ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark...
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Transcript of ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark...
![Page 1: ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark Braverman, Young Kun Ko, and Omri Weinstein.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1af97f8b9ab059983275/html5/thumbnails/1.jpg)
ETH-hardness of Densest-k-Subgraph
with Perfect Completeness
Aviad Rubinstein (UC Berkeley)
Mark Braverman, Young Kun Ko, and Omri Weinstein
![Page 2: ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark Braverman, Young Kun Ko, and Omri Weinstein.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1af97f8b9ab059983275/html5/thumbnails/2.jpg)
A confession…
rest of workshop this talk
vs vs
SETH: SAT requires ETH: SAT requires
Reduction size: Reduction size:
this talk isn’t really about low-polynomial time
(and this isn’t really Grandma)
![Page 3: ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark Braverman, Young Kun Ko, and Omri Weinstein.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1af97f8b9ab059983275/html5/thumbnails/3.jpg)
Densest k-Subgraphwith perfect completeness
![Page 4: ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark Braverman, Young Kun Ko, and Omri Weinstein.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1af97f8b9ab059983275/html5/thumbnails/4.jpg)
k-CLIQUE
G
|S|=k
does G contain a k-clique?
(NP-hard [Karp72])
![Page 5: ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark Braverman, Young Kun Ko, and Omri Weinstein.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1af97f8b9ab059983275/html5/thumbnails/5.jpg)
relax “k”?
G|S|≥k
NP-hard [FGLSS96 … Zuckerman07]
vsG
|S| < k/n1-ε
![Page 6: ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark Braverman, Young Kun Ko, and Omri Weinstein.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1af97f8b9ab059983275/html5/thumbnails/6.jpg)
relax “clique”?
G|S|=k
Quasi-poly algorithms [FS97, Barman15]
vsG
|S|=k
den(S)=1
den(S)<1-δ
![Page 7: ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark Braverman, Young Kun Ko, and Omri Weinstein.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1af97f8b9ab059983275/html5/thumbnails/7.jpg)
relax both?
G|S|≥k
Our main result: ETH-hard
vsG
den(S) = 1
|S| < k/nε/loglogn
den(S) < 1-δ
![Page 8: ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark Braverman, Young Kun Ko, and Omri Weinstein.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1af97f8b9ab059983275/html5/thumbnails/8.jpg)
related works on “sparse vs very sparse”
Gvs
G
den(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
![Page 9: ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark Braverman, Young Kun Ko, and Omri Weinstein.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1af97f8b9ab059983275/html5/thumbnails/9.jpg)
Main technique: Birthday Repetition
![Page 10: ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark Braverman, Young Kun Ko, and Omri Weinstein.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1af97f8b9ab059983275/html5/thumbnails/10.jpg)
Recent applications of B-day RepQuasi-poly time hardness for…
• [AIM14]: AM with k provers (“something quantum”)• Dense CSP’s• Free games
• [BKW15]: ε-best ε-Nash• [BPR16]: ε-Nash• [R15] / [BCKS]: Signaling
• [this talk]: Densest k-Subgraph
• [ you! ]: ???
![Page 11: ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark Braverman, Young Kun Ko, and Omri Weinstein.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1af97f8b9ab059983275/html5/thumbnails/11.jpg)
Recent applications of B-day RepQuasi-poly time hardness for…
• [AIM14]: AM with k provers (“something quantum”)• Dense CSP’s• Free games
• [BKW15]: ε-best ε-Nash• [BPR16]: ε-Nash• [R15] / [BCKS]: Signaling
• [this talk]: Densest k-Subgraph
• [ you! ]: graph isomorphism??
![Page 12: ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark Braverman, Young Kun Ko, and Omri Weinstein.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1af97f8b9ab059983275/html5/thumbnails/12.jpg)
Recent applications of B-day RepQuasi-poly time hardness for…
• [AIM14]: AM with k provers (“something quantum”)• Dense CSP’s• Free games
• [BKW15]: ε-best ε-Nash• [BPR16]: ε-Nash• [R15] / [BCKS]: Signaling
• [this talk]: Densest k-Subgraph
Tight by [FS97], [LMM03], [Barman15], [MM15],
[CCDEHT15], etc.
![Page 13: ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark Braverman, Young Kun Ko, and Omri Weinstein.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1af97f8b9ab059983275/html5/thumbnails/13.jpg)
Reduction in 1 slide
variables
xi
constraints
Reduction from 2CSP (e.g. 3COL)
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Birthday Paradox: every pair (u,v) should test a constraint
Analysis (soundness) in 1 slide
If 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?
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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 structure
• we 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
![Page 16: ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark Braverman, Young Kun Ko, and Omri Weinstein.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1af97f8b9ab059983275/html5/thumbnails/16.jpg)
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 structure
• we 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
![Page 17: ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark Braverman, Young Kun Ko, and Omri Weinstein.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1af97f8b9ab059983275/html5/thumbnails/17.jpg)
Analysis in 1 slide: “counting entropy”
G|S|=k
choice of
variables
choice of assignment
s
many CSP
violations!many
consistencyviolations!
low density,QED!
![Page 18: ETH-hardness of Densest-k-Subgraph with Perfect Completeness Aviad Rubinstein (UC Berkeley) Mark Braverman, Young Kun Ko, and Omri Weinstein.](https://reader035.fdocuments.us/reader035/viewer/2022062306/5a4d1af97f8b9ab059983275/html5/thumbnails/18.jpg)
Open problems
• “Warmup”: same result for Densest k-Bi-Subgraph
• Den(S)=1 vs den(S)=δ
• Stronger (NP?) hardness for “sparse vs very sparse”
• Fixed parameter (k) tractability?