SODA21 (joint work with Shuichi Hirahara)

1. Nearly Optimal Average-Case
Complexity of Counting Bicliques
Under SETH
Shuichi Hirahara and Nobutaka Shimizu
National Institute of Informatics The University of Tokyo

2. /27
Average-Case Complexity
• Distributinoal Problem (Π,
𝒟
)
2
• is a problem. = answer for input
Π Π(x) x
• is a sequence of distributions.
𝒟
= (
𝒟
n)n∈ℕ
A (deterministic) algorithm solves with success
probability if
A (Π,
𝒟
)
δ
[Levin, 1986]
∀n ∈ ℕ, Pr
x∼
𝒟
n
[A(x) = Π(x)] ≥ δ
• Hardness for a random input

3. /27
3
- Hamilton cycle : in time ( with
- chromatic number : 2-approx. in time [Grimmett, McDiarmid, 1975]
- max clique : 2-approx. in time [Karp, 1976]
- max matching: in time [Motwani, 1994]
O(n(log n)2
) G(n, p) p ≈ C log n/n
O(n3
)
O(n2
)
O(m log n)
Easy Problems on Random Graph
[Angluin and Valiant, 1979]

4. /27
Hard Problems on Random Graphs
• Maximum clique admits 2-approx. poly-time algorithm
→ Conjecture that -approx. is hard in poly-time [Karp, 1976]
(2 − ϵ)
4
・Subgraph counting problems [Dalirrooyfard, Lincoln, Williams, 2020]
• -Clique counting [Goldreich, Rothblum, 2018] [Boix-Adserà, Brennan, Bresler, 2019]
k
• Planted clique

5. /27
Hard Problems on Random Graphs
• Maximum clique admits 2-approx. poly-time algorithm
→ Conjecture that -approx. is hard in poly-time [Karp, 1976]
(2 − ϵ)
5
・Subgraph counting problems [Dalirrooyfard, Lincoln, Williams, 2020]
• -Clique counting [Goldreich, Rothblum, 2018] [Boix-Adserà, Brennan, Bresler, 2019]
k
• Planted clique
based on worst-case hardness

6. /27
Worst-Case to Average-Case
6
・ such that is reducible to [Goldreich, Rothblum, 2018]
・ is reducible to with overhead [Boix-Adserà, Brennan, Bresler, 2019]
∃
𝒟
#Ka (#Ka,
𝒟
)
#Ka (#Ka, G(n, p)) O(p−1
)
Worst-case to average-case reduction for graph problems:
Theorem
Under (randomized) ETH, any -time algorithm for has success
prob. for any constant .
no(a)
(#Ka, G(n, p))
≤ 1 − 1/polylog(n) p
[Boix-Adserà, Brennan, Bresler, 2019]
・For general subgraph , is reducible to
H #H (#H, G(n, p))
[Dalirrooyfard, Lincoln, Williams, 2020]

7. /27
Issue: Success Probability
7
Theorem
Under (randomized) ETH, any -time algorithm for has success
prob. for any constant .
no(a)
(#Ka, G(n, p))
≤ 1 − 1/polylog(n) p
[Boix-Adserà, Brennan, Bresler, 2019]

8. /27
Issue: Success Probability
• The result above looks at small fraction of random graphs.
8
1/polylog(n)
We want hardness of “almost all” random graphs
• Overcome this issue by hardness ampli cation.
Theorem
Under (randomized) ETH, any -time algorithm for has success
prob. for any constant .
no(a)
(#Ka, G(n, p))
≤ 1 − 1/polylog(n) p
[Boix-Adserà, Brennan, Bresler, 2019]

9. /27
Hardness Ampli cation
• Based on a mildly hard problem , construct a strongly hard
problem
(Π,
𝒟
)
(Π′

,
𝒟
′

)
9
(Π,
𝒟
) (Π′

,
𝒟
′

)
[Yao, 1982]
• Application in cryptography and derandomization
• Well-studied in coarse-grained complexity

10. /27
Direct Product
• Direct Product : basic technique of hardness ampli cation
10
For and , let a
s
・
,
・ is a sequence of distributions, where each is the distribution
of i.i.d. samples from .
(Π,
𝒟
) k ∈ ℕ (Π,
𝒟
)k
:= (Πk
,
𝒟
k
)
Πk
(x1, …, xk) = (Π(x1), …, Π(xk))
𝒟
k
= (
𝒟
k
n)n∈ℕ
𝒟
k
n
k
𝒟
n
[Yao, 1982]
→ If is hard, then is hard. Indeed, the converse holds.
(Π,
𝒟
)k
(Π,
𝒟
)
(Direct Product Theorem)
[Yao, 1982]
• If can be solved with success prob. , then can be solved
with success prob. with a overhead of factor .
(Π,
𝒟
) δ (Πk
,
𝒟
k
)
δk
k
De
fi
nition (Direct Product of )
(Π,
𝒟
)

11. Our Result

12. Input: graph
.
Output: # of -subgraphs in
#Ka,b
G
Ka,b G
a b
• -Detection
Ka,b
• NP-complete (a and b are given as input
)
• W[1]-hard (parameteriszed by
)
• -Detection requires time under ET
H
• time algo
a = b
Ka,a nΩ( a)
O*(1.6914n
)
[Garey, Johnson 1979]
[Lin, 2015]
[Binkele-Raible, Fernau, Gaspers, Liedloff, 2010]
• #Ka,b
• tim
e
• on bipartite grap
h
• time
O*(1.6107)
O*(1.4423n
)
O(1.2491n
) [Kutzkov, 2012]
[Gaspers, Kratsch, Liedloff 2012]
[Couturier, Kratsch 2012]
• enumerate Ka,b
pattern mining [Agrawal, Srikant, 1994]
bioinformatics [Driskell, Ané, Burleigh, McMahon, Sanderson, 2004]
• small : time, or time if
a O(na+1
) O(nω
) a = 2
12 /27
[Lin, 2015]

13. /27
Theorem
Theorem
Result 1
13
Under (randomized) SETH, for any and , there exists such tha
any time algorithm for has a success prob. .
a ≥ 3 ϵ > 0 b
O(na−ϵ
) (#Ka,b,
𝒦
a,b,n) ≤ 1 − 1/polylog(n)
For any and , can be solved in time .
a ≥ 8 b ∈ ℕ #Ka,b bna+o(1)

14. /27
Theorem
Theorem
Under (randomized) SETH, for any and , there exists such tha
any time algorithm for has a success prob. .
a ≥ 3 ϵ > 0 b
O(na−ϵ
) (#Ka,b,
𝒦
a,b,n) ≤ 1 − 1/polylog(n)
For any and , can be solved in time .
a ≥ 8 b ∈ ℕ #Ka,b bna+o(1)
Result 1
14
vert
𝒦
a,b,n
vertices
α vertices
β
: distribution of a random bipartite graph
w.p. 1/2
Let α ∼ Unif(1,…, a), β ∼ Unif(1,…, b)

15. /27
Theorem (Fine-Grained Hardness Ampli cation)
Result 2
Under SETH, for any and , there are and such tha
any time algorithm for has success prob. .
a ≥ 3 ϵ > 0 b ∈ ℕ k = polylog(n)
O(na−ϵ
) (#Ka,b,
𝒦
a,b,n)k
≤ n−O(ϵ)
・By combining Goldreich-Levin technique
15
1/polylog(n) 1 − n−ϵ
Result 1 Result 2
[Goldreich, Levin, 1989]
we amplify the hardness of computing parity , which corresponds to
#Ka,b
Yao’s XOR lemma [Yao, 1982], [Levin, 1987]

16. /27
Related Work
・ , requires under ET
- is the distribution of “arti
fi
cial” random graphs
・ requires time under ET
- Issue in success probability:
・Worst-case to average-case reduction for (H is a general graph
- Issue in success probability:
∃
𝒟
(#Ka,
𝒟
) nΩ(a)
𝒟
(#Ka, G(n, p)) nΩ(k)
1 − 1/polylog(n)
#H
1 − 1/polylog(n)
16
[Goldreich Rothblum, 2018]
[Boix-Adserà, Brennan, Bresler, 2019]
[Dalirrooyfard, Lincoln, Williams, 2020]

17. Techniques

18. /27
Theorem (Reminder)
Hardness Ampli cation
Under SETH, for any and , there are and such tha
any time algorithm for has success prob. .
a ≥ 3 ϵ > 0 b ∈ ℕ k = polylog(n)
O(na−ϵ
) (#Ka,b,
𝒦
a,b,n)k
≤ n−O(ϵ)
・We combine the direct product theorem of
18
1/polylog(n) 1 − n−ϵ
Result 1 Result 2
[Impagliazzo, Jaiswal, Kabanets, Wigderson, 2010]
with our new interactive proof system with low query complexity

19. /27
Theorem (Informal; Impagliazzo et al. 2010)
Direct Product Theorem
Let be a -time algorithm that solves with success probability
Then, there is a list of algorithms such tha
・For some , solves with success probability
・Each runs in time .
A T(n) (Π,
𝒟
)k
≈ δk
(M1, …, Mm)
i ∈ [m] MA
i (Π,
𝒟
) ≈ δ
MA
j ≈ T(n)
• We apply DPT of [Impagliazzo, Jaiswal, Kabanets, Wigderson, 2010]
Here, , and the list can be computed in time .
m = O((1/δ)k
) T(n)
(each algorithm is represented as an oracle circuit)
19

20. /27
Our Setting: #Ka,b
• : list of oracle algorithms such that
∃M1, …, Mm
Which solves ?
Mi (#Ka,b,
𝒦
a,b,n)
・For some , solves with success prob
・Each runs in time
・
i MA
i (#Ka,b,
𝒦
a,b,n) ≥ 1 − n−ϵ
Mj ≈ na−ϵ
m = poly log n
• To identify , we construct an interactive proof system
Mi
20
• : algorithm for with success prob. .
A (#Ka,b,
𝒦
a,b,n)polylogn
n−ϵ
and simulate it using each as a prover.
Mj
We want to solve
in time .
#Ka,b
na−ϵ+o(1)

21. /27
Theorem (IP for )
#Ka,b
IP with low query complexity
There is an -round IP for such that
1. Veri
fi
er runs in time
.
2. Veri
fi
er makes at most queries
3. Each query is of the form “
4. If Prover solves correctly, Veri
fi
er accepts w.p. 1
5. Otherwise, Veri
fi
er rejects w.p. .
O(log n) #Ka,b
n2
poly log n
poly log n
#Ka,b(G) = ?
#Ka,b
≥ 2/3
21
• Goldreich and Rothblum constructed an IP for with query complexity .
#Ka Θ(n)
[Goldreich and Rothblum, 2018]
• Based on sum-check protocol for permanent of [Lund, Fortnow, Karloff, Nisan, 1990]
• We construct IP for #Colored- .
H

22. /27
Identi
fi
cation using IP
• : list of algorithms such that
∃M1, …, Mm
• Which solves ?
Mi (#Ka,b,
𝒦
a,b,n)
・For some , solves with success prob
・Each runs in time
・
i Mi (#Ka,b,
𝒦
a,b,n) ≥ 1 − n−ϵ
Mj ≈ na−ϵ
m = poly log n
• Idea: Run IP using each as Prover
Mj
22

23. /27
Identi
fi
cation using IP
• Let be the input of .
G #Ka,b
• Run IP using as Prover.
Mj
Prover Veri
fi
er
?
#Ka,b(Gj) =
G1
#Ka,b(G1)
Note that is reducible to .
Therefore, we can implement Prover using
#Ka,b (#Ka,b,
𝒦
a,b,n)
Mj
• If Veri
fi
er accepts for Prover , then is the right algorithm!
Mi Mi
23

24. /27
Running Time
• Each runs in time .
Mj O(na−ϵ
)
• Veri
fi
er makes queries.
poly log n
• Run IP for times.
m = polylogn
• Total running time = na−ϵ
polylogn
24
Prover Veri
fi
er
G1
#Ka,b(G1)

25. /27
Running Time
• Each runs in time .
Mj O(na−ϵ
)
• Run IP for times.
m = polylogn
• Total running time = na−ϵ
polylogn
25
Prover Veri
fi
er
?
#Ka,b(Gj) =
G1
#Ka,b(G1)
• Veri
fi
er makes queries.
poly log n
If Veri
fi
er makes queries, then
total running time =
n
na+1−ϵ
poly log n

26. /27
Conclusion
26
• sharp threshold result
• Hardness ampli
fi
cation in
fi
ne-grained complexity
-time algo
∃ na+o(1)
-time algo
∀ na−ϵ
hardness ampli
fi
cation for -time algorithms
na−ϵ
・Contributes to
fi
ne-grained average-case complexity
[Ball, Rosen, Sabin, Vasudevan, 2017]