On the Hardness of Approximating the Network Coding Capacity

• Published in: IEEE transactions on information theory Vol. 57; no. 2; pp. 1008 - 1014
• Main Authors: Langberg, Michael, Sprintson, Alex
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.02.2011; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Approximation; Approximation methods; capacity; Coding; Color; complexity; Decoding; Encoding; Graph coloring; Graphs; index coding; Mathematical analysis; Network coding; Networks; Nonlinearity; Polynomials; Scalars; Vectors; Vectors (mathematics)
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2010.2094910

This work addresses the computational complexity of achieving the capacity of a general network coding instance. 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 paper we address the notion of approximation in the context of both linear and nonlinear network coding. Loosely speaking, we show that given an instance of the general network coding problem of capacity C , constructing a code of rate α C 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. Our results refer to scalar linear, vector linear, and nonlinear encoding functions and are the first results that address the computational complexity of achieving the network coding capacity in both the vector linear and general network coding scenarios. In addition, we consider the problem of determining the (scalar) linear capacity of a planar network coding instance (i.e., an instance in which the underlying graph is planar). We show that even for planar networks this problem remains NP-hard.