Connectivity and Tiling Algorithms - math.lsu.edusvanzwam/talks/20170824.pdfStefan van Zwam Connectivity and Tiling Algorithms Matroids everywhere! Matroid circuits generalize

Connectivity andTiling Algorithms

Stefan van Zwam

Department of MathematicsLouisiana State University

Phylanx Kick-off meetingBaton Rouge, August 24, 2017

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Stefan van Zwam Connectivity and Tiling Algorithms

Part IMy research area: matroid theory

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Matroids everywhere!Matroid circuits generalize

• Minimal linearly dependent subsets of vectors

• Cycles in graphs

• Min-weight codewords in error-correcting codes

• . . .

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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

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Connectivity in graphs and matroidsDefinition.A graph is vertically k-connected if it has no vertexcut of size < k.

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Definition.A matroid is k-connected if it has no -separation oforder < k.

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Definition.A matroid is k-connected if it has no -separation oforder < k.

Definition.A k-separation is a partition (A,B) with |A|, |B| ≥ kand λ(A) < k, where

λ(A) = r(A) + r(B) − r(A ∪ B)�

≥ r(A ∩ B)�

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Connectivity in graphs and matroids

BA BA

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ApplicationTheorem (Whitney).3-connected planar graphs have a uniqueplane embedding.

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ApplicationTheorem (Whitney).3-connected planar graphs have a uniqueplane embedding.

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Obtaining 3-connectivityDecompose graph/matroid into 3-connected pieces:

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Globally highly connected matroids

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Tangles, the idea

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TangleDefinition.

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 6= E(M)• E − {e} 6∈ T for all e ∈ E(M)

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TangleDefinition.

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 6= E(M)• E − {e} 6∈ T for all e ∈ E(M)

Definition.Branch width is maximum θ such that M has tangleof order θ.

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Algorithmic consequences• Small branch width =⇒ thin class of graphs, dy-

namic programming

• Large branch width =⇒ large grid minor =⇒redundant vertex

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Algorithmic consequences• Small branch width =⇒ thin class of graphs, dy-

namic programming

• Large branch width =⇒ large grid minor =⇒redundant vertex

Theorem (Geelen, Gerards, Whittle 2008).Let M be representable over GF(q). If branch widthsufficiently large, then M has k × k grid minor.

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Tangle matroidTheorem (Geelen, Gerards, Robertson, Whit-tle).

ρ(X) :=§

min{λ(Y) : X ⊆ Y ∈ T } if X ⊆ Y ∈ Tθ otherwise.

Then ρ is the rank function of a matroid, M(T ).

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Tangle matroidTheorem (Geelen, Gerards, Robertson, Whit-tle).

ρ(X) :=§

min{λ(Y) : X ⊆ Y ∈ T } if X ⊆ Y ∈ Tθ 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.

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The Structure of Highly Connected Ma-troidsGeelen, Gerards, Whittle announced the following:

Theorem. Let M be proper minor-closed class ofbinary matroids. There exist k, t such that every k-connected matroid M ∈ M has M or M∗ equal to arank-t perturbation of a graphic matroid.

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The Structure of Highly Connected Ma-troidsGeelen, Gerards, Whittle announced the following:

Theorem. Let M be proper minor-closed class ofbinary matroids. There exist k, t such that every k-connected matroid M ∈ M has M or M∗ equal to arank-t perturbation of a graphic matroid.

Perturbation: add low-rank matrix to representa-tion. Matroidal view: small number of lifts and pro-jections.

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Application: error-correcting codes

Noise

Channel

e

+ e

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Noise

Channel DecoderEncoder y

e

y + e

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Asymptotically good codes• Family C1, C2, . . . of linear codes with parameters[n, k, d] is asymptotically good if, for someϵ > 0:

(i) Growing size: n→∞ as →∞(ii) Constant rate: k/n ≥ ϵ(iii) Growing minimum distance: d/n ≥ ϵ

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Asymptotically good codes• Family C1, C2, . . . of linear codes with parameters[n, k, d] is asymptotically good if, for someϵ > 0:

(i) Growing size: n→∞ as →∞(ii) Constant rate: k/n ≥ ϵ(iii) Growing minimum distance: d/n ≥ ϵ

Theorem. Asymptotically good codes exist.

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Asymptotically good codes: structure?Operations on a code:

• Puncturing: C\ , remove th coordinate fromeach word

• Shortening: C/, take {c ∈ C : c = 0}, thenremove th coordinate.

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Asymptotically good codes: structure?Operations on a code:

• Puncturing: C\ , remove th coordinate fromeach word

• Shortening: C/, take {c ∈ C : c = 0}, thenremove th coordinate.

Theorem (Nelson, vZ 2015). Let M be a class ofbinary linear codes closed under puncturing, short-ening. If M contains an asymptotically good se-quence, then M contains all codes.

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Computational matroid theoryUse computer to

• Generate all small members of matroid classes

• Explore structure

• Perform finite case analysis

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SageMathSageMath is

• A computer algebra system similar to Maple,Mathematica

• Open source

• Common interface to lots of specialized software

• Well-supported:

É bug trackingÉ [email protected]É AskSage

• In the cloud

• Google Summer of Code Mentor Organization

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SageMathN = 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))]

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Part IITiling algorithms

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The Spartan system• NumPy-like programming language, implement-

ing 50+ NumPy built-ins

• Lazy evaluation captures these in expressiongraph

• Evaluate when a variable is used, or a user en-forces execution

• Tiling heuristic: greedy. Tile node with mostneighbors first.

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Spartan’s Tiling Performance

(source: Huang, Chen, Wang, Power, Ortiz, Li, Xiao2015)

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The Spartan problemTILING(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 arecompatible and sum of edge costs of “activated”edges is less than K?

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The Spartan problemTILING(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 arecompatible and sum of edge costs of “activated”edges is less than K?

MIN-TILING:Find a tiling that minimizes the sum of edge costs.

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

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

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

Proof: Reduction to NAE-3SAT.

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Approximation AlgorithmsLet P be a minimization problem, usually NP-hard.

Definition.A c-approximation algorithm for P is an efficient al-gorithm that, for every instance of P with optimumvalue OPT(), outputs a feasible solution with cost atmost c ·OPT().

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Approximation AlgorithmsLet P be a minimization problem, usually NP-hard.

Definition.A c-approximation algorithm for P is an efficient al-gorithm that, for every instance of P with optimumvalue OPT(), outputs a feasible solution with cost atmost c ·OPT().

c can be:

• (F)PTAS: c = 1+ ϵ, but running time depends on ϵ;

• Constant

• Function of input size

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Tools• Approximation-preserving reductions

• Randomized algorithms

• Primal-dual algorithms

• Ad-hoc techniques

Great success when submodularity appears in prob-lem description (work by Vondrák, Iwata, manyothers)

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Question.Is there an approximation-preserving NP-hardnessreduction for MIN-TILING?

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Question.Is there an approximation-preserving NP-hardnessreduction for MIN-TILING?Question.Does MIN-TILING admit an FPTAS? PTAS? Constant-factor approximation?

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Other analysis: FPT (Fixed-ParameterTractability)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 branchwidth.

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Other analysis: FPT (Fixed-ParameterTractability)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 branchwidth.

Question.Does MIN-TILING admit an FPT algorithm? With re-spect to which parameters?

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Other analysis: Online Algorithms• Have to make decisions in real time as input gets

slowly revealed

• k-competitive: solution produced by online algo-rithm is at most k times worse than optimal.

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Other analysis: Online Algorithms• Have to make decisions in real time as input gets

slowly revealed

• k-competitive: solution produced by online algo-rithm is at most k times worse than optimal.

Question.For which k does MIN-TILING admit a k-competitivealgorithm?

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Meta-questions• Is TILING the appropriate encoding of the prob-

lem?

• 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 de-cision?

• Is there a huge difference between average-case(practical applications) and worst-case inputs?

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Connectivity and Tiling Algorithms - math.lsu.edusvanzwam/talks/20170824.pdfStefan van Zwam Connectivity and Tiling Algorithms Matroids everywhere! Matroid circuits generalize