Aditya Bhaskara (Princeton) Moses Charikar (Princeton) Venkatesan Guruswami (CMU) Aravindan Vijayaraghavan (Princeton) Yuan Zhou (CMU)
Feb 07, 2016
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!