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2nd International Conference on Control, Decision and Information Technologies (CoDIT)

作者： Ebrahimi, Javad B. Siavoshani, Mahdi Jafari Chinese Univ Hong Kong Inst Network Coding Hong Kong Hong Kong Peoples R China Sharif Univ Technol Dept Comp Engn Tehran Iran

ISBN: (纸本)9781479967735

In [1], [2] it is shown that the minimum broadcast rate of a linear index code over a finite field F-q is equal to an algebraic invariant of the underlying digraph, called minrank(q). In [3], it is proved that for F-2 and any positive integer k, minrank(q) (G) <= k if and only if there exists a homomorphism from the complement of the graph G to the complement of a particular undirected graph family called "graph family {G(k)}". As observed in [2], by combining these two results one can relate the linear index coding problem of undirected graphs to the graph homomorphism problem. In [4], a direct connection between linear index coding problem and graph homomorphism problem is introduced. In contrast to the former approach, the direct connection holds for digraphs as well and applies to any field size. More precisely, in [4], a graph family {H-k(q)} has been introduced and shown that whether or not the scalar linear index of a digraph G is less than or equal to k is equivalent to the existence of a graph homomorphism from the complement of G to the complement of H-k(q). In this paper, we first study the structure of the digraphs H-k(q) defined in [4]. Analogous to the result of [2] about undirected graphs, we prove that H-k(q)'s are vertex transitive digraphs. Using this, and by applying a lemma of Hell and Nesetril [5], we derive a class of necessary conditions for digraphs G to satisfy lind(q) (G) <= k. Particularly, we obtain new lower bounds on lind(q) (G). Our next result is about the computational complexity of scalar linear index of a digraph. It is known that deciding whether the scalar linear index of an undirected graph is equal to k or not is NP-complete for k >= 3 and is polynomially decidable for k = 1, 2 [3]. For digraphs, it is shown in [6] that for the binary alphabet, the decision problem for k = 2 is NP-complete. We use graph homomorphism framework to extend this result to arbitrary alphabet.

关键词： index coding linear index coding graph homomorphism minrank of a graph computational complexity of the minrank

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The Minrank of Random Graphs

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IEEE TRANSACTIONS ON INFORMATION THEORY 2018年第11期64卷 6990-6995页

作者： Golovnev, Alexander Regev, Oded Weinstein, Omri Yahoo Res New York NY 10036 USA NYU Courant Inst Math Sci 251 Mercer St New York NY 10012 USA Columbia Univ New York NY 10027 USA

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 information-theoretic problems of (linear) index coding (Bar-Yossef et al.), network coding (Effros et al.), and distributed storage (Mazumdar, ISIT, 2014). We prove tight bounds on the minrank of directed Erdos-Renyi random graphs G(n, p) for all regimes of p is an element of[0, 1]. In particular, for any constant p, we show that minrk(G) = Theta (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 Omega (root n) (Haviv and Langberg), and partially settles an open problem raised by Lubetzky and Stav. Our lower bound matches the well-known upper bound obtained by the "clique covering" solution and settles the linear index coding problem for random knowledge graphs.

关键词： index coding minrank linear index coding

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The Minrank of Random Graphs 20

The Minrank of Random Graphs

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21st International Workshop on Randomization and Computation

作者： Golovnev, Alexander Regev, Oded Weinstein, Omri Yahoo Res New York NY 10036 USA NYU Courant Inst Math Sci 251 Mercer St New York NY 10012 USA Columbia Univ New York NY 10027 USA

ISBN: (纸本)9783959770446

关键词： index coding minrank linear index coding

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