TY - GEN

T1 - On the hardness of approximating the network coding capacity

AU - Langberg, Michael

AU - Sprintson, Alex

PY - 2008

Y1 - 2008

N2 - 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 αC for any universal (i.e., independent of the size of the instance) constant α 1 is "hard". 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.

AB - 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 αC for any universal (i.e., independent of the size of the instance) constant α 1 is "hard". 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.

UR - http://www.scopus.com/inward/record.url?scp=52349103512&partnerID=8YFLogxK

U2 - 10.1109/ISIT.2008.4594999

DO - 10.1109/ISIT.2008.4594999

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AN - SCOPUS:52349103512

SN - 9781424422579

T3 - IEEE International Symposium on Information Theory - Proceedings

SP - 315

EP - 319

BT - Proceedings - 2008 IEEE International Symposium on Information Theory, ISIT 2008

T2 - 2008 IEEE International Symposium on Information Theory, ISIT 2008

Y2 - 6 July 2008 through 11 July 2008

ER -