Alexander Golovnev, Oded Regev, Omri Weinstein

نتاج البحث: فصل من :كتاب / تقرير / مؤتمرمنشور من مؤتمرمراجعة النظراء

ملخص

The minrank of a directed graph G is the minimum rank of a matrix M that can be obtained from the adjacency matrix of G by switching some ones to zeros (i.e., deleting edges) and then setting all diagonal entries to one. This quantity is closely related to the fundamental informationtheoretic problems of (linear) index coding (Bar-Yossef et al., FOCS'06), network coding and distributed storage, and to Valiant's approach for proving superlinear circuit lower bounds (Valiant, Boolean Function Complexity '92). We prove tight bounds on the minrank of directed Erdos-Rényi random graphs G(n, p) for all regimes of p 2 [0, 1]. In particular, for any constant p, we show that minrk(G) = (n/ log n) with high probability, where G is chosen from G(n, p). This bound gives a near quadratic improvement over the previous best lower bound of (p n) (Haviv and Langberg, ISIT'12), and partially settles an open problem raised by Lubetzky and Stav (FOCS '07). Our lower bound matches the wellknown upper bound obtained by the "clique covering" solution, and settles the linear index coding problem for random graphs. Finally, our result suggests a new avenue of attack, via derandomization, on Valiant's approach for proving superlinear lower bounds for logarithmic-depth semilinear circuits.

اللغة الأصليةإنجليزيّة أمريكيّة
عنوان منشور المضيفApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017
المحررونJose D. P. Rolim, Klaus Jansen, David P. Williamson, Santosh S. Vempala
ناشرSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
رقم المعيار الدولي للكتب (الإلكتروني)9783959770446
المعرِّفات الرقمية للأشياء
حالة النشرنُشِر - 1 أغسطس 2017
منشور خارجيًانعم
الحدث20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017 - Berkeley, الولايات المتّحدة
المدة: ١٦ أغسطس ٢٠١٧١٨ أغسطس ٢٠١٧

سلسلة المنشورات

الاسمLeibniz International Proceedings in Informatics, LIPIcs
مستوى الصوت81

!!Conference

!!Conference20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017
الدولة/الإقليمالولايات المتّحدة
المدينةBerkeley
المدة١٦/٠٨/١٧١٨/٠٨/١٧

All Science Journal Classification (ASJC) codes

  • !!Software

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