Title :
On the hardness of approximating the network coding capacity
Author :
Langberg, Michael ; Sprintson, Alex
Author_Institution :
Comput. Sci. Div., Open Univ. of Israel, Raanana
Abstract :
This work addresses the computational complexity of achieving the capacity of a general network coding instance. We focus on the linear capacity, namely the capacity of the given instance when restricted to linear encoding functions. It has been shown [Lehman and Lehman, SODA 2005] that determining the (scalar) linear capacity of a general network coding instance is NP-hard. In this work we initiate the study of approximation in this context. Namely, we show that given an instance to the general network coding problem of linear capacity C, constructing a linear code of rate alphaC for any universal (i.e., independent of the size of the instance) constant alphales1 is ldquohardrdquo. Specifically, finding such network codes would solve a long standing open problem in the field of graph coloring. In addition, we consider the problem of determining the (scalar) linear capacity of a planar network coding instance (i.e., a general instance in which the underlying graph is planar). We show that even for planar networks this problem remains NP-hard.
Keywords :
channel capacity; channel coding; computational complexity; graph colouring; linear codes; computational complexity; graph coloring; linear encoding functions; network coding capacity; planar networks; Computational complexity; Computer science; Decoding; Encoding; Linear code; NP-hard problem; Network coding; Polynomials; Vectors;
Conference_Titel :
Information Theory, 2008. ISIT 2008. IEEE International Symposium on
Conference_Location :
Toronto, ON
Print_ISBN :
978-1-4244-2256-2
Electronic_ISBN :
978-1-4244-2257-9
DOI :
10.1109/ISIT.2008.4594999