Fast Deterministic Algorithms for Matrix Completion Problems

Fast Deterministic Algorithmsfor 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-SymmetricMatrices

6 Conclusion

1 Introduction

6 Conclusion

Matrix Completion

Matrix CompletionF: Field

Input Matrix A(x1, . . . , xn) over F(x1, . . . , xn) with indeterminatesx1, . . . , xn

Find α1, . . . , αn ∈ F maximizing rank A(α1, . . . , αn).

ExampleF = Q,

[1 + x1 2 + x2

]−→ A ′ =

[2 21 0

](x1 := 1, x2 := 0, x3 := 1, x4 := 0)

Matrix Completion

Matrix CompletionF: Field

Input Matrix A(x1, . . . , xn) over F(x1, . . . , xn) with indeterminatesx1, . . . , xn

Find α1, . . . , αn ∈ F maximizing rank A(α1, . . . , αn).

ExampleF = Q,

[1 + x1 2 + x2

]−→ A ′ =

[2 21 0

](x1 := 1, x2 := 0, x3 := 1, x4 := 0)

Backgrounds

A variety of combinatorial optimization problems can be formulated bymatrices with indeterminates:

Maximum matching,

Structural rigidity,

Network coding, etc.

Previous WorksMatrix completion for general matrices is solvable in polynomial timeby 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.

Backgrounds

A variety of combinatorial optimization problems can be formulated bymatrices with indeterminates:

Maximum matching,

Structural rigidity,

Network coding, etc.

Previous WorksMatrix completion for general matrices is solvable in polynomial timeby 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 ResultsDeterministic polynomial time algorithms for the following matrixcompletion 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 byrank-two skew-symmetric matrices

— the first deterministic polynomial time algorithm

They are working over an arbitrary field!

Our Results

Our ResultsDeterministic polynomial time algorithms for the following matrixcompletion 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 byrank-two skew-symmetric matrices

— the first deterministic polynomial time algorithm

They are working over an arbitrary field!

1 Introduction

6 Conclusion

Problem Definition

Matrix Completion by Rank-One MatricesMatrix completion for A = B0 + x1B1 + · · ·+ xnBn, where B1, . . . ,Bn are ofrank one.

Example

[1 00 0

], B1 =

[1 10 0

], B2 =

[2 01 0

[1 + x1 + 2x2 x1

Problem Definition

Matrix Completion by Rank-One MatricesMatrix completion for A = B0 + x1B1 + · · ·+ xnBn, where B1, . . . ,Bn are ofrank one.

Example

[1 00 0

], B1 =

[1 10 0

], B2 =

[2 01 0

[1 + x1 + 2x2 x1

Previous Works

In the case of B0 = 0:

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

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

For the general case:

Ivanyos, Karpinski & Saxena ’10

An optimal solution can be found in O(m4.37n) time.

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

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

Previous Works

For A = B0 + x1B1 + · · ·+ xnBn (Bi = uiv>i (i = 1, . . . , n))

1. . .

v>1...

1. . .

0 u1 · · · un B0

rank A = 2n + rank A

For A = B0 + x1B1 + · · ·+ xnBn (Bi = uiv>i (i = 1, . . . , n))

1. . .

v>1...

1. . .

0 u1 · · · un B0

rank A = 2n + rank A

Algorithm

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

Harvey, Karger & Murota ’05

Matrix completion for a mixed matrix can be done in O(m2.77) time.

↓ Apply to A

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

Algorithm

↓ Apply to A

Algorithm

↓ Apply to A

Min-Max Theorem

TheoremFor A = B0 + x1B1 + · · ·+ xnBn,

max{rank A : x1, . . . , xn}

0 [vj : j < J]>

[uj : j ∈ J] B0

]: J ⊆ {1, . . . , n}

Corollary (Lovasz ’89)If B0 = 0, then

max{rank A : x1, . . . , xn}

=min{dim〈uj : j ∈ J〉+ dim〈vj : j < J〉 : J ⊆ {1, . . . , n}}

Min-Max Theorem

TheoremFor A = B0 + x1B1 + · · ·+ xnBn,

max{rank A : x1, . . . , xn}

0 [vj : j < J]>

[uj : j ∈ J] B0

]: J ⊆ {1, . . . , n}

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

Simultaneous Matrix Completion by Rank-One MatricesF: Field

Input CollectionA of matrices in the form of B0 + x1B1 + · · ·+ xnBn

Find Value assignments αi ∈ F for each indeterminate xi

maximizing the rank of every matrix in A

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

Simultaneous Matrix Completion by Rank-One Matrices

Simultaneous Matrix Completion by Rank-One MatricesF: Field

Input CollectionA of matrices in the form of B0 + x1B1 + · · ·+ xnBn

Find Value assignments αi ∈ F for each indeterminate xi

maximizing the rank of every matrix in A

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

1 Introduction

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 linearlycorrelated

Each sink node demands the originalmessages x1 & x2

TheoremA solution of this multicast can be found in polynomial time.

Approach: simultaneous matrix completion by rank-one matrices.

1 Introduction

6 Conclusion

Problem Definition

Mixed Skew-Symmetric Matrix CompletionMatrix completion for a skew-symmetric matrix s.t. each indeterminateappears twice (mixed skew-symmetric matrix).

Example

0 −1 11 0 0−1 0 0

+ 0 x 0−x 0 y0 −y 0

= 0 −1 + x 11 − x 0 y−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 inO(m4) time.

Our ResultMatrix completion for an m ×m mixed skew-symmetric matrix can be donein 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 inO(m4) time.

Our ResultMatrix completion for an m ×m mixed skew-symmetric matrix can be donein 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 4 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 12 0 0 3−1 0 0 21 −3 −2 0

Pfaffian:

pf A :=∑

M:perfect matching in G

±∏ij∈M

= A12A34 − A13A24

Lemmadet A = (pf A)2

Support Graph and Pfaffian

Support graph:

0 −2 1 12 0 0 3−1 0 0 21 −3 −2 0

Pfaffian:

pf A :=∑

M:perfect matching in G

±∏ij∈M

= A12A34 − A13A24

Lemmadet A = (pf A)2

Sketch of Algorithm

Algorithm1: Find an optimal solution FQ and FT for linear delta-covering.2: Find a perfect matching M in the support graph of T [FT ].

3: for each ij ∈ M do4: Substitute α to Tij so that Q[FQ ∪ {i, j}] will be nonsingular after

substitution.5: FQ := FQ ∪ {i, j}6: end for7: Substitute 0 to the rest of indeterminates8: return the resulting matrix

Sketch of Algorithm

Algorithm1: Find an optimal solution FQ and FT for linear delta-covering.2: Find a perfect matching M in the support graph of T [FT ].3: for each ij ∈ M do4: 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 indeterminates8: return the resulting matrix

Sketch of Algorithm

Algorithm1: Find an optimal solution FQ and FT for linear delta-covering.2: Find a perfect matching M in the support graph of T [FT ].3: for each ij ∈ M do4: Substitute α to Tij so that Q[FQ ∪ {i, j}] will be nonsingular after

substitution.5: FQ := FQ ∪ {i, j}6: end for7: Substitute 0 to the rest of indeterminates8: return the resulting matrix

Sketch of Algorithm

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

A = Q + T

A ′ = Q ′ + T ′

LemmaQ ′: modified matrix of Q as Q ′ij := Qij + α, Q ′ji := Qji − α

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

Sketch of Algorithm

A = Q + T A ′ = Q ′ + T ′

Sketch of Algorithm

A = Q + T A ′ = Q ′ + T ′

Sketch of Algorithm

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

TheoremMatrix completion for an m ×m mixed skew-symmetric matrix can be donein O(m4) time.

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

1 Introduction

6 Conclusion

Problem Definition

Skew-Symmetric Matrix Completion by Rank-Two Skew-SymmetricMatricesMatrix completion for A = B0 + x1B1 + · · ·+ xnBn,where B0 is skew-symmetric and B1, . . . ,Bn are rank-two skew-symmteric

Our Result

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

solvable in O(m3n) time using the algorithm of Gabow & Stallman ’86.

Our ResultAn 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)

Our Result

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

solvable in O(m3n) time using the algorithm of Gabow & Stallman ’86.

Our ResultAn 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

6 Conclusion

Conclusion

Our ResultsFaster algorithm and Min-Max theorem for matrix completion byrank-one matrices.

Application for multicast problem with linearly correlated sources.

First deterministic polynomial time algorithm for mixedskew-symmetric matrix completion.

First deterministic polynomial time algorithm for skew-symmetricmatrix completion by rank-two skew-symmetric matrices.

Future WorksApplication of skew-symmetric matrix completion

Matrix completion for other types of matrices

Conclusion

Our ResultsFaster algorithm and Min-Max theorem for matrix completion byrank-one matrices.

Application for multicast problem with linearly correlated sources.

First deterministic polynomial time algorithm for mixedskew-symmetric matrix completion.

First deterministic polynomial time algorithm for skew-symmetricmatrix completion by rank-two skew-symmetric matrices.

Future WorksApplication of skew-symmetric matrix completion

Matrix completion for other types of matrices

Fast Deterministic Algorithms for Matrix Completion Problems

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