Streaming Complexity of Approximating Max 2CSP ...

Streaming Complexity of Approximating Max 2CSP and Max Acyclic Subgraph Venkatesan Guruswami1 Ameya Velingker2 Santhoshini Velusamy3 1 CMU, Pittsburgh 2 EPFL, Switzerland

3 IIT

Madras, Chennai

August 16, 2017

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Streaming model

Data comes in a stream (single pass). Space limit:

poly (log n).

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Streaming model

Data comes in a stream (single pass). Space limit:

poly (log n).

γ -approximation algorithm (maximisation) in this model, γ ∈ [0, 1] Randomised algorithm that outputs a value in Success probability

[γ × opt, opt].

> 0.9.

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Streaming model

Data comes in a stream (single pass). Space limit:

poly (log n).

γ -approximation algorithm (maximisation) in this model, γ ∈ [0, 1] Randomised algorithm that outputs a value in Success probability

[γ × opt, opt].

> 0.9.

Example S

≡

(1, d ) index

(2, a)

(1, c )

(3, b )

(3, f )

data

L1 (S )

...

time

= |d + c | + |a| + |b + f |.

No naive algorithm to compute

L1 −norm

in this model! 2 / 23

Our results

Result 1
No 12 +  -approximation algorithm for Max 2CSP for any  > 0.

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Our results

Result 1
No 12 +  -approximation algorithm for Max 2CSP for any  > 0. Result 2 ( 25 − δ)-approximation algorithm for Max 2CSP for any δ > 0.

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Our results

Result 1
No 12 +  -approximation algorithm for Max 2CSP for any  > 0. Result 2 ( 25 − δ)-approximation algorithm for Max 2CSP for any δ > 0. Result 3
No 78 +  -approximation algorithm for Max Acyclic Subgraph for any  > 0.

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Max CUT

A cut of size 4 G

1

4

2

5

3

6

Problem: Output the size of the largest cut in

G.

NP Hard. Notation:

m

is number of edges in

G, n

is number of vertices in

G.

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Max CUT

Streaming: edges come in a stream e1 (u1 , v1 )

e2 (u2 , v2 )

e3 (u3 , v3 )

e4 (u4 , v4 )

...

time

Trivial 21 -approximation algorithm: Output (m/2). Any graph has a cut of size at least m/2.

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Max CUT

Streaming: edges come in a stream e1 (u1 , v1 )

e2 (u2 , v2 )

e3 (u3 , v3 )

e4 (u4 , v4 )

...

time

Trivial 21 -approximation algorithm: Output (m/2). Any graph has a cut of size at least m/2. No ( 12 + )-approximation algorithm for Max CUT! [Kapralov et al., 2015]

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Max DICUT

Dicut: a directed cut of size 3 G

0

1

4

2

5

3

6

Problem: Output the size of the largest directed cut in

G

0

.

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( 12 + )-approximation

is hard

Reduction from Max CUT to Max DICUT 1

4

1

4

2

5

2

5

3

6

3

6

H

Max CUT of

H

H

0

= Max DICUT of

H

0

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( 12 + )-approximation

is hard

Reduction from Max CUT to Max DICUT 1

4

1

4

2

5

2

5

3

6

3

6

H

Max CUT of

H

H

0

= Max DICUT of

H

0

Trivial 14 -approximation algorithm! Can we beat the trivial algorithm? 7 / 23

Bias

Denition: Bias(G ) ,

P v ∈V (G ) |outv − inv |.

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Bias

Denition: Bias(G ) ,

P v ∈V (G ) |outv − inv |.

Range: 0 ≤ Bias(G ) ≤ 2m. High Bias: If Bias(G ) = 2m, then either outv = 0 or inv = 0.

A

perfect dicut!

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Bias

Denition: Bias(G ) ,

P v ∈V (G ) |outv − inv |.

Range: 0 ≤ Bias(G ) ≤ 2m. High Bias: If Bias(G ) = 2m, then either outv = 0 or inv = 0.

A

perfect dicut!

Theorem Bias(G ) ≥ (1 − )2m,  ∈ [0, 1].

1

If

2

If

Bias(G ) ≤ (1 − )2m,  ∈ [0, 1].

then Max DICUT of G

≥ (1 − )m,

then Max DICUT of G

≤ (1 − /2)m,

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Bias - Theorem

1

If Bias(G ) ≥ (1 − )2m, then Max DICUT of  ∈ [0, 1].

G

≥ (1 − )m,

Greedy Dicut

v

u

Left set (outv ≥ inv )

Right set (outu < inu )

Number of edges not in the dicut ≤ 2

P v ∈V (G ) min{outv , inv }.

If Bias(G ) ≤ (1 − )2m, then Max DICUT of  ∈ [0, 1]. For every dicut, at least in the dicut.

(

G

≤ (1 − /2)m,

P v ∈V (G ) min{outv , inv })/2

edges are not

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Computing Bias

S

≡

(1, d ) index

(2, a)

(1, c )

(3, b )

data

(3, f )

...

time

When index ∈ [n] and data ∈ {−M , −(M − 1), · · · , M }, where M is a constant, then, for any α > 0, there exists a streaming algorithm with output ∈ [(1 − α)L1 (S ), (1 + α)L1 (S )], 2 space: O(log n/α ), success probability

> 0.9.

[Indyk, 2006]

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Computing Bias

S

≡

(1, d ) index

(2, a)

(1, c )

(3, b )

data

(3, f )

...

time

When index ∈ [n] and data ∈ {−M , −(M − 1), · · · , M }, where M is a constant, then, for any α > 0, there exists a streaming algorithm with output ∈ [(1 − α)L1 (S ), (1 + α)L1 (S )], 2 space: O(log n/α ), success probability

> 0.9.

[Indyk, 2006]

For computing Bias: e (u , v ) → (u , 1) (v , −1).

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2 5 -approximation algorithm Assume that there is no error in computing Bias. If Bias(G ) = (1 − )2m, then (1 − )m ≤ Max DICUT ≤ (1 − /2)m. Max DICUT

m

Bias/2

m /4 0

2m

m/2

Bias

Algorithm: If Bias ≥ m/2, output Bias/2, else output m/4 as the Max DICUT. 2 When Bias ≥ m/2, approximation ratio ≥ (1 − )/(1 − /2) ≥ 5 for  ∈ [0, 34 ].

When Bias ≥ 25 .

< m/2,

Max DICUT

≤ 5m/8

and hence approximation

ratio

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Max 2CSP

Denition Collection of literals: x1 , x2 , . . . , xn ∈ {0, 1} and 2-ary Boolean predicates P1 , P2 , . . . , Pm dened on the literals. Table 2.1 :

Example of a 2-ary Boolean predicate

P (x , y )

x

y

P (x , y )

0 0 1 1

0 1 0 1

0 1 1 0

P (x , y )

= (¬x ∧ y ) ∨ (x ∧ ¬y )

Problem: What is the maximum number of predicates that can be satised?

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Max 2AND

An instance of Max 2CSP where every predicate is an AND clause.

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Max 2AND

An instance of Max 2CSP where every predicate is an AND clause. Convert Max 2CSP to Max 2AND instance with the same optimal value!

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Max 2AND

An instance of Max 2CSP where every predicate is an AND clause. Convert Max 2CSP to Max 2AND instance with the same optimal value! Max DICUT is an instance of Max 2AND!

u

v

0

1

≡ (¬xu ∧ xv = 1)

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Max 2AND

An instance of Max 2CSP where every predicate is an AND clause. Convert Max 2CSP to Max 2AND instance with the same optimal value! Max DICUT is an instance of Max 2AND!

u

v

0

1

≡ (¬xu ∧ xv = 1)

Pn i =1 |negi − posi |. The Bias theorems and the 52 -approximation algorithm for Max DICUT hold for Max 2AND as well! Bias of a Max 2AND instance =

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Maximum Acyclic Subgraph (MAS)

Problem: What is the size of the largest acyclic subgraph of a given directed graph? 2

3

1

4 5

Streaming model: Edges of the graph come in a stream. Trivial 21 -approximation algorithm: Output m/2. Either pick all the forward (u < v ) edges or all the backward (u > v ) edges, we get an acyclic subgraph of size ≥ m/2.

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Boolean Hidden Matching (BHM)

Denition Alice:

x

∈ {0, 1}n .

Bob: M ∈ {0, 1} 2 ×n , an undirected perfect matching (edge incidence matrix) on n vertices, and w ∈ {0, 1}n/2 .   .. .   xi1 ⊕ xi2 ⊕ wi . If ei = (i1 , i2 ), then Mx ⊕ w =    .. . n

A promise problem: Output if Mx ⊕ w = 1n/2 .

YES

if

Mx

⊕ w = 0n/2 and output

NO

Any randomised one-way communication (Alice to Bob) protocol √ that solves BHM with success probability > 0.9 has complexity Ω( n). [Gavinsky et al., 2007] 15 / 23

( 78 + )-approximation

is hard

Alice's edges

i

b

i

a

i

d

i

c

a

c

x

i =0

i

b

i

i

d

i

x

i =1

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( 78 + )-approximation Bob's edges when

w

is hard

i =0

a1

i

b1

i

a2

i

b2

i

a1

i

b1

i

a2

i

b2

i

d1

i

c2

i

d2

i

c1

i

d1

i

c2

i

d2

c1

i = 0,

x1

i =0

i = 0,

x2

x1

i i

i =1

x2

a1

i

b1

i

a2

i

b2

i

a1

i

b1

i

a2

i

b2

i

d1

i

c2

i

d2

i

c1

i

d1

i

c2

i

d2

c1

i = 1,

x1

i =0

x2

i = 1,

x1

i i

i =1

x2

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( 78 + )-approximation Bob's edges when

w

is hard

i =1

a1

i

b1

i

i

d1

i

a2

i

c1

i = 0,

x1

i

i

b2

i

i

i = 0,

x1

i

i

i

b2

i d i2 i =0

x2

i d i2 = 1 i

c2

x2

b1

i

a2

i

i

c1 c2

i

b2

i

a1

d1

i = 1,

i

a2

i d i2 = 0 i

i

x1

i

d1

x2

a2

i

b1

c2

b1

c1

i

c1

i

a1

a1

i

b2

d1

i = 1,

x1

i d i2 i =1

c2

x2

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The gap

If

Mx

If

Mx

⊕ w = 0n/2 , then MAS value is ⊕ w = 1n/2 , then MAS value is

7 8

m.

m.

A better-than- 8 -approximation algorithm can distinguish the two instances. 7

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Summary

No

1 2


+  -approximation algorithm for Max 2CSP for any  > 0.

No streaming algorithm can distinguish graphs with Max DICUT

m/2 from those with Max DICUT  ∈ [0, 1/4]. [Kapralov et al., 2015]

value

value at most (1/4 + )m,

By computing Bias, we can distinguish between graphs with Max

DICUT value more than (1/2 + 8)m from those with Max DICUT value at most (1/4 + )m,

 ∈ (0, 1/16).

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Summary

No

1 2


+  -approximation algorithm for Max 2CSP for any  > 0.

No streaming algorithm can distinguish graphs with Max DICUT

m/2 from those with Max DICUT  ∈ [0, 1/4]. [Kapralov et al., 2015]

value

value at most (1/4 + )m,

By computing Bias, we can distinguish between graphs with Max

DICUT value more than (1/2 + 8)m from those with Max DICUT value at most (1/4 + )m, 2 5

 ∈ (0, 1/16).


− δ -approximation algorithm for Max 2CSP for any δ > 0.

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Summary

No

1 2


+  -approximation algorithm for Max 2CSP for any  > 0.

No streaming algorithm can distinguish graphs with Max DICUT

m/2 from those with Max DICUT  ∈ [0, 1/4]. [Kapralov et al., 2015]

value

value at most (1/4 + )m,

By computing Bias, we can distinguish between graphs with Max

DICUT value more than (1/2 + 8)m from those with Max DICUT value at most (1/4 + )m,

 ∈ (0, 1/16).


− δ -approximation algorithm for Max 2CSP for any δ > 0.
No 87 +  -approximation algorithm for Max Acyclic Subgraph for any  > 0. 2 5

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Open questions

Can we get a better than 52 -approximation algorithm for Max 2CSP?

21 / 23

Open questions

Can we get a better than 52 -approximation algorithm for Max 2CSP? Can we improve the ( 12 + ) hardness of approximation for Max 2CSP?

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Open questions

Can we get a better than 52 -approximation algorithm for Max 2CSP? Can we improve the ( 12 + ) hardness of approximation for Max 2CSP? Can we show that there is no ( 12 + )-approximation algorithm for Max Acyclic Subgraph?

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References Dmitry Gavinsky, Julia Kempe, Iordanis Kerenidis, Ran Raz, and Ronald de Wolf. Exponential separations for one-way quantum communication complexity, with applications to cryptography. In Proceedings of the Thirty-ninth Annual ACM Symposium on Theory of Computing, STOC '07, pages 516525, New York, NY, USA, 2007. ACM. ISBN 978-1-59593-631-8. doi: 10.1145/1250790.1250866. URL http://doi.acm.org/10.1145/1250790.1250866. Piotr Indyk. Stable distributions, pseudorandom generators, embeddings, and data stream computation. J. ACM, 53(3):307323, May 2006. ISSN 0004-5411. doi: 10.1145/1147954.1147955. URL http://doi.acm.org/10.1145/1147954.1147955. Michael Kapralov, Sanjeev Khanna, and Madhu Sudan. Streaming lower bounds for approximating max-cut. In Proceedings of the Twenty-sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '15, pages 12631282, Philadelphia, PA, USA, 2015. Society for Industrial and Applied Mathematics. URL http://dl.acm.org/citation.cfm?id=2722129.2722213. 22 / 23

Thank you! Any questions?

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