Improving the performance guarantee for approximate graph coloring
Journal of the ACM (JACM)
New approximation algorithms for graph coloring
Journal of the ACM (JACM)
An Õ(n3/14)-coloring algorithm for 3-colorable graphs
Information Processing Letters
Approximate graph coloring by semidefinite programming
Journal of the ACM (JACM)
On the power of unique 2-prover 1-round games
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Coloring k-colorable graphs using relatively small palettes
Journal of Algorithms
Some NP-complete geometric problems
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
Improved Inapproximability Results for MaxClique, Chromatic Number and Approximate Graph Coloring
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
On the Hardness of 4-Coloring a 3-Colorable Graph
SIAM Journal on Discrete Mathematics
Near-optimal algorithms for unique games
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
New approximation guarantee for chromatic number
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Index Coding with Side Information
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Approximation Algorithms Using Hierarchies of Semidefinite Programming Relaxations
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Broadcasting with Side Information
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Non-local analysis of sdp-based approximation algorithms
Non-local analysis of sdp-based approximation algorithms
Conditional Hardness for Approximate Coloring
SIAM Journal on Computing
Nonlinear index coding outperforming the linear optimum
IEEE Transactions on Information Theory
On the conditional hardness of coloring a 4-colorable graph with super-constant number of colors
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Distributed source coding for satellite communications
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Hi-index | 0.00 |
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 et al., FOCS, 2006). 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 (J. ACM, 1998) for graph coloring 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.