The minrank of random graphs

Alexander Golovnev, Oded Regev, Omri Weinstein

פרסום מחקרי: פרק בספר / בדוח / בכנספרסום בספר כנסביקורת עמיתים

תקציר

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.

שפה מקוריתאנגלית אמריקאית
כותר פרסום המארחApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017
עורכיםJose D. P. Rolim, Klaus Jansen, David P. Williamson, Santosh S. Vempala
מוציא לאורSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
מסת"ב (אלקטרוני)9783959770446
מזהי עצם דיגיטלי (DOIs)
סטטוס פרסוםפורסם - 1 אוג׳ 2017
פורסם באופן חיצוניכן
אירוע20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017 - Berkeley, ארצות הברית
משך הזמן: 16 אוג׳ 201718 אוג׳ 2017

סדרות פרסומים

שםLeibniz International Proceedings in Informatics, LIPIcs
כרך81

כנס

כנס20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017
מדינה/אזורארצות הברית
עירBerkeley
תקופה16/08/1718/08/17

ASJC Scopus subject areas

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