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  • Количество слайдов: 22

Aditya Bhaskara (Princeton) Moses Charikar (Princeton) Venkatesan Guruswami (CMU) Aravindan Vijayaraghavan (Princeton) Yuan Zhou Aditya Bhaskara (Princeton) Moses Charikar (Princeton) Venkatesan Guruswami (CMU) Aravindan Vijayaraghavan (Princeton) Yuan Zhou (CMU)

The Densest k-Subgraph (Dk. S) problem graph G of size n H of size The Densest k-Subgraph (Dk. S) problem graph G of size n H of size k • Problem description Given G, find a subgraph H of size k of max. number of induced edges • No constant approximation algorithm known

Related problems • Max-density subgraph – no size restriction for the subgraph – find 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 cybercommunities [KRRT '99] – Algorithmic applications • Social networks. Trawling the web for emerging cybercommunities [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: ratio [BCCFV '10] approximation • Mostly used as Hardness applications • Best approximation algorithm: ratio [BCCFV '10] approximation • 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 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 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 Our results • Polynomial integrality gaps for strong SDP relaxation hierarchies • Theorem. gap for SA+ (Sherali-Adams+ SDP) hierarchy • Theorem. hierarchy gap for levels of Lasserre levels of

Implications of the SA+ SDP gap • Beating the best known approximation factor is 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 Dk. Implications of the Lasserre SDP gap • A strong (and first) evidence that Dk. S 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 [EH 05] [Tul 09] K-Coloring [KP 06] [Tul 09] Balanced Seperator, Uniform Sparest Cut 1 [GSZ'11] Dk. S 1 this work

Lasserre SDP gap for Dk. S Lasserre SDP gap for Dk. S

Outline • Gap reduction from [Tulsiani '09] (linear round Lasserre gap for Max K-CSP) Outline • Gap reduction from [Tulsiani '09] (linear round Lasserre gap for Max K-CSP) gap instance for Max K-CSP SDP gap instance for Dk. S SDP – Vector completeness: perfect solution for good solution for Max K-CSP SDP Dk. S SDP – Soundness: there is no good integer solution (w. h. p. )

The bipartite version of Dk. S • The Dense (k 1, k 2)-subgraph problem. The bipartite version of Dk. S • The Dense (k 1, k 2)-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 (k 1, k 2)-subgraph problem implies Lasserre gap of Dk. S • Only need to show Lasserre gap of Dense (k 1, k 2)subgraph problem

The new road map Lasserre Gap for Max K-CSP SDP Lasserre Gap for Dense The new road map Lasserre Gap for Max K-CSP SDP Lasserre Gap for Dense (k 1, k 2)-subgraph Lasserre Gap for Dense k-subgraph

The Max K-CSP instance • A linear code: • Alphabet: [q] = {0, 1, 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 [Tul 09] • Given C as a dual code Integrality gap for Max K-CSP [Tul 09] • Given C as a dual code of dist >= 3, for a random Max KCSP 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 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

Integrality gap • Vector Completeness. Max K-CSP instance is perfect satisfiable (in Lasserre) Dense Integrality gap • Vector Completeness. Max K-CSP instance is perfect satisfiable (in Lasserre) Dense (m, n)Subgraph (in Lasserre) – 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"

Integrality gap (cont'd) • Vector Completeness. Max K-CSP instance is perfect satisfiable (in Lasserre) Integrality gap (cont'd) • Vector Completeness. Max K-CSP instance is perfect satisfiable (in Lasserre) Dense (m, n)Subgraph (in Lasserre) • 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

Parameter selection • Take – C as the dual of Hamming code (i. e. 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

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

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