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An Approximation of the Zero Error Capacity by a Greedy Algorithm

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Combinatorial Optimization and Applications (COCOA 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12577))

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Abstract

We present a greedy algorithm that determines a lower bound on the zero error capacity. The algorithm has many new advantages, e.g., it does not store a whole product graph in a computer memory and it uses the so-called distributions in all dimensions to get a better approximation of the zero error capacity. We also show an additional application of our algorithm.

This article was partially supported by the Narodowe Centrum Nauki under grant DEC-2011/02/A/ST6/00201.

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Notes

  1. 1.

    The number of vertices and edges of G we often denote by n and m, respectively, thus \(|V(G)|=n\) and \(|E(G)|=m\).

  2. 2.

    We assume that W is non-empty.

  3. 3.

    It is worth to note that the number of maximal cliques in G is at most exponential with respect to |V(G)| [11], while the number of edges in G is at most quadratic with respect to |V(G)|.

  4. 4.

    A graph is vertex transitive if for any two vertices u and v of this graph, there is an automorphism such that the image of u is v.

  5. 5.

    There is a better upper bound on \(\varTheta (G)\), the so-called Lovász theta function [28].

  6. 6.

    This part of the proposition was found by my colleague [30].

  7. 7.

    Our algorithm works for an arbitrary graph product, as well as for an arbitrary single graph.

  8. 8.

    A fullerene graph is the graph formed from the vertices and edges of a convex polyhedron, whose faces are all pentagons or hexagons and all vertices have degree equal to three.

  9. 9.

    We can use the so-called Lovász theta function instead of \(\alpha ^*\), since it is also multiplicative with respect to the strong product for all graphs [28].

  10. 10.

    We assume that a network (graph) G is non-empty, i.e., \(|E(G)|\ne 0\).

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Authors and Affiliations

  1. Faculty of Electronics, Telecommunications and Informatics, Gdańsk University of Technology, Narutowicza 11/12, 80-233, Gdańsk, Poland

    Marcin Jurkiewicz

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  1. Marcin Jurkiewicz
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Correspondence to Marcin Jurkiewicz .

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  1. The University of Texas at Dallas, Richardson, TX, USA

    Weili Wu

  2. Xiamen University, Xiamen, China

    Zhongnan Zhang

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Jurkiewicz, M. (2020). An Approximation of the Zero Error Capacity by a Greedy Algorithm. In: Wu, W., Zhang, Z. (eds) Combinatorial Optimization and Applications. COCOA 2020. Lecture Notes in Computer Science(), vol 12577. Springer, Cham. https://doi.org/10.1007/978-3-030-64843-5_7

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  • DOI: https://doi.org/10.1007/978-3-030-64843-5_7

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