The Complexity of Matrix Completion

1
The Complexityof Matrix Completion

• Nick Harvey
• David Karger
• Sergey Yekhanin

2
What is matrix completion?

• Given matrix containing variables, substitute
  values for the variables to get full rank

3
Why should I care? Combinatorics

• Many combinatorial problems relate to matrices of
  variables

Relation to Algebra
Problem
Tutte 47, Edmonds 67, Lovasz 79
Graph Matching
Tomizawa-Iri 74, Murota 00
Matroid Intersection
God
Counting paths in DAG
(i.e., the BOOK)
Gessel-Viennot 85
4
Why should I care? Algorithms

• Often yields highly efficient algorithms

Algorithms
Problem
RNC KUW86, MVV87 Sequential O(n2.38) time
MS04, H06
Graph Matching
O(nr1.38) time H06
Matroid Intersection
Random Network CodesKoetter-Medard 03,Ho et
al. 03
Counting paths in DAG
5
Why should I care? Complexity

• Depending on parameters, can beNP-complete, in
  RP, or in P
• Key parametersField size, variables,
  occurrences of each variable
• Contains polynomial identity testing as special
  case (Valiant 79)
• Derandomizing PIT implies strong circuit lower
  bounds (Kabanets-Impagliazzo 03)

6
Field Size

• Why care about field size?
• Relevant to complexityrandom works over large
  fields
• Understanding smaller fields may provide insight
  to derandomization
• Important for network coding efficiency(i.e.,
  complexity of routers)

7
Complexity Regions
NP Hard
9
Lovasz 79
8
Buss et al. 99
7
6
RP
Occurences of an variable
5
4
3
Geelen 99
2
P
1
H., Karger,Murota 05
2
3
5
7
n1
22
Field Size
8
Complexity Regions
This Paper
NP Hard
9
8
NP Hard
7
6
RP
Occurences of an variable
5
4
3
2
P
1
2
3
5
7
n1
22
Field Size
9
VariantSimultaneous Completion

• We have set of matrices A A1, , Ad
• Each variable appears at most once per matrix
• An variable can appear in several matrices
• Def A simultaneous completion for A assigns
  values to variables whilepreserving the rank of
  all matrices
• RP algorithm still works over large field
• Application to Network Coding usesSimultaneous
  Completion

10
Relationship to Single Matrix Completion

• Hardness for Simultaneous Completion? Hardness
  for Single Matrix Completion w/many
  occurrences of variables

Simultaneous Completion
Single Matrix Completion
11
Simultaneous Completion Algorithm

• Simple self-reducibility algorithm
• Operates over field Fq, where d matrices lt q

• Input d matrices
• Compute rank of all matrices
• Pick an variable x
• for i ? 0,,d
• Set x i
• If all matrices have unchanged rank
• Recurse ( variables has decreased)

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A Sharp Threshold

• Simple self-reducibility algorithm
• Operates over field Fq, where d matrices lt q

• Thm Simultaneous completion for dmatrices over
  Fq is
• in P if q gt d HKM 05
• NP-hard if q d This paper

13
A Sharp Threshold

• Thm Simultaneous completion for dmatrices over
  Fq is
• in P if q gt d HKM 05
• NP-hard if q d This paper
• Cor Single matrix completion with d occurrences
  of variables over Fqis NP-hard if q d

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Approach

• Reduction from Circuit-SAT

A
C
NAND
B
C ? ( A ? B )
15
What have we shown so far?

• Simultaneous completion of an unbounded number of
  matrices over F2 is NP-hard
• Can we use fewer?
• Combine small matrices into huge matrix?
• Problem Variables appear too many times
• Need to somehow make copies of a variable
• Coming up next
• completing two matrices over F2 is NP-hard

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A Curious Matrix
1 1 1 1 0
x1 1 1 1 1
x2 1 1 1
x3 1 1

xn 1
Rn
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A Curious Matrix
Thm det Rn
1 1 1 1 0
x1 1 1 1 1
x2 1 1 1
x3 1 1

xn 1
Rn
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Linearity of Determinant
1 1 1 1 0
x1 1 1 1 1
x2 1 1 1
x3 1 1

xn 1
det
1 1 1 1 1
x1 1 1 1 1
x2 1 1 1
x3 1 1

xn 1
1 1 1 1 -1
x1 1 1 1 0
x2 1 1 0
x3 1 0

xn 0


det
det
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Column Expansion
1 1 1 1 1
x1 1 1 1 1
x2 1 1 1
x3 1 1

xn 1
1 1 1 1 -1
x1 1 1 1 0
x2 1 1 0
x3 1 0

xn 0

det
det
x1 1 1 1
x2 1 1
x3 1

xn
(-1)n1 det


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1 1 1 1 1
x1 1 1 1 1
x2 1 1 1
x3 1 1

xn 1
det
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Schur Complement Identity
1 1 1 1 1
x1 1 1 1 1
x2 1 1 1
x3 1 1

xn 1
det
x1 1 1 1
x2 1 1
x3 1

xn
1
1
1

1
-


det
1 1 1 1
1
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Applying Outer Product
x1 1 1 1
x2 1 1
x3 1

xn
1
1
1

1
-
det


1 1 1 1
1
1-x1
1 1-x2
1 1 1-x3

1 1 1 1-xn
det
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Finishing up
1-x1
1 1-x2
1 1 1-x3

1 1 1 1-xn
det

QED
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Replicating Variables

• Corollary
• If x1, x2, , xn in 0,1
• then det Rn ? 0 ? xi xj ?i,j

Proof det Rn
, which is arithmetization of So either all
variables true, or all false.
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Replicating Variables

• Corollary
• If x1, x2, , xn in 0,1
• then det Rn ? 0 ? xi xj ?i,j

Consequence over F2, need only 2 matrices
NAND
Rn
A
B
NAND
Rn
NAND
Rn
26
What have we shown so far?

• Simultaneous completion of
• an unbounded number of matricesover F2 is
  NP-hard
• two matrices over F2 is NP-hard
• Next
• q matrices over Fq is NP-hard

27
Handling Fields Fq

• Previous gadgets only work if each x ? 0,1.How
  can we ensure this over Fq?
• Introduce q-2 auxiliary variables
  xx(1), x(2), , x(q-1)
• Sufficient to enforce thatx(i) ? x(j) ?i,j
  and x(i) ? 0,1 ?i ? 2

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Handling Fields Fq

• x(i) ? x(j) ?i,j and x(i) ? 0,1 ?i
  ? 2

0
1
x(1)
x(q-1)
x(2)
x(3)
x(4)
Edge indicates endpoints non-equal
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Handling Fields Fq

• x(i) ? x(j) ?i,j and x(i) ? 0,1 ?i
  ? 2

• Pack these constraints into few matrices
• Each variable used once per matrix
• Amounts to edge-coloring
• From ?(Kn), conclude that q matrices suffice

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What have we shown so far?

• Simultaneous completion of
• an unbounded number of matricesover F2 is
  NP-hard
• two matrices over F2 is NP-hard
• q matrices over Fq is NP-hard

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Main Results
Thm A simultaneous completion for dmatrices
over Fq is NP-hard if q d Cor Completion of
single matrix, variables appearing d timesis
NP-hard if q d Cor Completion of
skew-symmetric matrix, variables appearing d
timesis NP-hard if q d
32
Open Questions

• Improved hardess results / algorithmsfor matrix
  completion?
• Lower bounds / hardness for field size in network
  coding?
• More combinatorial uses of matrix completion