Exploiting Symmetry in Computing Polyhedral Bounds on Network Coding Rate Regions

Transcription

1 Exploiting Symmetry in Computing Polyhedral Bounds on Network Coding Rate Regions Jayant Apte John Walsh Department of Electrical and Computer Engineering Drexel University, Philadelphia NetCod, 205 NSF support under CCF and CCF / 9

2 Outline Motivation and Background Explicit Expressions for MSNC Rate Regions Symmetry Can Help in Polyhedral Computation Symmetry in Network Coding Classes of Polyhedral Symmetries 2 Symmetries of polyhedral Bounds and NSG NSG and Rate Region Symmetries Convex Hull method (CHM) Basic ideas behind CHM 4 Symmetry Exploiting CHM (symchm) Symmetry Exploitation in symchm Comparison of CHM and symchm 2 / 9

3 Outline Motivation and Background Explicit Expressions for MSNC Rate Regions Symmetry Can Help in Polyhedral Computation Symmetry in Network Coding Classes of Polyhedral Symmetries 2 Symmetries of polyhedral Bounds and NSG NSG and Rate Region Symmetries Convex Hull method (CHM) Basic ideas behind CHM 4 Symmetry Exploiting CHM (symchm) Symmetry Exploitation in symchm Comparison of CHM and symchm / 9

4 Problems of Interest MSNC Multisource Network Coding Described by tuple (G, S, T, β) Sources Network Sinks β : T 2 S \ allows arbitrary subset demands 4 / 9

5 Problems of Interest MSNC Multisource Network Coding Described by tuple (G, S, T, β) Sources Network Sinks X n RVs associated with a MSNC instance n S + E h : 2 Xn \ R 0 is the entropy function ω s, s S and R e, e E are the source and edge rates β : T 2 S \ allows arbitrary subset demands 4 / 9

6 Problems of Interest MSNC Multisource Network Coding Described by tuple (G, S, T, β) Sources Network Destinations. Paths.. In(i) i Out(i).. Y. s s. t β(t) ω. Ue e Re. s.. S T S T Entropy function must satisfy: (L ) s S h s = h S (L 2 ) a h In(i) = h In(i) Out(i)), i V (L ) h e R e, h s ω s a L 2 can be partitioned into L 2, L 2 2.L 2 (constraints associated with sources, intermmediate nodes and sinks resp.) 5 / 9

7 Implicit characterization Yan, Yeung and Zhang s Characterization The set of all source rate- edge rate tuples (ω, r) is given as R = proj ω,r (con(γ n L ) L ) () Notation L = L L 2 L 2, L = L L 2 proj ω,r ( ) stands for linear projection onto rate variables con( ) stands for conic hull, ( ) stands for closure Γ n stands for the unknown entropy function region: set of all 2 n dimensional vectors arising from entropy function 6 / 9

8 Explicit Characterization? Shannon entropy subspaces matroids class of polymatroids bound comment polymatroidal set functions necessary but not sufficient more restrictive polymatroidal set functions arising from random variables Arising from a column partition of matrices over finite field of size q necessary and sufficient every possible code sufficient but not necessary, vector linear codes Arising from matrices over finite field of size q sufficient but not necessary, scalar linear codes R k = proj ω,r (Γ k n L 2), k {in,out} 7 / 9

9 Explicit Characterization? Shannon entropy subspaces matroids class of polymatroids bound comment polymatroidal set functions necessary but not sufficient more restrictive polymatroidal set functions arising from random variables Arising from a column partition of matrices over finite field of size q necessary and sufficient every possible code sufficient but not necessary, vector linear codes Arising from matrices over finite field of size q sufficient but not necessary, scalar linear codes R k = proj ω,r (Γ k n L 2), k {in,out} Polyhedral Projection 7 / 9

10 Leave no stone unturned We developed a C software library for polyhedral projection called chm Accepts arbitrary polyhedra as input, uses rational arithmetic, rational LP solver QSopt ex, and gives exact answers The underlying algorithm Convex Hull Method was first used by Xu, Wang and Sun* for computing non-shannon information inequalities We have used it for computing non-shannon information inequalities and LP/Shannon outer bounds on rate regions Can also be used for solving interesting problems related to quantum information called marginal problems chm currently turns a blind eye to symmetries of input polyhedra *W. Xu, J. Wang, J. Sun, A projection method for derivation of non-shannon-type information inequalities, The proceedings of IEEE International Symposium on Information Theory (ISIT), pages , / 9

11 Outline Motivation and Background Explicit Expressions for MSNC Rate Regions Symmetry Can Help in Polyhedral Computation Symmetry in Network Coding Classes of Polyhedral Symmetries 2 Symmetries of polyhedral Bounds and NSG NSG and Rate Region Symmetries Convex Hull method (CHM) Basic ideas behind CHM 4 Symmetry Exploiting CHM (symchm) Symmetry Exploitation in symchm Comparison of CHM and symchm 9 / 9

12 Symmetry and Polyhedral Computation Symmetry can be exploited in polyhedral representation conversion (given extreme points compute inequalities or vice versa) Symmetry can be exploited in linear programming 2 D. Bremner and M. Sikiric and A. Schürmann. Polyhedral representation conversion up to symmetries, Proceedings of the 2006 CRM workshop on polyhedral computation, AMS/CRM Lecture Notes 48 (2009), R. Bödi and K. Herr, Symmetries in linear and integer programs, arxiv: [math.co] 0 / 9

13 Symmetry and Polyhedral Computation Symmetry can be exploited in polyhedral representation conversion (given extreme points compute inequalities or vice versa) Symmetry can be exploited in linear programming 2 What about polyhedral projection (esp. CHM)? D. Bremner and M. Sikiric and A. Schürmann. Polyhedral representation conversion up to symmetries, Proceedings of the 2006 CRM workshop on polyhedral computation, AMS/CRM Lecture Notes 48 (2009), R. Bödi and K. Herr, Symmetries in linear and integer programs, arxiv: [math.co] 0 / 9

14 Outline Motivation and Background Explicit Expressions for MSNC Rate Regions Symmetry Can Help in Polyhedral Computation Symmetry in Network Coding Classes of Polyhedral Symmetries 2 Symmetries of polyhedral Bounds and NSG NSG and Rate Region Symmetries Convex Hull method (CHM) Basic ideas behind CHM 4 Symmetry Exploiting CHM (symchm) Symmetry Exploitation in symchm Comparison of CHM and symchm / 9

15 Network Symmetry Group* Those permutations of {,..., n} that keep network constraints L 2 invariant Form a group called Network Symmetry Group (NSG) Can be computed using algorithms for computing automorphism groups of graphs *Jayant Apte, and John MacLaren Walsh, Symmetry in Network Coding, extended version, 2 / 9

16 Example: Butterfly Network source independence h +h 2 =h {,2} (,2)()(4,5)(6,7)(8,9) h +h 2 =h {,2} h =h {,4,6} h 2 =h {2,5,7} 2 node constraints h 2 =h {2,5,7} h {4,5} =h {,4,5} h =h {,8,9} h {6,8} =h {2,6,8} h =h {,4,6} h {4,5} =h {,4,5} h =h {,8,9} h {7,9} =h {,7,9} h {7,9} =h {,7,9} h {6,8} =h {2,6,8} rate constraints / 9

17 Example: M-Network Example An MSNC instance The Network Symmetry group is generated by: g =(, )(2, 4)(5, 7)(6, 8) (9, 2)(0, )(, 2) (4, 2)(5, 22)(6, 24)(8, 9) g 2 =(, 4)(7, 8)(, 4)(5, 6) (7, 8)(9, 20)(2, 22)(2, 24) g =(, 2)(5, 6)(, 5)(4, 6) (7, 9)(8, 20)(2, 2)(22, 24) A group of order 8 4 / 9

18 Example: Independent Distributed Source Coding 2 2 a a 2 2 b 2 b 2 c c <(a,b),(b,c)> <(2,)(b,c),(,2)(a,b)> 4 a,2 5 6 a,2 2 b, 2 b, 2 c 2, c 2, <(2,)(a,b),(,2)(b,c)> 2 2 <(2,)(b,c),(,2)(a,b)> 2 a b c a b c 2 2 <(2,)(a,b),(,2)(b,c)> <(2,)(b,c),(,2)(a,b)> 5 / 9

19 Outline Motivation and Background Explicit Expressions for MSNC Rate Regions Symmetry Can Help in Polyhedral Computation Symmetry in Network Coding Classes of Polyhedral Symmetries 2 Symmetries of polyhedral Bounds and NSG NSG and Rate Region Symmetries Convex Hull method (CHM) Basic ideas behind CHM 4 Symmetry Exploiting CHM (symchm) Symmetry Exploitation in symchm Comparison of CHM and symchm 6 / 9

20 Combinatorial Symmetries combinatorial symmetries 7 / 9

21 Combinatorial Symmetries combinatorial symmetries Very difficult to compute (require representation conversion*): *Volker Kaibel and Alexander Schwartz. On the complexity of polytope isomorphism problems. Graphs and Combinatorics, 9(2):25 20, 2007 / 9

22 Restricted Symmetries A subgroup of combinatorial symmetry group: combinatorial symmetries restricted symmetries (0,2) (0,) is a restricted symmetry of the triangle below (0,0) (,0) 8 / 9

23 Restricted Symmetries A subgroup of combinatorial symmetry group: combinatorial symmetries restricted symmetries (0,2) (0,) is a restricted symmetry of the triangle below (0,0) (,0) Can be obtained by computing automorphism group of an edge colored graph*: *Computing symmetry groups of polyhedra. David Bremner, Mathieu Dutour Sikiric, Dmitrii V. Pasechnik, Thomas Rehn, Achill Schrmann. arxiv: v [math.co], / 9

24 Permutational Symmetries A subgroup of restricted symmetry group: combinatorial symmetries restricted symmetries All! permutations of standard basis vectors of are permutational symmetries of the cube permutational symmetries Can be thought of as restricted symmetry arising from permutation matrices 9 / 9

25 Outline Motivation and Background Explicit Expressions for MSNC Rate Regions Symmetry Can Help in Polyhedral Computation Symmetry in Network Coding Classes of Polyhedral Symmetries 2 Symmetries of polyhedral Bounds and NSG NSG and Rate Region Symmetries Convex Hull method (CHM) Basic ideas behind CHM 4 Symmetry Exploiting CHM (symchm) Symmetry Exploitation in symchm Comparison of CHM and symchm 20 / 9

26 Symmetries of Rate Regions NSG can be interpreted as permutational symmetry group of rate region is achievable rate tuple iff is achievable rate tuple (bpnu), (bpnu), 0,0 (bpnu) (bpnu) (bpnu: bits per network use) 2 / 9

27 Symmetries of Rate Regions Let L 2 be the intersection of halfspaces and hyperplanes associated with network constraints NSG can be interpreted as permutational symmetry group of rate region R NSG can be interpreted as permutational symmetry group of Γ in n L 2, Γout n L 2 provided that Γ in n and Γ out n are setwise invariant under random variable permutations (which is indeed the case with most known bounds on entropy) 22 / 9

28 Symmetries of Rate Regions NSG can be interpreted as permutational symmetry group of proj ω,r (Γ in n L 2 ), proj ω,r(γ out n L 2 ) (,2) is a permutational symmetry (,2) is still a permutational symmetry after projection 2 / 9

29 Outline Motivation and Background Explicit Expressions for MSNC Rate Regions Symmetry Can Help in Polyhedral Computation Symmetry in Network Coding Classes of Polyhedral Symmetries 2 Symmetries of polyhedral Bounds and NSG NSG and Rate Region Symmetries Convex Hull method (CHM) Basic ideas behind CHM 4 Symmetry Exploiting CHM (symchm) Symmetry Exploitation in symchm Comparison of CHM and symchm 24 / 9

30 Key Idea ) Solve LP over with 0 in non-projection cost vector entries 2) Project extreme point of attaining the optimum ) Obtain an extreme point of projection 25 / 9

31 Key Idea 2 full-dimensional inner bound facet of inner bound that is not a facet of projection can be identified by solving an LP over facet Not a facet else is not a facet of projection is a facet of projection gives previously unknown vertex 26 / 9

32 Key Idea previously known full-dimensional inner bound new and improved improved fulldimensional inner bound Update the inner bound using an iteration of double description method* previously unknown vertex 27 / 9

33 Iterations of CHM 28 / 9

34 Outline Motivation and Background Explicit Expressions for MSNC Rate Regions Symmetry Can Help in Polyhedral Computation Symmetry in Network Coding Classes of Polyhedral Symmetries 2 Symmetries of polyhedral Bounds and NSG NSG and Rate Region Symmetries Convex Hull method (CHM) Basic ideas behind CHM 4 Symmetry Exploiting CHM (symchm) Symmetry Exploitation in symchm Comparison of CHM and symchm 29 / 9

35 Three uses of symmetry symmetric improvements symmetry exploitation symmetric linear programs symmetric updates 0 / 9

36 Exploitation : Symmetric Improvements symmetric improvements CHM iterations symmetric linear programs symmetry exploitation symmetric updates symchm iterations,0,,,0 If we know 0,, one of these then we know all of these! projection of a hypercube to -d / 9

37 Exploitation 2: Symmetric Updates symmetric improvements,0, 0,, facets incident to (0,,) symmetry exploitation,0,,,0 0,, facets incident to (,0,) symmetric linear programs symmetric updates (,2),,0 0,,,0, 2 / 9

38 Exploitation : Symmetric LPs symmetric improvements symmetry exploitation permutation of standard bases is a restricted symmetry of this polytope it also leaves cost invariant maximize symmetric linear programs symmetric updates there exists a solution of this LP in fixed space: the subspace of points mapping to themselves under the basis vector permutation / 9

39 Outline Motivation and Background Explicit Expressions for MSNC Rate Regions Symmetry Can Help in Polyhedral Computation Symmetry in Network Coding Classes of Polyhedral Symmetries 2 Symmetries of polyhedral Bounds and NSG NSG and Rate Region Symmetries Convex Hull method (CHM) Basic ideas behind CHM 4 Symmetry Exploiting CHM (symchm) Symmetry Exploitation in symchm Comparison of CHM and symchm 4 / 9

40 Comparison of CHM (top) and symchm (bottom) Initial Hull New vertex Old vertex Initial Hull 5 / 9

41 Complexity Comparison (No. and dimension of linear programs) 2 a b c <(a,b),(b,c)> No. of LPs solved CHM symchm CHM symchm CHM symchm CHM symchm CHM symchm CHM symchm d=64 d=2 d=28 d=24 d=20 d=6 2 2 a b c 2 2 <(2,)(b,c),(,2)(a,b)> a 25 4 a,2 2 2 b, b 2 20 c 2, c <(2,)(a,b),(,2)(b,c)> 5 <(2,)(a,b),(,2)(b,c)> a a,2 2 b 2 2 b, 5 c c 2, 0 <(2,)(b,c),(,2)(a,b)> <(2,)(b,c),(,2)(a,b)> I-DSC instance 6 / 9

42 Conclusion In this paper, We related network symmetries to symmetries of polyhedral bounds Developed a symmetry exploiting variant of Convex Hull Method Future work: A Python implementation of symchm in SageMath A C implementation of symchm to go with the Entropic Vectors Computation Package Extensive computational experimemnts with arbitrary symmetric polyhedra 7 / 9

43 Selected References I D. Bremner and M. Sikiric and A. Schürmann. Polyhedral representation conversion up to symmetries, Proceedings of the 2006 CRM workshop on polyhedral computation, AMS/CRM Lecture Notes 48 (2009), R. Bödi and K. Herr, Symmetries in linear and integer programs, [Online]. Available: 8 / 9

44 Questions 9 / 9