Kernels for storage capacity and dual index coding
The storage capacity of a graph measures the maximum amount of information that can be stored across its vertices, such that the information at any vertex can be recovered from the information stored at its neighborhood. The study of this graph quantity is motivated by applications in distributed st...
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| Published in: | Journal of combinatorial theory. Series A Vol. 216; p. 106059 |
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| Abstract | The storage capacity of a graph measures the maximum amount of information that can be stored across its vertices, such that the information at any vertex can be recovered from the information stored at its neighborhood. The study of this graph quantity is motivated by applications in distributed storage and by its intimate relations to the index coding problem from the area of network information theory. In the latter, one wishes to minimize the amount of information that has to be transmitted to a collection of receivers, in a way that enables each of them to discover its required data using some prior side information.
In this paper, we initiate the study of the ▪ and ▪ problems from the perspective of parameterized complexity. We prove that the ▪ problem parameterized by the solution size admits a kernelization algorithm producing kernels of linear size. We also provide such a result for the ▪ problem, in the linear and non-linear settings, where it is parameterized by the dual value of the solution, i.e., the length of the transmission that can be saved using the side information. A key ingredient in the proofs is the crown decomposition technique due to Chor, Fellows, and Juedes [14,11]. As an application, we significantly extend an algorithmic result of Dau, Skachek, and Chee [13]. |
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| AbstractList | The storage capacity of a graph measures the maximum amount of information that can be stored across its vertices, such that the information at any vertex can be recovered from the information stored at its neighborhood. The study of this graph quantity is motivated by applications in distributed storage and by its intimate relations to the index coding problem from the area of network information theory. In the latter, one wishes to minimize the amount of information that has to be transmitted to a collection of receivers, in a way that enables each of them to discover its required data using some prior side information.
In this paper, we initiate the study of the ▪ and ▪ problems from the perspective of parameterized complexity. We prove that the ▪ problem parameterized by the solution size admits a kernelization algorithm producing kernels of linear size. We also provide such a result for the ▪ problem, in the linear and non-linear settings, where it is parameterized by the dual value of the solution, i.e., the length of the transmission that can be saved using the side information. A key ingredient in the proofs is the crown decomposition technique due to Chor, Fellows, and Juedes [14,11]. As an application, we significantly extend an algorithmic result of Dau, Skachek, and Chee [13]. |
| ArticleNumber | 106059 |
| Author | Haviv, Ishay |
| Author_xml | – sequence: 1 givenname: Ishay orcidid: 0000-0002-2903-076X surname: Haviv fullname: Haviv, Ishay email: ishayhav@mta.ac.il organization: The Academic College of Tel Aviv-Yaffo, Tel Aviv 61083, Israel |
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| Cites_doi | 10.1007/BF01261326 10.3390/a16030144 10.1016/j.ic.2017.06.001 10.1109/TIT.2019.2912184 10.1137/23M155760X 10.1109/TIT.2019.2910026 10.1137/070683933 10.1109/TIT.2010.2094910 10.1109/TIT.2006.874540 10.1109/TIT.2023.3347296 10.1109/TIT.2010.2103753 10.1109/TIT.2024.3446000 10.1109/TIT.2013.2295331 10.1109/TIT.2015.2472521 |
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| Keywords | Index coding Storage capacity Minrank Parameterized complexity Kernelization |
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| SubjectTerms | Index coding Kernelization Minrank Parameterized complexity Storage capacity |
| Title | Kernels for storage capacity and dual index coding |
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