The Complexityof Matrix Completion

Nick HarveyDavid KargerSergey Yekhanin

What is matrix completion?

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

1 x

1 y

1 1

1 0

x=1, y=0

1 x

1 y

1 1

1 1

x=1, y=1Bad

Why should I care? Combinatorics

Many combinatorial problems relate to matrices of variables

Tutte ’47, Edmonds ’67, Lovasz ’79

Relation to Algebra

Tomizawa-Iri ’74, Murota ’00

Gessel-Viennot ’85

Graph Matching

Matroid Intersection

Counting paths in DAG

Problem

God(i.e., the BOOK)

Why should I care? Algorithms

Often yields highly efficient algorithms

RNC: KUW’86, MVV’87Sequential O(n2.38) time: MS’04, H’06

O(nr1.38) time: H’06

Random Network Codes:Koetter-Medard ’03,

Ho et al. ’03

Graph Matching

Matroid Intersection

Counting paths in DAG

AlgorithmsProblem

Why should I care? Complexity

Depending on parameters, can beNP-complete, in RP, or in PKey parameters:

Field 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)

Field Size

Why care about field size?Relevant to complexity:

random works over large fieldsUnderstanding smaller fields may provide

insight to derandomization Important for network coding efficiency

(i.e., complexity of routers)

Complexity Regions

Field Size

# Occurencesof an variable

2 3 5 7 n+1

1

2

3

4

5

6

7

8

9

22

NPHard

RP

P

Buss et al. ‘99Lovasz ‘79

H., Karger,Murota ‘05

P

Geelen ‘99? ? ? ? ? ?

Complexity Regions

# Occurencesof an variable

2 3 5 7 n+1

1

2

3

4

5

6

7

8

9

22

NPHard

RP

NPHard

Field Size

P P

Variant:Simultaneous Completion We have set of matrices A := {A1, …, Ad}

Each variable appears at most once per matrixAn 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 uses

Simultaneous Completion

Relationship to Single Matrix Completion Hardness for Simultaneous Completion

Hardness for Single Matrix Completion w/many occurrences of variables

1 AB C

Simultaneous Completion

1 AD E

1 BC D

Single Matrix Completion

Simultaneous Completion Algorithm

Input: d matrices

Compute rank of all matricesPick an variable x

for i {0,…,d}Set x := iIf all matrices have unchanged rank Recurse (# variables has decreased)

Simple self-reducibility algorithm Operates over field Fq, where d := # matrices < q

Non-trivial!Murota ’93.

A Sharp Threshold Simple self-reducibility algorithm Operates over field Fq, where d := # matrices < q

Thm: Simultaneous completion for dmatrices over Fq is:

in P if q > d [HKM ’05] NP-hard if q ≤ d [This paper]

A Sharp ThresholdThm: Simultaneous completion for d

matrices over Fq is:

in P if q > d [HKM ’05] NP-hard if q ≤ d [This paper]

Cor: Single matrix completion with d occurrences of variables over Fq

is NP-hard if q ≤ d

Approach Reduction from Circuit-SAT

ANAND

BC

C = ( A B )

C = 1 - A ∙ B (if A, B, C {0, 1})

det 01 A

B C

(if A, B, C {0, 1})

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 timesNeed to somehow make “copies” of a variable

Coming up next: completing two matrices over F2 is NP-hard

A Curious Matrix

1 1 1 1 0

x1 1 1 1 1

x2 1 1 1

x3 1 1

xn 1

Rn :=

A Curious Matrix

1 1 1 1 0

x1 1 1 1 1

x2 1 1 1

x3 1 1

xn 1

Rn :=

Thm: det Rn = i

ii

i xx )()1(

Linearity of Determinant1 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

det

1 1 1 1 -1

x1 1 1 1 0

x2 1 1 0

x3 1 0

xn 0

det+=

Column Expansion

ii

n x1)1(

x1 1 1 1

x2 1 1

x3 1

xn

(-1)n+1 det= =

1 1 1 1 1

x1 1 1 1 1

x2 1 1 1

x3 1 1

xn 1

det

1 1 1 1 -1

x1 1 1 1 0

x2 1 1 0

x3 1 0

xn 0

det+

ii

n x1)1(

1 1 1 1 1

x1 1 1 1 1

x2 1 1 1

x3 1 1

xn 1

det

Schur Complement Identity1 1 1 1 1

x1 1 1 1 1

x2 1 1 1

x3 1 1

xn 1

det

= det

x1 1 1 1

x2 1 1

x3 1

xn

1 1 1 11

1

1

1

1

∙ ∙ -

ii

n x1)1(

ii

n x1)1(

Applying Outer Product

= det

1-x1

1 1-x2

1 1 1-x3

1 1 1 1-xn

= det

x1 1 1 1

x2 1 1

x3 1

xn

1 1 1 11

1

1

1

1

∙ ∙ -

ii

n x1)1(

ii

n x1)1(

Finishing up

= det

1-x1

1 1-x2

1 1 1-x3

1 1 1 1-xn

i

ii

i xx )()1(=

QED

ii

n x1)1(

Proof:

det Rn = ,

which is arithmetization of

So either all variables true, or all false.

Replicating VariablesCorollary:

If {x1, x2, …, xn} in {0,1}

then det Rn 0 xi = xj i,j

i

ii

i xx )()1(

xi xi. i i

Replicating VariablesCorollary:

If {x1, x2, …, xn} in {0,1}

then det Rn 0 xi = xj i,j

Consequence: over F2, need only 2 matrices

NAND

NAND

NAND

A :=

Rn

Rn

Rn

B :=

What have we shown so far?

Simultaneous completion of: an unbounded number of matrices

over F2 is NP-hard

two matrices over F2 is NP-hard

Next: q matrices over Fq is NP-hard

Handling Fields Fq

Previous gadgets only work if each x {0,1}.How can we ensure this over Fq?

Introduce q-2 auxiliary variables: x=x(1), x(2), …, x(q-1)

Sufficient to enforce that:x(i) x(j) i,j and x(i) {0,1} i 2

det 01 1

x(i) x(j) etc.

Handling Fields Fq

x(i) x(j) i,j and x(i) {0,1} i 2

x(2)

0 1

x(1)

x(3)x(4)

x(q-1)

Edge indicates endpoints non-equal

Handling Fields Fq

x(i) x(j) i,j and x(i) {0,1} i 2

x(2)

0 1

x(1)

x(3)x(4)

x(q-1)

Pack these constraints into few matrices

Each variable used once per matrix

Amounts to edge-coloring From (Kn), conclude that

q matrices suffice

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

Main ResultsThm: A simultaneous completion for d

matrices 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

Open Questions Improved hardess results / algorithms

for matrix completion? Lower bounds / hardness for field size in

network coding? More combinatorial uses of matrix

completion