Subgraph with Perfect Completeness Aviad Rubinstein UC Berkeley Mark Braverman Young Kun Ko and Omri Weinstein A confession rest of workshop this talk vs vs SETH SAT requires ID: 1001160
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1. ETH-hardness of Densest-k-Subgraph with Perfect CompletenessAviad Rubinstein (UC Berkeley)Mark Braverman, Young Kun Ko, and Omri Weinstein
2. A confession…rest of workshopthis talk vs vs SETH: SAT requires ETH: SAT requires Reduction size: Reduction size: rest of workshopthis talkthis talk isn’t really about low-polynomial time(and this isn’t really Grandma)
3. Densest k-Subgraphwith perfect completeness
4. k-CLIQUEG|S|=kdoes G contain a k-clique?(NP-hard [Karp72])
5. relax “k”?G|S|≥kNP-hard [FGLSS96 … Zuckerman07]vsG|S| < k/n1-ε
6. relax “clique”?G|S|=kQuasi-poly algorithms [FS97, Barman15]vsG|S|=kden(S)=1den(S)<1-δ
7. relax both?G|S|≥kOur main result: ETH-hardvsGden(S) = 1|S| < k/nε/loglognden(S) < 1-δ
8. related works on “sparse vs very sparse”GvsGden(S)≥c(n)den(S)<s(n)0<s(n)<c(n)≪1[Feige02, AAMMW11] – random k-CNF[BCVGZ12] – SDP relaxations[RS10] – Unique Games with expansion
9. Main technique: Birthday Repetition
10. Recent applications of B-day RepQuasi-poly time hardness for…[AIM14]: AM with k provers (“something quantum”)Dense CSP’sFree games[BKW15]: ε-best ε-Nash[BPR16]: ε-Nash[R15] / [BCKS]: Signaling[this talk]: Densest k-Subgraph[ you! ]: ???
11. Recent applications of B-day RepQuasi-poly time hardness for…[AIM14]: AM with k provers (“something quantum”)Dense CSP’sFree games[BKW15]: ε-best ε-Nash[BPR16]: ε-Nash[R15] / [BCKS]: Signaling[this talk]: Densest k-Subgraph[ you! ]: graph isomorphism??
12. Recent applications of B-day RepQuasi-poly time hardness for…[AIM14]: AM with k provers (“something quantum”)Dense CSP’sFree games[BKW15]: ε-best ε-Nash[BPR16]: ε-Nash[R15] / [BCKS]: Signaling[this talk]: Densest k-SubgraphTight by [FS97], [LMM03], [Barman15], [MM15], [CCDEHT15], etc.
13. Reduction in 1 slidevariablesxiconstraints Reduction from 2CSP (e.g. 3COL)
14. Birthday Paradox: every pair (u,v) should test a constraintAnalysis (soundness) in 1 slideIf every assignment to 2CSP violates -fraction of the constrains,the corresponding k-subgraph should be missing -fraction of edges.QED? what about k-subgraphs that don’t correspond to any assignment?
15. Not so simple…“Typically” birthday repetition is easy Queries to Alice and Bob are independent.More subtle for Densest k-Subgraph…Very simple problem – hard to enforce structurewe only know that the subgraph is large + dense“like proving a parallel repetition theorem,when Alice and Bob choose which queries to answer”If it were easy, we would get hardness for den(S)=1 vs den(S)=δ
16. So what can we do?“Typically” birthday repetition is easy Queries to Alice and Bob are independent.More subtle for Densest k-Subgraph…Very simple problem – hard to enforce structurewe only know that the subgraph is large + dense“like proving a parallel repetition theorem,when Alice and Bob choose which queries to answer”If it were easy, we would get hardness for den(S)=1 vs den(S)=δ
17. Analysis in 1 slide: “counting entropy”G|S|=k choice of variableschoice of assignments many CSP violations!many consistencyviolations!low density,QED!
18. Open problems“Warmup”: same result for Densest k-Bi-SubgraphDen(S)=1 vs den(S)=δStronger (NP?) hardness for “sparse vs very sparse”Fixed parameter (k) tractability?