From Wireless Network Coding to Matroids. Rico Zenklusen

Transcription

1 From Wireless Network Coding to Matroids Rico Zenklusen

2 A sketch of my research areas/interests Computer Science Combinatorial Optimization Matroids & submodular funct. Rounding algorithms Applications in Engineering & Operations Research Wireless network coding Probability, Statistics Graph Theory, Combinatorics Network optimization Personal rapid transit (vehicle routing) Continuous Optimization Train routing Power grid maintenance 2 / 21

3 Computer Science Combinatorial Optimization Matroids & submodular funct. Rounding algorithms Applications in Engineering & Operations Research Wireless network coding Probability, Statistics Graph Theory, Combinatorics Network optimization Personal rapid transit (vehicle routing) Continuous Optimization Train routing Power grid maintenance 2 / 21

4 Computer Science Combinatorial Optimization Matroids & submodular funct. Rounding algorithms Applications in Engineering & Operations Research Wireless network coding Probability, Statistics Graph Theory, Combinatorics Network optimization Personal rapid transit (vehicle routing) Continuous Optimization Train routing Power grid maintenance 2 / 21

5 Wireless information flows Features of wireless information flows Broadcasting Superposition Complex signal interactions. 3 / 21

6 Wireless information flows Features of wireless information flows Broadcasting Superposition Complex signal interactions. Classical model: Multiuser Gaussian Channel Unknown how to determine capacity of network. 3 / 21

7 Wireless information flows Features of wireless information flows Broadcasting Superposition Complex signal interactions. Classical model: Multiuser Gaussian Channel Unknown how to determine capacity of network. The ADT model (Avestimehr, Diggavi, Tse [2007]) A deterministic model to approximate Multiuser Gaussian Channels. 3 / 21

8 The ADT information flow model 4 / 21

9 The ADT information flow model 4 / 21

10 The ADT information flow model 4 / 21

11 The ADT information flow model Task: Send maximum number of signals from s to t. A signal is an element of F 2. 4 / 21

12 The ADT information flow model 4 / 21

13 The ADT information flow model 4 / 21

14 The ADT information flow model 4 / 21

15 The ADT information flow model 4 / 21

16 The ADT information flow model 4 / 21

17 The ADT information flow model 4 / 21

18 The ADT information flow model Interference between the two signals! Interference is modelled as XOR. 4 / 21

19 The ADT information flow model Interference between the two signals! Interference is modelled as XOR. 4 / 21

20 The ADT information flow model Receiver gets signals (x, x + y). Due to linear independence, received signals can be decoded to get (x, y). 4 / 21

21 The ADT information flow model 4 / 21

22 The ADT information flow model Received signals are linearly dependent. Receiver cannot decode. 4 / 21

23 The ADT information flow model Goal Route maximum number of decodable (i.e., linearly indep.) signals from s to t. 4 / 21

24 What was known previously? Theorem ([Avestimehr et al., 2007b]) There is a notion of ADT cut (think of s-t cut) such that Max ADT flow (coding) = Min ADT cut. However, they do not show how to find a max flow or a min cut. Theorem ([Amaudruz and Fragouli, 2009]) A maximum flow and a minimum cut can be found in polynomial time. Algorithm is rather involved and computationally expensive. 5 / 21

25 Our contribution to ADT flows We present a strong link between the ADT model and matroids. Results/techniques from matroid theory translate into ADT setting. 6 / 21

26 Our contribution to ADT flows We present a strong link between the ADT model and matroids. Results/techniques from matroid theory translate into ADT setting. Implications to ADT model (Goemans, Iwata, Z. [2009]) Classical matroid algorithms can be used to find optimal coding. Yields fast algorithms to optimize ADT networks. Leads to interesting extensions (e.g., min-cost flows). Polyhedral descriptions of all feasible solutions. Max-flow min-cut theorem as a simple consequence. 6 / 21

27 Our contribution to ADT flows We present a strong link between the ADT model and matroids. Results/techniques from matroid theory translate into ADT setting. Implications to ADT model (Goemans, Iwata, Z. [2009]) Classical matroid algorithms can be used to find optimal coding. Yields fast algorithms to optimize ADT networks. Leads to interesting extensions (e.g., min-cost flows). Polyhedral descriptions of all feasible solutions. Max-flow min-cut theorem as a simple consequence. Further generalizations (Goemans, Iwata, Z. [2012]) We present a very general matroidal model containing ADT. allows for modeling a node-capacitated ADT model. 6 / 21

28 Moving towards a cleaner description of codings 7 / 21

29 Moving towards a cleaner description of codings An ADT flow can be represented by set of used vertices. Information about exact wiring is lost. Does not matter since wiring always preserves linear independence. 7 / 21

30 Moving towards a cleaner description of codings 7 / 21

31 Feasibility for red vertex sets Propagation of signals from second to third layer: ( ) 1 0 (x, y) = (x + y, y). 1 1 }{{} Induced adjacency matrix 8 / 21

32 Feasibility for red vertex sets 1st row 1st col 2nd row 2nd col Propagation of signals from second to third layer: ( ) 1 0 (x, y) = (x + y, y). 1 1 }{{} Induced adjacency matrix 8 / 21

33 Feasibility for red vertex sets 1st row 1st col 2nd row 2nd col Propagation of signals from second to third layer: ( ) 1 0 (x, y) = (x + y, y). 1 1 }{{} Induced adjacency matrix This matrix has to be full rank. 8 / 21

34 Feasibility for red vertex sets Set of is feasible if and only if: i) each node has same number of red inputs and outputs, ii) induced adjacency matrices are full rank. 8 / 21

35 Feasibility for red vertex sets Feasible ADT flow feasible set of red vertices. 8 / 21

36 Feasibility for red vertex sets Feasible ADT flow feasible set of red vertices. To better describe feasible red vertex sets we use matroids. 8 / 21

37 Computer Science Combinatorial Optimization Matroids & submodular funct. Rounding algorithms Applications in Engineering & Operations Research Wireless network coding Probability, Statistics Graph Theory, Combinatorics Network optimization Personal rapid transit (vehicle routing) Continuous Optimization Train routing Power grid maintenance 9 / 21

38 Computer Science Combinatorial Optimization Matroids & submodular funct. Rounding algorithms Applications in Engineering & Operations Research Wireless network coding Probability, Statistics Graph Theory, Combinatorics Network optimization Personal rapid transit (vehicle routing) Continuous Optimization Train routing Power grid maintenance 9 / 21

39 What are matroids? Matroids formalize problems that can be solved by the greedy algorithm. 10 / 21

40 What are matroids? Matroids formalize problems that can be solved by the greedy algorithm. Maximum weight spanning trees Graph: G = (V, E) Forests: I = {F E F forest} / 21

41 What are matroids? Matroids formalize problems that can be solved by the greedy algorithm. Maximum weight spanning trees Graph: G = (V, E) Forests: I = {F E F forest} What structure of the problem makes the greedy algorithm work? 10 / 21

42 Formal definition Definition: matroid M = (N, I) N: finite set (ground set) I 2 N : Non-empty family of subsets of N (i) If I I and J I, then J I. (ii) If I, J I and I > J, then s I \ J with J {s} I. I: independent sets (forests), given by independence oracle. Maximal independent sets: bases (spanning trees). Because of (ii) all bases of a matroid have the same cardinality. 11 / 21

43 Formal definition Definition: matroid M = (N, I) N: finite set (ground set) I 2 N : Non-empty family of subsets of N (i) If I I and J I, then J I. (ii) If I, J I and I > J, then s I \ J with J {s} I. I: independent sets (forests), given by independence oracle. Maximal independent sets: bases (spanning trees). Because of (ii) all bases of a matroid have the same cardinality. Another example: linear matroid N R m, N < (finite set of vectors). I = {I N vectors in I are linearly independent}. 11 / 21

44 Why care about matroids? It is surprising what can be optimized with a simple greedy algorithm. Rich algorithmic theory, going far beyond the greedy algorithm. (matroid intersection, matroid union/matroid partition, matroid duality, submodular functions,... ) Toolbox with numerous applications. 12 / 21

45 Why care about matroids? It is surprising what can be optimized with a simple greedy algorithm. Rich algorithmic theory, going far beyond the greedy algorithm. (matroid intersection, matroid union/matroid partition, matroid duality, submodular functions,... ) Toolbox with numerous applications. 12 / 21

46 Beyond greedy: matroid union Finding disjoint spanning trees / 21

47 Beyond greedy: matroid union Definition: union of matroids Union of matroids M i = (N, I i ) i [k] is k i=1 M i = (N, n i=1 I i), where k I i = { k i=1i i I i I i i [k]}. i=1 Finding disjoint spanning trees / 21

48 Beyond greedy: matroid union Definition: union of matroids Union of matroids M i = (N, I i ) i [k] is k i=1 M i = (N, n i=1 I i), where k I i = { k i=1i i I i I i i [k]}. i=1 Finding disjoint spanning trees M = k i=1 M i is a matroid. Independence oracle for M can be constructed out of the ones for M i. Partition for indep. set in M can be found efficiently / 21

49 Computer Science Combinatorial Optimization Matroids & submodular funct. Rounding algorithms Applications in Engineering & Operations Research Wireless network coding Probability, Statistics Graph Theory, Combinatorics Network optimization Personal rapid transit (vehicle routing) Continuous Optimization Train routing Power grid maintenance 14 / 21

50 Describing relationships between layers by matroids We distinguish between relationships V i V i+1 i odd i even 15 / 21

51 Describing full rank submatrices by matroids v 1 w 1 v 2 w 2 A = v 1 v 2 v 3 v 4 w 1 w 2 w 3 w v 3 v 4 w 3 w 4 M = (N, I) is a matroid, where N = V i V i+1 and I = {R C R V i, C V i+1, A[R, V i+1 \ C] has full row rank}. B N is basis of M A[B V i, V i+1 \ B] is (square and) full rank. 16 / 21

52 Describing wiring by matroids M = (N, I) is a matroid, where N = V i V i+1 and I = {U W U V i, W V i+1, U can be wired within V i+1 \ W }. B N is basis of M B V i can be wired with V i+1 \ B. 17 / 21

53 Putting things together 18 / 21

54 Putting things together 18 / 21

55 Putting things together 18 / 21

56 Putting things together 18 / 21

57 Putting things together 18 / 21

58 Putting things together M 1 M 2 M 3 M 4 M 5 18 / 21

59 Putting things together M 1 M 2 M 3 M 4 M 5 Each color is a basis in the matroid M i describing the relation V i V i+1. All colored vertices are a basis in M 1 M 2 M 3 M 4 M / 21

60 ADT flows via matroid partitioning M 1 M 2 M 3 M 4 M 5 Let M = M 1 M 2 M 3 M 4 M / 21

61 ADT flows via matroid partitioning M 1 M 2 M 3 M 4 M 5 Let M = M 1 M 2 M 3 M 4 M 5. There is a one-to-one correspondence between: ADT flows Bases B (and corresponding color-partition) of M covering all middle layers (V 2, V 3, V 4, V 5 ). 19 / 21

62 ADT flows via matroid partitioning M 1 M 2 M 3 M 4 M 5 Let M = M 1 M 2 M 3 M 4 M 5. There is a one-to-one correspondence between: ADT flows Bases B (and corresponding color-partition) of M covering all middle layers (V 2, V 3, V 4, V 5 ). Number of indep. signals: B V / 21

63 Getting optimal coding via matroid union M 1 M 2 M 3 M 4 M 5 Goal: Find basis B in M covering all middle layers and maximizing B V 1. Solution: Use greedy algorithm to find maximum weight basis in M, using follows weights: first layer (V 1 ) : 1, middle layers (V 2, V 3, V 4, V 5 ) : 1 + ɛ, last layer (V 6 ) : 1 ɛ. 20 / 21

64 Conclusion The ADT model is just a special case of matroid union/partitioning. It is pretty surprising what can be solved by a simple greedy algorithm. Matroids are a powerful tool in the algorithmist s toolkit. 21 / 21

65 References I Amaudruz, A. and Fragouli, C. (2009). Combinatorial algorithms for wireless information flow. In SODA 09: Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages Avestimehr, A. S., Diggavi, S. N., and Tse, D. N. C. (2007a). A deterministic approach to wireless relay networks. In Proceedings of Allerton Conference on Communication, Control, and Computing. Avestimehr, A. S., Diggavi, S. N., and Tse, D. N. C. (2007b). Wireless network information flow. In Proceedings of Allerton Conference on Communication, Control, and Computing. Goemans, M. X., Iwata, S., and Zenklusen, R. (2009). An algorithmic framework for wireless information flow. In Proceedings of the Forty-Seventh Annual Allerton Conference on Communication, Control, and Computing. Goemans, M. X., Iwata, S., and Zenklusen, R. (2011). A flow model based on polylinking systems. Mathematical Programming, Series A. Accepted for publication.