Improved Approximation Algorithms for Index Coding

The index coding problem is concerned with broadcasting encoded information to a collection of receivers in a way that enables each receiver to discover its required data based on its side information, which comprises the data required by some of the others. Given the side information map, represent...

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Published in:IEEE transactions on information theory Vol. 70; no. 11; pp. 8266 - 8275
Main Authors: Chawin, Dror, Haviv, Ishay
Format: Journal Article
Language:English
Published: IEEE 01.11.2024
Subjects:
Approximation algorithms
clique cover
Codes
Encoding
Index coding
Indexes
Receivers
Size measurement
Upper bound
ISSN:0018-9448, 1557-9654
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Abstract The index coding problem is concerned with broadcasting encoded information to a collection of receivers in a way that enables each receiver to discover its required data based on its side information, which comprises the data required by some of the others. Given the side information map, represented by a graph in the symmetric case and by a digraph otherwise, the goal is to devise a coding scheme of minimum broadcast length. We present a general method for developing efficient algorithms for approximating the index coding rate for prescribed families of instances. As applications, we obtain polynomial-time algorithms that approximate the index coding rate of graphs and digraphs on n vertices to within factors of <inline-formula> <tex-math notation="LaTeX">O(n/\log ^{2} n) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">O(n/\log n) </tex-math></inline-formula> respectively. This improves on the approximation factors of <inline-formula> <tex-math notation="LaTeX">O(n/\log n) </tex-math></inline-formula> for graphs and <inline-formula> <tex-math notation="LaTeX">O(n \cdot \log \log n/\log n) </tex-math></inline-formula> for digraphs achieved by Blasiak, Kleinberg, and Lubetzky. For the family of quasi-line graphs, we exhibit a polynomial-time algorithm that approximates the index coding rate to within a factor of 2. This improves on the approximation factor of <inline-formula> <tex-math notation="LaTeX">O(n^{2/3}) </tex-math></inline-formula> achieved by Arbabjolfaei and Kim for graphs on n vertices taken from certain sub-families of quasi-line graphs. Our approach is applicable for approximating a variety of additional graph and digraph quantities to within the same approximation factors. Specifically, it captures every graph quantity sandwiched between the independence number and the clique cover number and every digraph quantity sandwiched between the maximum size of an acyclic induced sub-digraph and the directed clique cover number.
AbstractList The index coding problem is concerned with broadcasting encoded information to a collection of receivers in a way that enables each receiver to discover its required data based on its side information, which comprises the data required by some of the others. Given the side information map, represented by a graph in the symmetric case and by a digraph otherwise, the goal is to devise a coding scheme of minimum broadcast length. We present a general method for developing efficient algorithms for approximating the index coding rate for prescribed families of instances. As applications, we obtain polynomial-time algorithms that approximate the index coding rate of graphs and digraphs on n vertices to within factors of <inline-formula> <tex-math notation="LaTeX">O(n/\log ^{2} n) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">O(n/\log n) </tex-math></inline-formula> respectively. This improves on the approximation factors of <inline-formula> <tex-math notation="LaTeX">O(n/\log n) </tex-math></inline-formula> for graphs and <inline-formula> <tex-math notation="LaTeX">O(n \cdot \log \log n/\log n) </tex-math></inline-formula> for digraphs achieved by Blasiak, Kleinberg, and Lubetzky. For the family of quasi-line graphs, we exhibit a polynomial-time algorithm that approximates the index coding rate to within a factor of 2. This improves on the approximation factor of <inline-formula> <tex-math notation="LaTeX">O(n^{2/3}) </tex-math></inline-formula> achieved by Arbabjolfaei and Kim for graphs on n vertices taken from certain sub-families of quasi-line graphs. Our approach is applicable for approximating a variety of additional graph and digraph quantities to within the same approximation factors. Specifically, it captures every graph quantity sandwiched between the independence number and the clique cover number and every digraph quantity sandwiched between the maximum size of an acyclic induced sub-digraph and the directed clique cover number.
Author Haviv, Ishay
Chawin, Dror
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SubjectTerms Approximation algorithms
clique cover
Codes
Encoding
Index coding
Indexes
Receivers
Size measurement
Upper bound
Title Improved Approximation Algorithms for Index Coding
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