103 research outputs found
Information Inequalities for Joint Distributions, with Interpretations and Applications
Upper and lower bounds are obtained for the joint entropy of a collection of
random variables in terms of an arbitrary collection of subset joint entropies.
These inequalities generalize Shannon's chain rule for entropy as well as
inequalities of Han, Fujishige and Shearer. A duality between the upper and
lower bounds for joint entropy is developed. All of these results are shown to
be special cases of general, new results for submodular functions-- thus, the
inequalities presented constitute a richly structured class of Shannon-type
inequalities. The new inequalities are applied to obtain new results in
combinatorics, such as bounds on the number of independent sets in an arbitrary
graph and the number of zero-error source-channel codes, as well as new
determinantal inequalities in matrix theory. A new inequality for relative
entropies is also developed, along with interpretations in terms of hypothesis
testing. Finally, revealing connections of the results to literature in
economics, computer science, and physics are explored.Comment: 15 pages, 1 figure. Originally submitted to the IEEE Transactions on
Information Theory in May 2007, the current version incorporates reviewer
comments including elimination of an erro
Private Information Retrieval in Graph-Based Replication Systems
In a Private Information Retrieval (PIR) protocol, a user can download a file from a database without revealing the identity of the file to each individual server. A PIR protocol is called t-private if the identity of the file remains concealed even if t of the servers collude. Graph based replication is a simple technique, which is prevalent in both theory and practice, for achieving robustness in storage systems. In this technique each file is replicated on two or more storage servers, giving rise to a (hyper-)graph structure. In this paper we study private information retrieval protocols in graph based replication systems. The main interest of this work is understanding the collusion structures which emerge in the underlying graph. Our main contribution is a 2-replication scheme which guarantees perfect privacy from acyclic sets in the graph, and guarantees partial-privacy in the presence of cycles. Furthermore, by providing an upper bound, it is shown that the PIR rate of this scheme is at most a factor of two from its optimal value for regular graphs. Lastly, we extend our results to larger replication factors and to graph-based coding, a generalization of graph based replication that induces smaller storage overhead and larger PIR rate in many cases
Hardness of robust graph isomorphism, Lasserre gaps, and asymmetry of random graphs
Building on work of Cai, F\"urer, and Immerman \cite{CFI92}, we show two
hardness results for the Graph Isomorphism problem. First, we show that there
are pairs of nonisomorphic -vertex graphs and such that any
sum-of-squares (SOS) proof of nonisomorphism requires degree . In
other words, we show an -round integrality gap for the Lasserre SDP
relaxation. In fact, we show this for pairs and which are not even
-isomorphic. (Here we say that two -vertex, -edge graphs
and are -isomorphic if there is a bijection between their
vertices which preserves at least edges.) Our second result is that
under the {\sc R3XOR} Hypothesis \cite{Fei02} (and also any of a class of
hypotheses which generalize the {\sc R3XOR} Hypothesis), the \emph{robust}
Graph Isomorphism problem is hard. I.e.\ for every , there is no
efficient algorithm which can distinguish graph pairs which are
-isomorphic from pairs which are not even
-isomorphic for some universal constant . Along the
way we prove a robust asymmetry result for random graphs and hypergraphs which
may be of independent interest
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