Connectivity and Tiling Algorithms

Transcription

1 Connectivity and Tiling Algorithms Department of Mathematics Louisiana State University Phylanx Kick-off meeting Baton Rouge, August 24, 2017

2 2/29 Part I My research area: matroid theory

3 3/29 Matroids everywhere! Matroid circuits generalize Minimal linearly dependent subsets of vectors Cycles in graphs Min-weight codewords in error-correcting codes...

4 3/29 Matroids everywhere! Matroid circuits generalize Minimal linearly dependent subsets of vectors Cycles in graphs Min-weight codewords in error-correcting codes... Concepts borrowed/generalized/unified: Deletion, contraction (=projection): minors Duality (orthogonality) Connectivity

5 4/29 Connectivity in graphs and matroids Definition. A graph is vertically k-connected if it has no vertex cut of size < k.

6 4/29 Connectivity in graphs and matroids Definition. A graph is vertically k-connected if it has no vertex cut of size < k. Definition. A matroid is k-connected if it has no -separation of order < k.

7 4/29 Connectivity in graphs and matroids Definition. A graph is vertically k-connected if it has no vertex cut of size < k. Definition. A matroid is k-connected if it has no -separation of order < k. Definition. A k-separation is a partition (A, B) with A, B k and λ(a) < k, where λ(a) = r(a) + r(b) r(a B) r(a B)

8 5/29 Connectivity in graphs and matroids A B A B

9 6/29 Application Theorem (Whitney). 3-connected planar graphs have a unique plane embedding.

10 6/29 Application Theorem (Whitney). 3-connected planar graphs have a unique plane embedding.

11 7/29 Obtaining 3-connectivity Decompose graph/matroid into 3-connected pieces:

12 8/29 Globally highly connected matroids

13 9/29 Tangles, the idea

14 10/29 Tangle Definition. λ(x) = r(x) + r(e X) r(m). Definition. T is a tangle of M of order θ if λ(x) < θ X T or E X T X T λ(x) < θ X, Y, Z T X Y Z = E(M) E {e} T for all e E(M)

15 10/29 Tangle Definition. λ(x) = r(x) + r(e X) r(m). Definition. T is a tangle of M of order θ if λ(x) < θ X T or E X T X T λ(x) < θ X, Y, Z T X Y Z = E(M) E {e} T for all e E(M) Definition. Branch width is maximum θ such that M has tangle of order θ.

16 11/29 Algorithmic consequences Small branch width = thin class of graphs, dynamic programming Large branch width = large grid minor = redundant vertex

17 11/29 Algorithmic consequences Small branch width = thin class of graphs, dynamic programming Large branch width = large grid minor = redundant vertex Theorem (Geelen, Gerards, Whittle 2008). Let M be representable over GF(q). If branch width sufficiently large, then M has k k grid minor.

18 12/29 Tangle matroid Theorem (Geelen, Gerards, Robertson, Whittle). min{λ(y) : X Y T } if X Y T ρ(x) := θ otherwise. Then ρ is the rank function of a matroid, M(T ).

19 12/29 Tangle matroid Theorem (Geelen, Gerards, Robertson, Whittle). min{λ(y) : X Y T } if X Y T ρ(x) := θ otherwise. Then ρ is the rank function of a matroid, M(T ). Question. Do tangles help analysis of big data sets? Observation (Whittle, Diestel 2016). Might help to identify features in images.

20 13/29 The Structure of Highly Connected Matroids Geelen, Gerards, Whittle announced the following: Theorem. Let M be proper minor-closed class of binary matroids. There exist k, t such that every k- connected matroid M M has M or M equal to a rank-t perturbation of a graphic matroid.

21 13/29 The Structure of Highly Connected Matroids Geelen, Gerards, Whittle announced the following: Theorem. Let M be proper minor-closed class of binary matroids. There exist k, t such that every k- connected matroid M M has M or M equal to a rank-t perturbation of a graphic matroid. Perturbation: add low-rank matrix to representation. Matroidal view: small number of lifts and projections.

22 14/29 Application: error-correcting codes e Noise Channel + e ˆ

23 15/29 Application: error-correcting codes e Noise Encoder y Channel y + e Decoder ˆ

24 16/29 Application: error-correcting codes

25 17/29 Asymptotically good codes Family C 1, C 2,... of linear codes with parameters [n, k, d ] is asymptotically good if, for some ϵ > 0: (i) Growing size: n as (ii) Constant rate: k /n ϵ (iii) Growing minimum distance: d /n ϵ

26 17/29 Asymptotically good codes Family C 1, C 2,... of linear codes with parameters [n, k, d ] is asymptotically good if, for some ϵ > 0: (i) Growing size: n as (ii) Constant rate: k /n ϵ (iii) Growing minimum distance: d /n ϵ Theorem. Asymptotically good codes exist.

27 18/29 Asymptotically good codes: structure? Operations on a code: Puncturing: C \, remove th coordinate from each word Shortening: C/, take {c C : c = 0}, then remove th coordinate.

28 18/29 Asymptotically good codes: structure? Operations on a code: Puncturing: C \, remove th coordinate from each word Shortening: C/, take {c C : c = 0}, then remove th coordinate. Theorem (Nelson, vz 2015). Let M be a class of binary linear codes closed under puncturing, shortening. If M contains an asymptotically good sequence, then M contains all codes.

29 19/29 Computational matroid theory Use computer to Generate all small members of matroid classes Explore structure Perform finite case analysis

30 20/29 SageMath SageMath is A computer algebra system similar to Maple, Mathematica Open source Common interface to lots of specialized software Well-supported: bug tracking sage-support@googlegroups.com AskSage In the cloud Google Summer of Code Mentor Organization

31 21/29 SageMath N = Matroid(field=GF(5), matrix=[[1,0,0,1,1], [0,1,0,1,0], [0,0,1,1,1]] ) L = [M for M in N.linear_extensions() if M.has_minor(matroids.Uniform(2,5))]

32 22/29 Part II Tiling algorithms

33 23/29 The Spartan system NumPy-like programming language, implementing 50+ NumPy built-ins Lazy evaluation captures these in expression graph Evaluate when a variable is used, or a user enforces execution Tiling heuristic: greedy. Tile node with most neighbors first.

34 24/29 Spartan s Tiling Performance (source: Huang, Chen, Wang, Power, Ortiz, Li, Xiao 2015)

35 25/29 The Spartan problem TILING(K): INPUT: Acyclic expression digraph node groups for each call to an operator Cost function on edges PROBLEM: Is there a tiling (representative choice) from each node group such that inputs/outputs are compatible and sum of edge costs of activated edges is less than K?

36 25/29 The Spartan problem TILING(K): INPUT: Acyclic expression digraph node groups for each call to an operator Cost function on edges PROBLEM: Is there a tiling (representative choice) from each node group such that inputs/outputs are compatible and sum of edge costs of activated edges is less than K? MIN-TILING: Find a tiling that minimizes the sum of edge costs.

37 26/29 Theorem (Huang, Chen, Wang, Power, Ortiz, Li, Xiao 2015). MIN-TILING is NP-hard.

38 26/29 Theorem (Huang, Chen, Wang, Power, Ortiz, Li, Xiao 2015). MIN-TILING is NP-hard. TILING(K) is NP-complete.

39 26/29 Theorem (Huang, Chen, Wang, Power, Ortiz, Li, Xiao 2015). MIN-TILING is NP-hard. TILING(K) is NP-complete. Proof: Reduction to NAE-3SAT.

40 27/29 Approximation Algorithms Let P be a minimization problem, usually NP-hard. Definition. A c-approximation algorithm for P is an efficient algorithm that, for every instance of P with optimum value OPT( ), outputs a feasible solution with cost at most c OPT( ).

41 27/29 Approximation Algorithms Let P be a minimization problem, usually NP-hard. Definition. A c-approximation algorithm for P is an efficient algorithm that, for every instance of P with optimum value OPT( ), outputs a feasible solution with cost at most c OPT( ). c can be: (F)PTAS: c = 1 + ϵ, but running time depends on ϵ; Constant Function of input size

42 Tools Approximation-preserving reductions Randomized algorithms Primal-dual algorithms Ad-hoc techniques Great success when submodularity appears in problem description (work by Vondrák, Iwata, many others) 28/29

43 Tools Approximation-preserving reductions Randomized algorithms Primal-dual algorithms Ad-hoc techniques Great success when submodularity appears in problem description (work by Vondrák, Iwata, many others) Question. Is there an approximation-preserving NP-hardness reduction for MIN-TILING? 28/29

44 Tools Approximation-preserving reductions Randomized algorithms Primal-dual algorithms Ad-hoc techniques Great success when submodularity appears in problem description (work by Vondrák, Iwata, many others) Question. Is there an approximation-preserving NP-hardness reduction for MIN-TILING? Question. Does MIN-TILING admit an FPTAS? PTAS? Constantfactor approximation? 28/29

45 29/29 Other analysis: FPT (Fixed-Parameter Tractability) Find the optimal solution in running time O(ƒ (k) n) where n is size of the input k is some parameter of the input, like branch width.

46 29/29 Other analysis: FPT (Fixed-Parameter Tractability) Find the optimal solution in running time O(ƒ (k) n) where n is size of the input k is some parameter of the input, like branch width. Question. Does MIN-TILING admit an FPT algorithm? With respect to which parameters?

47 30/29 Other analysis: Online Algorithms Have to make decisions in real time as input gets slowly revealed k-competitive: solution produced by online algorithm is at most k times worse than optimal.

48 30/29 Other analysis: Online Algorithms Have to make decisions in real time as input gets slowly revealed k-competitive: solution produced by online algorithm is at most k times worse than optimal. Question. For which k does MIN-TILING admit a k-competitive algorithm?

49 31/29 Meta-questions Is TILING the appropriate encoding of the problem? Do we have the right cost functions on the edges? Can we analyze loops? What running times are acceptable? In the online setting, how long can we delay a decision? Is there a huge difference between average-case (practical applications) and worst-case inputs?

50 32/29 The End