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E-raamat: Submodular Rate Region Models for Multicast Communication in Wireless Networks

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Submodular Rate Region Models for Multicast Communication in Wireless NetworksSuurem pilt
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?This book proposes representations of multicast rate regions in wireless networks based on the mathematical concept of submodular functions, e.g., the submodular cut model and the polymatroid broadcast model. These models subsume and generalize the graph and hypergraph models. The submodular structure facilitates a dual decomposition approach to network utility maximization problems, which exploits the greedy algorithm for linear programming on submodular polyhedra. This approach yields computationally efficient characterizations of inner and outer bounds on the multicast capacity regions for various classes of wireless networks.

1 Introduction
1(14)
1.1 Preliminaries and Notation
7(2)
1.2 List of Symbols and Operators
9(6)
References
11(4)
2 Submodular Information Flow Models for Multicast Communication
15(52)
2.1 Cut Model
16(3)
2.2 Graph Model
19(4)
2.3 Hypergraph Model
23(4)
2.4 Polymatroid Broadcast Model
27(5)
2.5 Transformation of Models
32(6)
2.6 Generalized Cut Model
38(2)
2.7 Penalized Polymatroid Broadcast Model
40(2)
2.8 Rate Region Properties and Equivalence
42(4)
2.9 Cut Rate Sandwiched Multicast Source Rate Regions
46(3)
2.10 Extension to Per-terminal Cut Models
49(1)
2.11 Proofs
50(17)
2.11.1 Polymatroid Max-Flow Min-Cut Theorem
50(2)
2.11.2 Transformation of Models
52(8)
2.11.3 Rate Region Properties and Equivalence
60(2)
2.11.4 Cut Rate Sandwiched Multicast Source Rate Regions
62(2)
References
64(3)
3 Network Utility Maximization via Submodular Dual Decomposition
67(38)
3.1 Concave Network Utility Maximization
69(2)
3.2 Dual Decomposition Approach for Min-Cut Rate Regions
71(4)
3.3 Dual Decomposition Approach for Max-Flow Regions
75(3)
3.4 Connections Between the Dual Decomposition Approaches
78(2)
3.5 Dual Decomposition Approach for Hyperarc Rate Regions
80(1)
3.6 Convexity and Comprehensiveness
81(5)
3.7 Upper Bound for Nonsubmodular Cut Rate Regions
86(2)
3.8 Counting Set Function Evaluations
88(5)
3.9 Discussion and Related Dual Decomposition Methods
93(1)
3.10 Extension to Per-terminal Cut Models
94(2)
3.11 Proofs and Appendices
96(9)
3.11.1 Utility Characterization of the Multicast Rate Region
96(2)
3.11.2 Network Utility Maximization Problem
98(1)
3.11.3 Dual Decomposition Approaches
98(3)
3.11.4 Convexity and Comprehensiveness
101(1)
References
102(3)
4 Network Coding Bounds and Submodularity
105(50)
4.1 Discrete Memoryless Multicast Networks
108(19)
4.1.1 Cut-Set Outer Bound
109(3)
4.1.2 Noisy Network Coding Inner Bound
112(8)
4.1.3 Elementary Hypergraph Decomposition Inner Bound
120(4)
4.1.4 Weighted Sum Multicast Rate Maximization
124(3)
4.2 Networks of Independent Broadcast Channels
127(13)
4.2.1 Cut-Set Outer Bound
128(2)
4.2.2 Noisy Network Coding Inner Bound
130(3)
4.2.3 Elementary Broadcast Decomposition Inner Bound
133(1)
4.2.4 Elementary Broadcast Decomposition for Less Noisy Channels
134(4)
4.2.5 Weighted Sum Multicast Rate Maximization
138(2)
4.3 Discrete Memoryless Networks with Known State Sequence
140(4)
4.3.1 Cut-Set Outer Bound
141(1)
4.3.2 Noisy Network Coding Inner Bound
142(2)
4.4 Proofs
144(11)
4.4.1 Cut-Set Outer Bound
144(1)
4.4.2 Noisy Network Coding Inner Bound
145(3)
4.4.3 Networks of Independent Broadcast Channels
148(3)
References
151(4)
5 Deterministic and Linear Finite Field Networks
155(20)
5.1 Deterministic Networks
157(3)
5.1.1 Bounds on the Multicast Capacity Region
157(1)
5.1.2 Weighted Sum Source Rate Maximization
158(2)
5.2 Networks of Independent Deterministic Broadcast Channels
160(2)
5.2.1 Broadcast Representation of the Capacity Region
160(1)
5.2.2 Insufficiency of the Hyperarc Model
161(1)
5.2.3 Weighted Sum Source Rate Maximization
162(1)
5.3 Noisy Linear Finite Field Networks
162(11)
5.3.1 Cut-Set Outer Bound
164(2)
5.3.2 Noisy Network Coding Inner Bound
166(1)
5.3.3 Tightness of Inner and Outer Bounds
167(3)
5.3.4 Deterministic Linear Finite Field Networks
170(1)
5.3.5 Weighted Sum Source Rate Maximization
171(2)
5.4 Proofs
173(2)
References
174(1)
6 Erasure Broadcast Networks
175(30)
6.1 Networks of Independent Erasure Broadcast Channels
177(9)
6.1.1 Cut-Set Outer Bound
177(2)
6.1.2 Noisy Network Coding
179(6)
6.1.3 Tightness of Inner and Outer Bounds in Packet Networks
185(1)
6.2 Networks of Erasure Broadcast Channels with States
186(3)
6.2.1 Cut-Set Outer Bound
188(1)
6.2.2 Noisy Network Coding Inner Bound
188(1)
6.3 Weighted Sum Multicast Rate Maximization
189(8)
6.3.1 Characterization of the Cut-Set Outer Bound
190(1)
6.3.2 Perfect Erasure Quantization
191(1)
6.3.3 Advanced Erasure Quantization Optimization Approaches
191(6)
6.4 Numerical Example
197(3)
6.5 Proofs
200(5)
References
202(3)
7 Network Coding Bounds for Gaussian Networks
205(42)
7.1 Gaussian Networks
208(24)
7.1.1 Cut-Set Outer Bound
208(1)
7.1.2 Loosening the Cut-Set Outer Bound
209(3)
7.1.3 Submodular Approximations of the Cut-Set Outer Bound
212(3)
7.1.4 Noisy Network Coding Inner Bound
215(2)
7.1.5 Tightness of Inner and Outer Bounds
217(6)
7.1.6 Asymptotic Analysis of Inner and Outer Bounds
223(3)
7.1.7 Weighted Sum Multicast Rate Maximization
226(6)
7.2 Networks of Gaussian Broadcast Channels
232(10)
7.2.1 Cut-Set Outer Bound
232(2)
7.2.2 Noisy Network Coding Inner Bound
234(2)
7.2.3 Elementary Broadcast Decomposition Inner Bound
236(1)
7.2.4 Elementary Broadcast Decomposition for Degraded Channels
237(2)
7.2.5 Weighted Sum Multicast Rate Maximization
239(3)
7.3 Proofs
242(5)
References
243(4)
8 Numerical Results for Gaussian Networks
247(32)
8.1 Random Networks Topology and Channel Model
248(3)
8.2 Sum Multicast Rate Results
251(19)
8.2.1 Bidirectional Communication
251(5)
8.2.2 Single-Source Multicast Communication
256(4)
8.2.3 Multiple Access Relay Networks
260(6)
8.2.4 Multi-source Multicast Communication
266(4)
8.3 Cut Rate Function Evaluation Results
270(9)
References
277(2)
9 Concluding Remarks
279
References
281
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