aditya bhaskara ( princeton ) moses charikar (princeton) venkatesan guruswami (cmu)
DESCRIPTION
Polynomial integrality gaps for strong SDP relaxtions of Densest k-Subgraph. Aditya Bhaskara ( Princeton ) Moses Charikar (Princeton) Venkatesan Guruswami (CMU) Aravindan Vijayaraghavan (Princeton) Yuan Zhou (CMU). The Densest k-Subgraph (DkS) problem. Problem description - PowerPoint PPT PresentationTRANSCRIPT
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Aditya Bhaskara (Princeton)Moses Charikar (Princeton)
Venkatesan Guruswami (CMU)Aravindan Vijayaraghavan (Princeton)
Yuan Zhou (CMU)
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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
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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]
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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]
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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
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Proof of hardness?
• Unfortunately, APX-hardness is not known for the Densest k-subgraph problem
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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
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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
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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
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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
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Lasserre SDP gap for DkS
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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
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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
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The new road map
Lasserre Gap for Max K-CSP SDP
Lasserre Gap for Dense (k1, k2)-
subgraph
Lasserre Gap for Dense k-subgraph
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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
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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
||
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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
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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)
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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)
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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
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• gap for -round Lasserre SDP ?
• gap for -round Sherali-Adams+ SDP ?
Furture directions
4/1n)(n
4/1n)(n
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Thank you!