Fast Deterministic Algorithms for Matrix Completion Problems

Fast Deterministic Algorithms
for Matrix Completion Problems

Tasuku Soma

Research Institute for Mathematical Sciences,
Kyoto Univ.

Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 1 / 29

1 Introduction

2 Matrix Completion by Rank-One Matrices

3 Application to Network Coding

4 Mixed Skew-Symmetric Matrix Completion

5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric
Matrices

6 Conclusion


1 Introduction




Matrices

6 Conclusion


Matrix Completion

Matrix Completion
F: Field
Input Matrix A (x1 , . . . , xn ) over F(x1 , . . . , xn ) with indeterminates
x1 , . . . , xn
Find α1 , . . . , αn ∈ F maximizing rank A (α1 , . . . , αn ).


Matrix Completion

Matrix Completion
F: Field
Input Matrix A (x1 , . . . , xn ) over F(x1 , . . . , xn ) with indeterminates
x1 , . . . , xn
Find α1 , . . . , αn ∈ F maximizing rank A (α1 , . . . , αn ).

Example
F = Q,

1 + x1 2 + x2 2 2
A= −→ A =
x3 x4 1 0
(x1 := 1, x2 := 0, x3 := 1, x4 := 0)


Backgrounds

A variety of combinatorial optimization problems can be formulated by
matrices with indeterminates:
Maximum matching,
Structural rigidity,
Network coding, etc.


Backgrounds

A variety of combinatorial optimization problems can be formulated by
matrices with indeterminates:
Maximum matching,
Structural rigidity,
Network coding, etc.

Previous Works
Matrix completion for general matrices is solvable in polynomial time
by a randomized algorithm if the field is sufficiently large.
Deterministic algorithms are known only for special matrices
(cf. polynomial identity testing)
NP hard over a general field.


Our Results

Our Results
Deterministic polynomial time algorithms for the following matrix
completion problems:
Matrix completion by rank-one matrices
— a faster algorithm than the previous one
Mixed skew-symmetric matrix completion
— the ﬁrst deterministic polynomial time algorithm
Skew-symmetric matrix completion by
rank-two skew-symmetric matrices


Our Results

Our Results
Deterministic polynomial time algorithms for the following matrix
completion problems:
Matrix completion by rank-one matrices
— a faster algorithm than the previous one
Mixed skew-symmetric matrix completion
Skew-symmetric matrix completion by
rank-two skew-symmetric matrices

They are working over an arbitrary ﬁeld!


1 Introduction




Matrices

6 Conclusion


Problem Deﬁnition

Matrix Completion by Rank-One Matrices
Matrix completion for A = B0 + x1 B1 + · · · + xn Bn , where B1 , . . . , Bn are of
rank one.


Problem Deﬁnition

Matrix Completion by Rank-One Matrices
Matrix completion for A = B0 + x1 B1 + · · · + xn Bn , where B1 , . . . , Bn are of
rank one.

Example
1 0 1 1 2 0
B0 = , B1 = , B2 =
0 0 0 0 1 0
1 + x1 + 2x2 x1
A=
x2 0


Previous Works

In the case of B0 = 0:

´
Lovasz ’89
This can be reduced to linear matroid intersection.

solvable in O (mn1.62 ) time using the algorithm of Gabow & Xu ’96

m: the larger of row and column sizes, n: # of indeterminates


Previous Works


´
Lovasz ’89


For the general case:

Ivanyos, Karpinski & Saxena ’10
An optimal solution can be found in O (m4.37 n) time.



Previous Works


´
Lovasz ’89



Ivanyos, Karpinski & Saxena ’10
An optimal solution can be found in O (m4.37 n) time.

Our Result
An optimal solution can be found in O ((m + n)2.77 ) time.



Idea

For A = B0 + x1 B1 + · · · + xn Bn (Bi = ui vi (i = 1, . . . , n))

 1 v1
 

.. .
 
0 .

 

. .


 




 


 



 1 vn 




 x1 
1

 


 
˜ :=  .. .. .
 
A
0
 
. .


 




 


xn 1


 




 




 


 
0
 
B0
 
u1 ··· un


 


 


Idea

For A = B0 + x1 B1 + · · · + xn Bn (Bi = ui vi (i = 1, . . . , n))

 1 v1
 

.. .
 
0 .

 

. .


 




 


 



 1 vn 




 x1 
1

 


 
˜ :=  .. .. .
 
A
0
 
. .


 




 


xn 1


 




 




 


 
0
 
B0
 
u1 ··· un


 


 

Lemma
˜
rank A = 2n + rank A


Algorithm

˜ ˜
Each indeterminate appears only once in A ! (A is a mixed matrix)


Algorithm

˜ ˜

Harvey, Karger & Murota ’05
Matrix completion for a mixed matrix can be done in O (m2.77 ) time.



Algorithm

˜ ˜

Harvey, Karger & Murota ’05
Matrix completion for a mixed matrix can be done in O (m2.77 ) time.

˜
↓ Apply to A

Theorem
Matrix completion by rank-one matrices can be done in O ((m + n)2.77 )
time.



Min-Max Theorem

Theorem
For A = B0 + x1 B1 + · · · + xn Bn ,

max{rank A : x1 , . . . , xn }
0 [vj : j J]
= min rank : J ⊆ {1, . . . , n} .
[uj : j ∈ J ] B0


Min-Max Theorem

Theorem
For A = B0 + x1 B1 + · · · + xn Bn ,

max{rank A : x1 , . . . , xn }
0 [vj : j J]
= min rank : J ⊆ {1, . . . , n} .
[uj : j ∈ J ] B0

´
Corollary (Lovasz ’89)
If B0 = 0, then

max{rank A : x1 , . . . , xn }
= min{dim uj : j ∈ J + dim vj : j J : J ⊆ {1, . . . , n}}


Simultaneous Matrix Completion by Rank-One Matrices

F: Field
Input Collection A of matrices in the form of B0 + x1 B1 + · · · + xn Bn
Find Value assignments αi ∈ F for each indeterminate xi
maximizing the rank of every matrix in A



F: Field
Input Collection A of matrices in the form of B0 + x1 B1 + · · · + xn Bn
Find Value assignments αi ∈ F for each indeterminate xi
maximizing the rank of every matrix in A

Theorem
A solution of simultaneous matrix completion by rank-one matrices can be
found in polynomial time, if |F| > |A|.


1 Introduction




Matrices

6 Conclusion


Network Coding

Network communication model s.t. intermediate nodes can perform coding

Classical model Network coding


Multicast Problem with Linearly Correlated Sources

Messages in source nodes are linearly
correlated
Each sink node demands the original
messages x1 & x2

Theorem
A solution of this multicast can be found in polynomial time.

Approach: simultaneous matrix completion by rank-one matrices.


1 Introduction




Matrices

6 Conclusion


Problem Deﬁnition

Mixed Skew-Symmetric Matrix Completion
Matrix completion for a skew-symmetric matrix s.t. each indeterminate
appears twice (mixed skew-symmetric matrix).

Example
     
 0 −1 1  0

  
  x 0  0
 
  −1 + x 1

A =1 0 0 + −x 0 y  = 1 − x 0 y

  
   
  



  
   
  

−1 0 0
 
0 −y 0
 
−1 −y

0


Our Result

There were no algorithms for this problem, but we can compute the rank.

Murota ’03 (←Geelen, Iwata & Murota ’03)
The rank of an m × m mixed skew-symmetric matrix can be computed in
O (m4 ) time.


Our Result

There were no algorithms for this problem, but we can compute the rank.

Murota ’03 (←Geelen, Iwata & Murota ’03)
The rank of an m × m mixed skew-symmetric matrix can be computed in
O (m4 ) time.

Our Result
Matrix completion for an m × m mixed skew-symmetric matrix can be done
in O (m4 ) time.


Rank of Mixed Skew-Symmetric Matrix

Lemma (Murota ’03)
For an m × m mixed skew-symmetric matrix A = Q + T
(Q: constant part, T : indeterminates part),

rank A = max |FQ FT | : both Q [FQ ], T [FT ] are nonsingular

RHS is linear delta-covering.
Optimal FQ and FT can be found in O (m4 )
time (Geelen, Iwata & Murota ’03).


Support Graph and Pfafﬁan

Support graph:

 0 −2 1
 
 1

2 
0 0 3

 

A =
 

−1 0



 0 2




 
1 −3 −2 0


Support Graph and Pfafﬁan

Support graph:

 0 −2 1
 
 1

2 
0 0 3

 

A =
 

−1 0



 0 2




 
1 −3 −2 0
Pfafﬁan:

pf A := ± Aij
M :perfect matching in G ij ∈M

= A12 A34 − A13 A24

Lemma
det A = (pf A )2


Sketch of Algorithm

Algorithm
1: Find an optimal solution FQ and FT for linear delta-covering.
2: Find a perfect matching M in the support graph of T [FT ].


Sketch of Algorithm

Algorithm
3: for each ij ∈ M do
4: Substitute α to Tij so that Q [FQ ∪ {i , j }] will be nonsingular after
substitution.
5: FQ := FQ ∪ {i , j }
6: end for


Sketch of Algorithm

Algorithm
3: for each ij ∈ M do
4: Substitute α to Tij so that Q [FQ ∪ {i , j }] will be nonsingular after
substitution.
5: FQ := FQ ∪ {i , j }
6: end for
7: Substitute 0 to the rest of indeterminates
8: return the resulting matrix


Sketch of Algorithm
How can we ﬁnd α s.t. Q [FQ ∪ {i , j }] will be nonsingular?

A =Q +T


Sketch of Algorithm

A =Q +T A =Q +T


Sketch of Algorithm

A =Q +T A =Q +T

Lemma
Q : modiﬁed matrix of Q as Qij := Qij + α, Qji := Qji − α

pf Q [FQ ∪ {i , j }] = pf Q [FQ ∪ {i , j }] ± α · pf Q [FQ ]


Sketch of Algorithm

Finally, we obtain Q s.t. rank Q = rank A .

Theorem
Matrix completion for an m × m mixed skew-symmetric matrix can be done
in O (m4 ) time.

Using delta-covering algortihm of Geelen, Iwata & Murota ’03


1 Introduction




Matrices

6 Conclusion


Problem Deﬁnition

Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric
Matrices
Matrix completion for A = B0 + x1 B1 + · · · + xn Bn ,
where B0 is skew-symmetric and B1 , . . . , Bn are rank-two skew-symmteric


Our Result


´
Lovasz ’89
This can be reduced to linear matroid parity.

solvable in O (m3 n) time using the algorithm of Gabow & Stallman ’86.


Our Result


´
Lovasz ’89
This can be reduced to linear matroid parity.

solvable in O (m3 n) time using the algorithm of Gabow & Stallman ’86.


Our Result
An optimal solution can be found in O ((m + n)4 ) time.

Idea: Reduction to mixed skew-symmetric matrix completion
(similar to matrix completion by rank-one matrices)


1 Introduction




Matrices

6 Conclusion


Conclusion

Our Results
Faster algorithm and Min-Max theorem for matrix completion by
rank-one matrices.
Application for multicast problem with linearly correlated sources.
First deterministic polynomial time algorithm for mixed
skew-symmetric matrix completion.
First deterministic polynomial time algorithm for skew-symmetric
matrix completion by rank-two skew-symmetric matrices.


Conclusion

Our Results
Faster algorithm and Min-Max theorem for matrix completion by
rank-one matrices.
Application for multicast problem with linearly correlated sources.
First deterministic polynomial time algorithm for mixed
skew-symmetric matrix completion.
First deterministic polynomial time algorithm for skew-symmetric
matrix completion by rank-two skew-symmetric matrices.

Future Works
Application of skew-symmetric matrix completion
Matrix completion for other types of matrices


Fast Deterministic Algorithms for Matrix Completion Problems

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