Polynomial Integrality Gaps in Strong SDP Relaxations of Densest k-Subgraph

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

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. 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 • Take • C as some generalized BCH code • carefully chosen q and K Get gap for -round Lasserre SDP

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

22. Thank you!