Exploiting symmetry in computing polyhedral bounds on network coding rate regions

Apte, Jayant; Walsh, John MacLaren

doi:10.1109/netcod.2015.7176793

Cited by 14 publications

(19 citation statements)

References 13 publications

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“…where (b) is by the sub-modularity of the entropy function, and (c) is because of (3). Now substituting (63) into (61) gives (28), which completes the proof.…”

Section: M + 2rmentioning

confidence: 72%

“…It should be noted that in [2], the region of interest was obtained by first finding a set of fine-spaced points on the boundary of the outer bound using the reduced LP, and then manually identifying the effective bounding segments using these boundary points. This task can however be accomplished more efficiently using an approach proposed by Lassez and Lassez [27], as pointed out in [28]. This prompted the author to implement this part of the computer program using this more efficient approach.…”

Section: A Computed-aided Approach To Find Outer Boundsmentioning

confidence: 99%

“…In the caching problem, the tradeoff is between two quantities M and R. We note here if there are more than two quantities which need to be considered in the tradeoff, the algorithm is more involved, and we refer the readers to [27] and [28] for more details on such settings.…”

Section: Appendix a Finding Corner Points Of The Lp Outer Boundsmentioning

confidence: 99%

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Symmetry, Outer Bounds, and Code Constructions: A Computer-Aided Investigation on the Fundamental Limits of Caching

Tian

2018

Entropy

112

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Abstract:We illustrate how computer-aided methods can be used to investigate the fundamental limits of the caching systems, which are significantly different from the conventional analytical approach usually seen in the information theory literature. The linear programming (LP) outer bound of the entropy space serves as the starting point of this approach; however, our effort goes significantly beyond using it to prove information inequalities. We first identify and formalize the symmetry structure in the problem, which enables us to show the existence of optimal symmetric solutions. A symmetry-reduced linear program is then used to identify the boundary of the memory-transmission-rate tradeoff for several small cases, for which we obtain a set of tight outer bounds. General hypotheses on the optimal tradeoff region are formed from these computed data, which are then analytically proven. This leads to a complete characterization of the optimal tradeoff for systems with only two users, and certain partial characterization for systems with only two files. Next, we show that by carefully analyzing the joint entropy structure of the outer bounds for certain cases, a novel code construction can be reverse-engineered, which eventually leads to a general class of codes. Finally, we show that outer bounds can be computed through strategically relaxing the LP in different ways, which can be used to explore the problem computationally. This allows us firstly to deduce generic characteristic of the converse proof, and secondly to compute outer bounds for larger problem cases, despite the seemingly impossible computation scale.

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“…where (b) is by the sub-modularity of the entropy function, and (c) is because of (3). Now substituting (63) into (61) gives (28), which completes the proof.…”

Section: M + 2rmentioning

confidence: 72%

Section: A Computed-aided Approach To Find Outer Boundsmentioning

confidence: 99%

See 1 more Smart Citation

Symmetry, Outer Bounds, and Code Constructions: A Computer-Aided Investigation on the Fundamental Limits of Caching

Tian

2018

Entropy

112

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“…Note that, because it leaves the sets of edges, decoder demands, and topology constraints set-wise invariant, the elements of the stabilizer subgroup G (Q,W) also leave the set of rate region constraints L A invariant. Such a group of permutations on sources and edges is called the network symmetry group [32], [33], which is further exploited in [18] and [23] to substantially reduce the complexity of computing the network's rate regions. This network symmetry group plays a role in the present study because, as depicted in Figures 3a and 3b, by the orbit stabilizer theorem mentioned above, it determines the number of networks equivalent to a given canonical network (the representative we will select from the orbit).…”

Section: B Expressing Network Equivalence With a Group Actionmentioning

confidence: 99%

On Multi-Source Networks: Enumeration, Rate Region Computation, and Hierarchy

Weber

Walsh

2017

IEEE Trans. Inform. Theory

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This paper investigates the enumeration, rate region computation, and hierarchy of general multi-source multi-sink hyperedge networks under network coding, which includes multiple network models, such as independent distributed storage systems and index coding problems, as special cases. A notion of minimal networks and a notion of network equivalence under group action are defined. An efficient algorithm capable of directly listing single minimal canonical representatives from each network equivalence class is presented and utilized to list all minimal canonical networks with up to 5 sources and hyperedges. Computational tools are then applied to obtain the rate regions of all of these canonical networks, providing exact expressions for 744,119 newly solved network coding rate regions corresponding to more than 2 trillion isomorphic network coding problems. In order to better understand and analyze the huge repository of rate regions through hierarchy, several embedding and combination operations are defined so that the rate region of the network after operation can be derived from the rate regions of networks involved in the operation.The embedding operations enable the definition and determination of a list of forbidden network minors for the sufficiency of classes of linear codes. The combination operations enable the rate regions of some larger networks to be obtained as the combination of the rate regions of smaller networks. The integration of both the combinations and embedding operators is then shown to enable the calculation of rate regions for many networks not reachable via combination operations alone.

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“…Li et al used a similar computational approach to tackle the multilevel diversity coding problem [17] and multi-source network coding problems with simple network topology [21] (also see Reference [22]); however, the main focus was to provide an efficient enumeration and classification of the large number of specific small instances (all instances considered require 7 or fewer random variables), where each instance itself poses little computation issue. Beyond computing outer bounds, the problem of computationally generating inner bounds was also explored [23,24].…”

Section: Literature Reviewmentioning

confidence: 99%

Computational Techniques for Investigating Information Theoretic Limits of Information Systems

et al. 2021

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Computer-aided methods, based on the entropic linear program framework, have been shown to be effective in assisting the study of information theoretic fundamental limits of information systems. One key element that significantly impacts their computation efficiency and applicability is the reduction of variables, based on problem-specific symmetry and dependence relations. In this work, we propose using the disjoint-set data structure to algorithmically identify the reduction mapping, instead of relying on exhaustive enumeration in the equivalence classification. Based on this reduced linear program, we consider four techniques to investigate the fundamental limits of information systems: (1) computing an outer bound for a given linear combination of information measures and providing the values of information measures at the optimal solution; (2) efficiently computing a polytope tradeoff outer bound between two information quantities; (3) producing a proof (as a weighted sum of known information inequalities) for a computed outer bound; and (4) providing the range for information quantities between which the optimal value does not change, i.e., sensitivity analysis. A toolbox, with an efficient JSON format input frontend, and either Gurobi or Cplex as the linear program solving engine, is implemented and open-sourced.

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Exploiting symmetry in computing polyhedral bounds on network coding rate regions

Cited by 14 publications

References 13 publications

Symmetry, Outer Bounds, and Code Constructions: A Computer-Aided Investigation on the Fundamental Limits of Caching

Symmetry, Outer Bounds, and Code Constructions: A Computer-Aided Investigation on the Fundamental Limits of Caching

On Multi-Source Networks: Enumeration, Rate Region Computation, and Hierarchy

Computational Techniques for Investigating Information Theoretic Limits of Information Systems

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