Searching for Capacity Factors is NP-Complete
YUAN LI
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Abstract
In this paper we investigate the problems of searching for the capacity factors and determining the capacity ranks of edges in a network coding-based network, which were first proposed in (K. Cai and P.Y. Fan, 2007). For the former problem, we prove that it is computationally hard by reducing the well known NP-complete SUB-SUM problem to the current problem. For the latter problem, we devise efficient algorithms in a special case of networks and conjecture that in general case the problem is also hard.
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Store-and-forward had been the predominant technique for transmitting information through a network until its optimality was refuted by network coding theory. Network coding offers a new paradigm for network communications and has generated abundant research interest in information and coding theory, networking, switching, wireless communications, cryptography, computer science, operations research, and matrix theory. We review the foundational work that has led to the development of network coding theory and discuss the theory for the transmission from a single source node to other nodes in the network. A companion issue discusses the theory when there are multiple source nodes each intending to transmit to a different set of destination nodes. Publisher's Note References to 'Part I' and 'Part II' in this issue refer to Foundations and Trends R in Communications and Information Technology Volume 2 Numbers 4 and 5 respectively. 3 Cyclic Networks 3.1 Non-equivalence between local and global descriptions 3.2 Convolutional network code 3.3 Decoding of convolutional network code 4 Network Coding and Algebraic Coding 4.1 The combination network 4.2 The Singleton bound and MDS codes 4.3 Network erasure/error correction and error detection vii 4.4 Further remarks a summary of the literature (see page 135) in the form of a table according to the following categorization of topics: 1. Linear coding 2. Nonlinear coding 3. Random coding 4. Static codes 5. Convolutional codes 6. Group codes 7. Alphabet size 8. Code construction 9. Algorithms/protocols 10. Cyclic networks 11. Undirected networks 12. Link failure/Network management 13. Separation theorem 14. Error correction/detection 15. Cryptography 16. Multiple sources 17. Multiple unicasts 18. Cost criteria 19. Non-uniform demand 20. Correlated sources 21. Max-flow/cutset/edge-cut bound 22. Superposition coding 23. Networking 24. Routing 25. Wireless/satellite networks 26. Ad hoc/sensor networks 27. Data storage/distribution 28. Implementation issues 29. Matrix theory 30. Complexity theory 31. Graph theory 32. Random graph 33. Tree packing 1.2. Some examples 245 Fig. 1.1 Multicasting over a communication network. Assume that we multicast two data bits b 1 and b 2 from the source node S to both the nodes Y and Z in the acyclic network depicted by Figure 1.1(a). Every channel carries either the bit b 1 or the bit b 2 as indicated. In this way, every intermediate node simply replicates and sends out the bit(s) received from upstream. The same network as in Figure 1.1(a) but with one less channel appears in Figures 1.1(b) and (c), which shows a way of multicasting 3 bits b 1 , b 2 and b 3 from S to the nodes Y and Z in 2 time units. This
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