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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Topological preliminaries Optimization Problem Modularity equivalence and its implications
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
Topological preliminaries Optimization Problem Modularity equivalence and its implications
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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Topological preliminaries Optimization Problem Modularity equivalence and its implications
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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Topological preliminaries Optimization Problem Modularity equivalence and its implications
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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Topological preliminaries Optimization Problem Modularity equivalence and its implications
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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Topological preliminaries Optimization Problem Modularity equivalence and its implications
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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Topological preliminaries Optimization Problem Modularity equivalence and its implications
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
Topological preliminaries Optimization Problem Modularity equivalence and its implications
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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Topological preliminaries Optimization Problem Modularity equivalence and its implications
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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Topological preliminaries Optimization Problem Modularity equivalence and its implications
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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Topological preliminaries Optimization Problem Modularity equivalence and its implications
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
Topological preliminaries Optimization Problem Modularity equivalence and its implications
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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Topological preliminaries Optimization Problem Modularity equivalence and its implications
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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Topological preliminaries Optimization Problem Modularity equivalence and its implications
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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Topological preliminaries Optimization Problem Modularity equivalence and its implications
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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Topological preliminaries Optimization Problem Modularity equivalence and its implications
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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Topological preliminaries Optimization Problem Modularity equivalence and its implications
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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Topological preliminaries Optimization Problem Modularity equivalence and its implications
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 !
Topological preliminaries Optimization Problem Modularity equivalence and its implications
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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