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 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)

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]

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

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

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

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

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

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.

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

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)

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

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

Thank you!