Aditya Bhaskara (Princeton)
Moses Charikar (Princeton)
Venkatesan Guruswami
(CMU)
Aravindan Vijayaraghavan (Princeton)
Yuan Zhou
(CMU)
The Densest k-Subgraph (DkS) problem
graph G of size n
H of size k
• Problem description
Given G,
find a subgraph H of
size k of max. number of
induced edges
• No constant
approximation algorithm
known
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 cybercommunities [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: O(n1/ 4 ) 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)
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 3SAT  BPTIME ( sube xp)
• [RS '10] No constant factor approximation assuming the
Small Set Expansion Conjecture
• [FS '97] Natural SDP has an (n1/ 3 ) integrality gap
– Doesn't serve as a "strong" evidence since stronger
SDP indeed improves the integrality gap [BCCFV '10]
Our results
• Polynomial integrality gaps for strong SDP relaxation
hierarchies
~
1/ 4

(
n
) gap for (log n / log log n) levels of
• Theorem.
SA+ (Sherali-Adams+ SDP) hierarchy
• Theorem. n
hierarchy
 ( )
gap for
n1 levels of Lasserre
Implications of the SA+ SDP gap
• Beating the best known n1/ 4 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
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
Max K-CSP
K-Coloring
2k / 2
n/2
Lasserre Gap
[EH05] 2 k / 2k
2k
(logn )3 / 4 [KP06]
[Tul09]
n
2
c log n log log n
[Tul09]
Balanced Seperator,
Uniform Sparest Cut
1
1 
[GSZ'11]
DkS
1
n
this work
Lasserre SDP gap for DkS
Outline
• Gap reduction from [Tulsiani '09] (linear round Lasserre
gap for Max K-CSP)
gap instance for Max
K-CSP SDP
gap instance for
DkS SDP
– Vector completeness:
perfect solution for
good solution for
Max K-CSP SDP
DkS SDP
– Soundness:
there is no good integer solution (w.h.p.)
The bipartite version of DkS
• The Dense (k1, k2)-subgraph problem.
– Given bipartite graph G = (V, W, E)
– Find two subsets A  V , B  W , such that
1) | A | k1 , | B | k2
2) | E  A  B | (# 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
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:
CF
K
q
• Alphabet: [q] = {0, 1, 2, ..., q-1}
• Variables: x1 , x2 ,, xn
• Constraints: C1 , C2 , Cm
– C i is over xi , xi , , xi , insisting
1
2
K
( xi1  b1(i ) , xi2  b2(i ) ,, xiK  bK(i ) )  C
– where
b (i )  FqK
• A random Max K-CSP instance:
(i )
– Choose i1 , i2 ,, iK and b
completely by random
Integrality gap for Max K-CSP [Tul09]
• Given C as a dual code of dist >= 3, for a random Max KCSP instance
• Vector completeness. For constant K, there exists
perfect solution for linear round Lasserre SDP w.h.p.
|C |
• Soundness. W.h.p. no solution satisfies more than
K
2
(fraction) clauses.
The gap reduction to Densest (m, n)-subgraph
• The constraint variable graph of Max K-CSP
– left vertices: constraint and satisfying assignment
pair
{( Ci ,  ) |  : {xi1 , xi2 , , xiK }  [q],  satisfies Ci }
– right vertices: all assignments for singletons
{ : {xi }  [q]}
– edges:
(Ci ,  ) is connected to a
right vertex  when 
is an sub-assignment of

C1

C2
x1  0
x1  1
x2  0
x2  1
x3  0
x3  1
Integrality gap
• Vector Completeness.
Max K-CSP instance is
perfect satisfiable
(in Lasserre)
Dense (m, n)Subgraph
(in Lasserre)
– 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"
Integrality gap (cont'd)
• Vector Completeness.
Max K-CSP instance is
perfect satisfiable (in
Lasserre)
Dense (m, n)Subgraph
(in Lasserre)
• 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
Parameter selection
• Take
– C as the dual of Hamming code (i.e. the Hadamard
code)

– q  n , K  n 2
Get
1 O (  )

-round Lasserre SDP
n gap for n
• Take
– C as some generalized BCH code
– carefully chosen q and K
Get
 ( )
-round Lasserre SDP
n 2 / 53 gap for n
Furture directions
• n1/ 4 gap for n  ( ) -round Lasserre SDP ?
• n1/ 4 gap for n  ( ) -round Sherali-Adams+ SDP ?
Thank you!