ARTICLEREVIEW OptimalHomologousCycles,TotalUnimodularity...

ARTICLE REVIEWOptimal Homologous Cycles, Total Unimodularity, and LinearProgrammingby Tamal K. Dey, Anil N. Hirani and Bala Krishnamoorthy

January 9, 2017

Etienne Moutot and Johanna Seif

Topological preliminaries Optimization Problem Modularity equivalence and its implications

OUTLINE

1. Topological preliminaries

2. Optimization Problem

3. Modularity equivalence and its implications

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INTRODUCTION

Problem: Optimal Homologous Chain Problem (OHCP).

It is NP-hard over Z2 [2]Here: p-chains over Z

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TOPOLOGICAL PRELIMINARIES


P-CHAIN

K is an oriented simplicial complex.

Definition

A p-chain in K with coefficients in Z is a function:

p-simplices → Z

Cp(K): p-chain group of K.

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BOUNDARY

Definition

The boundary operator ∂p : Cp(K) → Cp−1(K) (where v̂i is thedeletion of vi in the vertex set) is defined by:

∂p[v0, . . . , vp] =

p∑i=0

(−1)i[v0, . . . , v̂i, . . . , vp]

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MATRIX REPRESENTATION

Definition

Let {σi}m−1i=0 and {τj}n−1

j=0 be the set of oriented (p− 1) and p sim-plices respectively in K. We write ∂pτj =

∑αiσi.

Thematrix representation [∂p] of the homeomorphism ∂p is :

[∂p]i,j = αi

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HOMOLOGY

Definition

Two p-chains c and c′ in K are homologous if there exists a(p + 1)-chain d in K such that

c = c′ + ∂p+1d.

If c′ = 0 then we says that c is homologous to zero.

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TORSION

Fundamental theorem of finitely generated groups [4]

Every finitely generated group G can be written as

G = F ⊕ T

where F is homeomorph to a direct sum of Z andT ∼= Z/t1 ⊕ · · · ⊕ Z/tk with ti > 1 and ti|ti+1.

We say that G is torsion free if T = 0.

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RELATIVE HOMOLOGY GROUP

Definition

We call relative chain Cp(L,L0) of L modulo a subcomplex L0 :

Cp(L)/Cp(L0)

Definition

→ ∂(L,L0)p : Cp(L,L0) → Cp−1(L,L0) = ∂p|L0

→ Zp(L,L0) = ker ∂(L,L0)p relative cycles

→ Bp(L,L0) = im ∂(L,L0)p+1 relative boundaries

→ Hp(L,L0) = Zp(L,L0)/Bp(L,L0): relative homology group

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OPTIMIZATION PROBLEM


OHCP FORMULATION

Optimal Homologous Chain Problem (OHCP)

Input: a p-chain c in K and the diagonal matrix W.Output: find c∗ homologous chain to c that minimize ∥Wc∗∥1

(with the classical definition of the 1-norm).

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LINEAR PROGRAM

minx,y

||Wx||1 such that x = c + [δp+1]y, and x ∈ Zm, y ∈ Zn

→ Take c∗ = (x1 . . . xn)

→ Replace |xi| with x+i − x−i

min∑

i|wi|(x+i + x−i )

subject to x+ − x− = c+ [∂p+1](y+ − y−)x+, x−, y+, y− ≥ 0

x+, x− ∈ Zm, y ∈ Zn

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Theorem (TU optimization)

We consider two vectors b ∈ Zm and f ∈ Rn and the integerlinear program:

min f Tx subject to Ax = b, x ≥ 0, x ∈ Zn

If A is Totally Unimodular (TU), this integer linear program canbe solved in polynomial time in the dimensions of A.

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THE MATRIX TO STUDY

The matrix in our linear program is:[I −I −[∂p+1] [∂p+1]

]Lemma

If [∂p+1] is TU then[I −I −[∂p+1] [∂p+1]

]is also TU.

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MODULARITY EQUIVALENCE AND ITSIMPLICATIONS


ORIENTABLE MANIFLODS

Theorem

For a finite simplicial complex triangulating a (p+1)-dimensionalcompact orientable manifold, [∂p+1] is totally unimodular irre-spective of the orientations of the simplices.

→ Proof independent of the main theorem.→ Rely on the link between [∂p+1] and the topological boundary.

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Theorem


→ Proof independent of the main theorem.

→ Rely on the link between [∂p+1] and the topological boundary.

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Theorem


→ Proof independent of the main theorem.→ Rely on the link between [∂p+1] and the topological boundary.

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MODULARITY EQUIVALENCE

Definition

A pure simplicial complex of dimension p is a simplicial com-plex formed by a collection of p-simplices and their proper faces.

Theorem (Main Theorem)

[∂p+1] is totally unimodular iff Hp(L,L0) is torsion-free for allpure subcomplexes L0, L of K of dimension p and p + 1 respec-tively, where L0 ⊂ L.

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Proof idea.

→ Relies on Smith normal form.→ S, T invertible, D = diag(d1, . . . , dl) (where di ≥ 1) such that:[

∂(L,L0)p+1

]= S

[D 00 0

]T

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TORSION-FREE DECISION PROBLEM

Corollary

For a simplicial complexK of dimension greater than p, there is apolynomial time algorithm for answering the following question:Is Hp(L,L0) torsion-free for all pure subcomplexes L0 and L ofdimensions p and (p + 1) such that L0 ⊂ L?

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DISCUSSION

→ Polynomial time algorithm for torsion-free spaces→ More precise complexity→ What when there is a torsion ?→ Is it still NP-hard in Z2 without torsion ?

→ Actual implementation vaguely presented→ Not much details given (source code, . . .)→ Performances of the implementation ?

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Thank you for your attention !


REFERENCES

Tamal K. Dey, Anil N. Hirani and Bala Krishnamoorthy»Optimal Homologous Cycles, Total Unimodularity, and LinearProgramming.«SIAM Journal on Computing, 2011

Chao Chen and Daniel Freedman»Hardness results for homology localization.«Discrete & Computational Geometry, 2011

Alireza Tahbaz-Salehi and Ali Jadbabaie»Distributed coverage verification in sen- sor networks withoutlocation information.«EEE Transactions on Automatic Control, 2010

James R. Munkres»Elements of algebraic topology«Addison–Wesley Publishing Company, Menlo Park, 1984

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