Multiple unicasts, graph guessing games, and non-Shannon inequalities

Baber, Rahil; Christofides, Demetres; Dang, Anh N.; Riis, Søren; Vaughan, Emil R.

doi:10.1109/netcod.2013.6570823

Cited by 16 publications

(23 citation statements)

References 8 publications

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“…Under some assumptions, it is possible to find all mimina in (1). Recall that the f -closure cl(I) of I ⊆ N consists of those i ∈ N that satisfy f (iI) = f (I).…”

Section: The Contraction Of a Polymatroid (mentioning

confidence: 99%

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Entropy Region and Convolution

Matúš

Csirmaz

2016

IEEE Trans. Inform. Theory

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Abstract. The entropy region is constructed from vectors of random variables by collecting Shannon entropies of all subvectors. Its shape is studied here by means of polymatroidal constructions, notably by convolution. The closure of the region is decomposed into the direct sum of tight and modular parts, reducing the study to the tight part. The relative interior of the reduction belongs to the entropy region. Behavior of the decomposition under selfadhesivity is clarified. Results are specialized to and completed for the region of four random variables. This and computer experiments help to visualize approximations of a symmetrized part of the entropy region. Four-atom conjecture on the minimization of Ingleton score is refuted.

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“…Under some assumptions, it is possible to find all mimina in (1). Recall that the f -closure cl(I) of I ⊆ N consists of those i ∈ N that satisfy f (iI) = f (I).…”

Section: The Contraction Of a Polymatroid (mentioning

confidence: 99%

“…The guessing numbers of games on directed graphs and entropies of the graphs can be related to the network coding [38,17] where non-Shannon inequalities provide sharper bounds [1]. Information-theoretic inequalities are under investigation in additive combinatorics [27].…”

Section: Introductionmentioning

confidence: 99%

Entropy Region and Convolution

Matúš

Csirmaz

2016

IEEE Trans. Inform. Theory

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“…and some additional results in [5]. Our first main result is that  ∩ [0, ] is finite for every integer ; more precisely, it is (2 6 2 ).…”

Section: Resultsmentioning

confidence: 94%

“…If contains 7 as an induced subgraph, then either = 7 and ( ) = 3.5 or has eight or more vertices, and an argument similar to that used in Lemma 3.6 indicates that ( ) ≥ 4. Thus, contains 5 as an induced subgraph, say by vertices 1 , … , 5 . The rest of the proof will show that the entropy of is 11∕3.…”

Section: We Say Is Entropy-minimal If For Anymentioning

confidence: 99%

On the possible values of the entropy of undirected graphs

Gadouleau

2017

Journal of Graph Theory

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The entropy of a digraph is a fundamental measure which relates network coding, information theory, and fixed points of finite dynamical systems. In this paper, we focus on the entropy of undirected graphs. We prove that for any integer k the number of possible values of the entropy of an undirected graph up to k is finite. We also determine all the possible values for the entropy of an undirected graph up to the value of four.

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“…It has been showed (argued) that solving many important problems (in network coding [3], secret sharing [4], database theory [5], graph guessing [6] and Index coding [7]) is something that strongly depends on our ability for computing finite checkable definitions of the entropic regions of all orders. Thus, computing finite checkable definitions of the entropic regions (and their polar cones) is an important problem.…”

Section: Introductionmentioning

confidence: 99%

Network coding and the model theory of linear information inequalities

Gomez

Mejía

Montoya

2014

2014 International Symposium on Network Coding (NetCod)

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Let n ≥ 4, can the entropic region of order n be defined by a finite list of polynomial inequalities? This question was first asked by Chan and Grant. We show that if it were the case one could solve many algorithmic problems coming from Network Coding, Index Coding and Secret Sharing. Unfortunately, it seems that the entropic regions of order larger than four are not semialgebraic. Actually, we guess that it is the case and we provide strong evidence supporting our conjecture.

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Multiple unicasts, graph guessing games, and non-Shannon inequalities

Cited by 16 publications

References 8 publications

Entropy Region and Convolution

Entropy Region and Convolution

On the possible values of the entropy of undirected graphs

Network coding and the model theory of linear information inequalities

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