On the Hardness of Approximating the Network Coding Capacity
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 noti...
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| Published in: | IEEE transactions on information theory Vol. 57; no. 2; pp. 1008 - 1014 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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New York
IEEE
01.02.2011
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9448, 1557-9654 |
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| Abstract | 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. |
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| AbstractList | 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 [Formula Omitted], constructing a code of rate for any universal (i.e., independent of the size of the instance) constant 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. 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. 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 alpha C for any universal (i.e., independent of the size of the instance) constant alpha less than or equal to 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. |
| Author | Sprintson, Alex Langberg, Michael |
| Author_xml | – sequence: 1 givenname: Michael surname: Langberg fullname: Langberg, Michael email: mikel@openu.ac.il organization: Comput. Sci. Div., Open Univ. of Israel, Raanana, Israel – sequence: 2 givenname: Alex surname: Sprintson fullname: Sprintson, Alex email: spalex@tamu.edu organization: Dept. of Electr. & Comput. Eng., Texas A&M Univ., College Station, TX, USA |
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| SubjectTerms | 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) |
| Title | On the Hardness of Approximating the Network Coding Capacity |
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