1. 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)

2. 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

3. 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]

4. 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]

5. 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)

6. Proof of hardness?
Unfortunately, APX-hardness is not known for the Densest k-subgraph problem

7. 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]

8. 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

9. 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

10. 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]
Balanced Seperator, Uniform Sparest Cut 1
[GSZ'11]
DkS
this work

11. Lasserre SDP gap for DkS

12. 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.)
gap instance for Max K-CSP SDP
gap instance for DkS SDP
perfect solution for Max K-CSP SDP
good solution for DkS SDP

13. 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

14. The new road map Lasserre Gap for Max K-CSP SDP
Lasserre Gap for Dense (k1, k2)-subgraph
Lasserre Gap for Dense k-subgraph

15. 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

16. 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.

17. 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

18. Max K-CSP instance is perfect satisfiable
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)

19. 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)

20. Parameter selection Take
C as the dual of Hamming code (i.e. the Hadamard code) ,
Get
gap for round Lasserre SDP
C as some generalized BCH code
carefully chosen q and K
gap for round Lasserre SDP

21. Furture directions gap for -round Lasserre SDP ?
gap for round Sherali-Adams+ SDP ?

22. Thank you!