Linear Index Coding via Semidefinite Programming
In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the goal is to broadcast an n-bit word to n receivers (one bit per receiver), where the receivers have side information represented by a graph G. The objective is to minimize the length of a codeword sent to all receivers which...
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| Published in: | Combinatorics, probability & computing Vol. 23; no. 2; pp. 223 - 247 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Cambridge, UK
Cambridge University Press
01.03.2014
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| ISSN: | 0963-5483, 1469-2163 |
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| Abstract | In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the goal is to broadcast an n-bit word to n receivers (one bit per receiver), where the receivers have side information represented by a graph G. The objective is to minimize the length of a codeword sent to all receivers which allows each receiver to learn its bit. For linear index coding, the minimum possible length is known to be equal to a graph parameter called minrank (Bar-Yossef, Birk, Jayram and Kol, IEEE Trans. Inform. Theory, 2011). We show a polynomial-time algorithm that, given an n-vertex graph G with minrank k, finds a linear index code for G of length Õ(nf(k)), where f(k) depends only on k. For example, for k = 3 we obtain f(3) ≈ 0.2574. Our algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani and Sudan for graph colouring (J. Assoc. Comput. Mach., 1998) and its refined analysis due to Arora, Chlamtac and Charikar (STOC, 2006). Since the SDP we use is not a relaxation of the minimization problem we consider, a crucial component of our analysis is an upper bound on the objective value of the SDP in terms of the minrank. At the heart of our analysis lies a combinatorial result which may be of independent interest. Namely, we show an exact expression for the maximum possible value of the Lovász ϑ-function of a graph with minrank k. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph. |
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| AbstractList | In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the goal is to broadcast an n-bit word to n receivers (one bit per receiver), where the receivers have side information represented by a graph G. The objective is to minimize the length of a codeword sent to all receivers which allows each receiver to learn its bit. For linear index coding, the minimum possible length is known to be equal to a graph parameter called minrank (Bar-Yossef, Birk, Jayram and Kol, IEEE Trans. Inform. Theory, 2011). We show a polynomial-time algorithm that, given an n-vertex graph G with minrank k, finds a linear index code for G of length Õ(n f(k)), where f(k) depends only on k. For example, for k = 3 we obtain f(3) [approximate] 0.2574. Our algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani and Sudan for graph colouring (J. Assoc. Comput. Mach., 1998) and its refined analysis due to Arora, Chlamtac and Charikar (STOC, 2006). Since the SDP we use is not a relaxation of the minimization problem we consider, a crucial component of our analysis is an upper bound on the objective value of the SDP in terms of the minrank. At the heart of our analysis lies a combinatorial result which may be of independent interest. Namely, we show an exact expression for the maximum possible value of the Lovász [vartheta]-function of a graph with minrank k. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph. [PUBLICATION ABSTRACT] In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the goal is to broadcast an n-bit word to n receivers (one bit per receiver), where the receivers have side information represented by a graph G. The objective is to minimize the length of a codeword sent to all receivers which allows each receiver to learn its bit. For linear index coding, the minimum possible length is known to be equal to a graph parameter called minrank (Bar-Yossef, Birk, Jayram and Kol, IEEE Trans. Inform. Theory, 2011). We show a polynomial-time algorithm that, given an n-vertex graph G with minrank k, finds a linear index code for G of length Õ(nf(k)), where f(k) depends only on k. For example, for k = 3 we obtain f(3) ≈ 0.2574. Our algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani and Sudan for graph colouring (J. Assoc. Comput. Mach., 1998) and its refined analysis due to Arora, Chlamtac and Charikar (STOC, 2006). Since the SDP we use is not a relaxation of the minimization problem we consider, a crucial component of our analysis is an upper bound on the objective value of the SDP in terms of the minrank. At the heart of our analysis lies a combinatorial result which may be of independent interest. Namely, we show an exact expression for the maximum possible value of the Lovász ϑ-function of a graph with minrank k. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph. In the index coding problem, introduced by Birk and Kol ( INFOCOM, 1998 ), the goal is to broadcast an n -bit word to n receivers (one bit per receiver), where the receivers have side information represented by a graph G . The objective is to minimize the length of a codeword sent to all receivers which allows each receiver to learn its bit. For linear index coding, the minimum possible length is known to be equal to a graph parameter called minrank (Bar-Yossef, Birk, Jayram and Kol, IEEE Trans. Inform. Theory , 2011). We show a polynomial-time algorithm that, given an n -vertex graph G with minrank k , finds a linear index code for G of length Õ ( n f ( k ) ), where f ( k ) depends only on k . For example, for k = 3 we obtain f (3) ≈ 0.2574. Our algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani and Sudan for graph colouring ( J. Assoc. Comput. Mach. , 1998) and its refined analysis due to Arora, Chlamtac and Charikar ( STOC , 2006). Since the SDP we use is not a relaxation of the minimization problem we consider, a crucial component of our analysis is an upper bound on the objective value of the SDP in terms of the minrank. At the heart of our analysis lies a combinatorial result which may be of independent interest. Namely, we show an exact expression for the maximum possible value of the Lovász ϑ-function of a graph with minrank k . This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph. In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the goal is to broadcast an n-bit word to n receivers (one bit per receiver), where the receivers have side information represented by a graph G. The objective is to minimize the length of a codeword sent to all receivers which allows each receiver to learn its bit. For linear index coding, the minimum possible length is known to be equal to a graph parameter called minrank (Bar-Yossef, Birk, Jayram and Kol, IEEE Trans. Inform. Theory, 2011). We show a polynomial-time algorithm that, given an n-vertex graph G with minrank k, finds a linear index code for G of length O(n super( )fk), where f(k) depends only on k. For example, for k = 3 we obtain f(3) approximately 0.2574. Our algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani and Sudan for graph colouring (J. Assoc. Comput. Mach., 1998) and its refined analysis due to Arora, Chlamtac and Charikar (STOC, 2006). Since the SDP we use is not a relaxation of the minimization problem we consider, a crucial component of our analysis is an upper bound on the objective value of the SDP in terms of the minrank. At the heart of our analysis lies a combinatorial result which may be of independent interest. Namely, we show an exact expression for the maximum possible value of the Lovasz [thetav]-function of a graph with minrank k. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph. |
| Author | CHLAMTÁČ, EDEN HAVIV, ISHAY |
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| References | Haemers (S0963548313000564_ref19) 1981 Feller (S0963548313000564_ref15) 1968 S0963548313000564_ref21 S0963548313000564_ref20 El Rouayheb (S0963548313000564_ref14) 2008 S0963548313000564_ref24 S0963548313000564_ref23 S0963548313000564_ref22 S0963548313000564_ref7 S0963548313000564_ref29 S0963548313000564_ref28 Knuth (S0963548313000564_ref25) 1994; 1 S0963548313000564_ref6 S0963548313000564_ref5 S0963548313000564_ref27 S0963548313000564_ref26 S0963548313000564_ref4 S0963548313000564_ref9 S0963548313000564_ref8 S0963548313000564_ref3 S0963548313000564_ref2 S0963548313000564_ref1 S0963548313000564_ref32 S0963548313000564_ref10 S0963548313000564_ref31 S0963548313000564_ref30 S0963548313000564_ref13 S0963548313000564_ref12 S0963548313000564_ref33 S0963548313000564_ref11 S0963548313000564_ref18 S0963548313000564_ref17 S0963548313000564_ref16 |
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| SubjectTerms | Algorithms Coding Combinatorial analysis Graph coloring Graphs Optimization Receivers Upper bounds |
| Title | Linear Index Coding via Semidefinite Programming |
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