Aditya Bhaskara (Princeton)Moses Charikar (Princeton)

Venkatesan Guruswami (CMU)Aravindan Vijayaraghavan (Princeton)

Yuan Zhou (CMU)

The Densest k-Subgraph (DkS) problem

• Problem description Given G, find a subgraph H of

size k of max. number of induced edges

• No constant approximation algorithm known

graph G of size n

H of size k

Related problems• Max-density subgraph

– no size restriction for the subgraph– find a subgraph of max. edge density (i.e. average

degree)– solvable in poly-time [GGT'87]

Algorithmic applications• Social networks. Trawling the web for emerging

cyber-communities [KRRT '99]– Web communities are characterized by dense

bipartite subgraphs

• Computational biology. Mining dense subgraphs across massive biological networks for functional discovery [HYHHZ '05]– Dense protein interaction subgraph corresponds

to a protein complex [BD '03]

Hardness applications• Best approximation algorithm:

approximation ratio [BCCFV '10]

• Mostly used as an (average case) hardness assumption

– [ABW '10] Variant was used as the hardness assumption in Public Key Cryptography

– [ABBG '10] Toxic assets can be hidden in complex financial derivatives to commit undetectable fraud

– [CMVZ '12] Derive inapproximability for many other problems (e.g. k-route cut)

)( 4/1 nO

Proof of hardness?

• Unfortunately, APX-hardness is not known for the Densest k-subgraph problem

Evidence of hardness?• [Feige '02] No PTAS under the Random 3-SAT

hypothesis

• [Khot '04] No PTAS unless

• [RS '10] No constant factor approximation assuming the Small Set Expansion Conjecture

• [FS '97] Natural SDP has an integrality gap– Doesn't serve as a "strong" evidence since

stronger SDP indeed improves the integrality gap [BCCFV '10]

)(3 xpsubeBPTIMESAT

)( 3/1n

Our results• Polynomial integrality gaps for strong SDP relaxation

hierarchies

• Theorem. gap for levels of SA+ (Sherali-Adams+ SDP) hierarchy

• Theorem. gap for levels of Lasserre hierarchy

)( 4/1~

n )loglog/(log nn

)(n 1n

Implications of the SA+ SDP gap

• Beating the best known approximation factor is a barrier for current techniques– Since the algorithm of [BCCFV '10] only uses

constant rounds of Sherali-Adams LP relaxation

• Natural distributions of instances are gap instances w.h.p.– We use Erdös-Renyi random graphs as gap

instances

4/1n

Implications of the Lasserre SDP gap• A strong (and first) evidence that DkS is hard to

approximate within polynomial factors– Reason: Very few problems have Lasserre gaps

stronger than known NP-Hardness results

NP-Hardness Lasserre Gap

Max K-CSP [EH05] [Tul09]

K-Coloring [KP06] [Tul09]

Balanced Seperator, Uniform Sparest Cut 1 [GSZ'11]

DkS 1 this work

kk 22/2 kk 2/24/3)(log2/ nn nnc

nlogloglog2

1

n

Lasserre SDP gap for DkS

Outline• Gap reduction from [Tulsiani '09] (linear round

Lasserre gap for Max K-CSP)

– Vector completeness:

– Soundness: there is no good integer solution (w.h.p.)

perfect solution for Max K-CSP SDP

good solution for DkS SDP

gap instance for Max K-CSP SDP

gap instance for DkS SDP

The bipartite version of DkS• The Dense (k1, k2)-subgraph problem.

– Given bipartite graph G = (V, W, E)– Find two subsets , such that

1) 2) (# of induced edges) is

maximized

• Lemma. Lasserre gap of Dense (k1, k2)-subgraph problem implies Lasserre gap of DkS

• Only need to show Lasserre gap of Dense (k1, k2)-subgraph problem

WBVA ,21 ||,|| kBkA

|| BAE

The new road map

Lasserre Gap for Max K-CSP SDP

Lasserre Gap for Dense (k1, k2)-

subgraph

Lasserre Gap for Dense k-subgraph

The Max K-CSP instance• A linear code:

• Alphabet: [q] = {0, 1, 2, ..., q-1}• Variables: • Constraints:

– is over , insisting

– where

• A random Max K-CSP instance:– Choose and completely by

random

KqFC

mCCC ,, 21

iC Kiii xxx ,,,21

Cbxbxbx iKi

ii

ii K

),,,( )()(2

)(1 21

Kq

i Fb )(

Kiii ,,, 21 )(ib

nxxx ,,, 21

Integrality gap for Max K-CSP [Tul09]• Given C as a dual code of dist >= 3, for a random

Max K-CSP instance

• Vector completeness. For constant K, there exists perfect solution for linear round Lasserre SDP w.h.p.

• Soundness. W.h.p. no solution satisfies more than (fraction) clauses.

K

C

2

||

The gap reduction to Densest (m, n)-subgraph• The constraint variable graph of Max K-CSP

– left vertices: constraint and satisfying assignment pair

– right vertices: all assignments for singletons

– edges: is connected to a right vertex when is an sub-assignment of

} satisfies],[},,,{:|),{(21 iiiii CqxxxC

K

]}[}{:{ qxi 1C

2C

01 x11 x02 x12 x03 x13 x

),( iC

Integrality gap • Vector Completeness.

– Intuition: translate the following argument (for integer solution) into Lasserre language

– Given an satisfying solution for Max K-CSP instance, we can choose m left vertices (one per constraint) and n right vertices (one per variable) agree with the solution, such that the subgraph is "dense"

Max K-CSP instance is perfect satisfiable

(in Lasserre)

Dense (m, n)-Subgraph

(in Lasserre)

Integrality gap (cont'd)• Vector Completeness.

• Soundness. W.h.p. there is no dense (m, n)-subgraph– Intuition: random bipartite graph does not have

dense (m, n)-subgraph w.h.p.

– Argue that our graph has enough randomness to rule out dense (m, n)-subgraph

Max K-CSP instance is perfect satisfiable

(in Lasserre)

Dense (m, n)-Subgraph

(in Lasserre)

Parameter selection• Take

– C as the dual of Hamming code (i.e. the Hadamard code)

– , Get gap for -round Lasserre SDP

• Take – C as some generalized BCH code– carefully chosen q and K

Get gap for -round Lasserre SDP

nq 2nK

n )(1 On

53/2n )(n

• gap for -round Lasserre SDP ?

• gap for -round Sherali-Adams+ SDP ?

Furture directions

4/1n)(n

4/1n)(n

Thank you!