1. “(More) Consequences of Falsifying SETH
and the Orthogonal Vectors Conjecture” or
“A Sample of Fine-grained Complexity”
Jesper Nederlof
joint work with Karl Bringmann, Amir Abboud, Holger Dell
(Slides partly by Karl Bringmann and Holger Dell)
Dauer: …

2. Classic Complexity Theory
Polynomial time
Not in polynomial time unless P=NP
Classic Complexity Theory
Integer Linear Programming
Satisfiability
Maximum Matching
Travelling Salesperson
Longest common subsequence
Clique
Linear Programming

3. Fine-grained Complexity
Multivariate Complexity k
O(nk)
Fixed-parameter tractable (FPT)
O(2kn)
Complexity in P n!
O*(2√n)
O*(1.34n)
O*(2n)
2O(n)
Exponential Time Algorithms
O(n)
O(n2)
... n
𝑂 ∗ (⋅) suppresses factors poly in input size

4. Fine-Grained Complexity
OV, 𝑛 2
APSP, 𝑛 3
3SUM, 𝑛 2 ⟹
SAT, 2 𝑛
SETH

5. Fine-Grained Complexity
Problem 𝒅-SAT:
Given formula in 𝑑-CNF with 𝑛 variables, is it satisfiable?
𝑥 1 ∨¬ 𝑥 2 ∨ 𝑥 3 ∧ 𝑥 2 ∨¬ 𝑥 3 ∨ 𝑥 5 ∧(¬ 𝑥 1 ∨ 𝑥 4 ∨ 𝑥 5 )
Strong Exponential Time Hypothesis:
SAT, 2 𝑛
[IP’01]
SETH
∀𝜀>0 ∃𝑑: 𝑑-SAT has no 𝑂 ∗ ( 2 1−𝜀 𝑛 )-time algorithm
Example Implications:
[CDLMNOPSW‘16]
Hitting Set, Set Splitting, NAE-SAT: 𝑂 ∗ ( 2 𝑛 ) but not 𝑂 ∗ ( (2−𝜀) 𝑛 )
Independent Set: 𝑂 ∗ ( 2 𝑡𝑤 ) but not 𝑂 ∗ ( (2−𝜀) 𝑡𝑤 )
[LMS‘11]
Subset Sum: 𝑂 (𝑛+𝑡) but not 𝑡 1−𝜀 2 𝑜(𝑛)
[B’17, ABHS’18+]

6. Fine-Grained Complexity
OV, 𝑛 2
Problem Orthogonal Vectors:
Given sets 𝐴,𝐵⊆ {0,1} 𝑑 of size 𝑛,
are any 𝑎∈𝐴,𝑏∈𝐵 orthogonal? ( 𝑖 𝑎 𝑖 ⋅ 𝑏 𝑖 =0) ⟹
OV-Hypothesis: (moderate dimension)
SAT, 2 𝑛
SETH
∀𝜀,𝛿>0: OV in 𝑑= 𝑛 𝛿 has no 𝑂( 𝑛 2−𝜀 )-time algorithm
Example Implications:
[BI’15,ABVW’15,BK’15,VWR’13,B’14,BI’16,BGL’17]
No 𝑂( 𝑛 2−𝜀 ) algorithm for: Edit Distance, LCS, Diameter-2, Frechet distance, RegExp Matching, ...
No 𝑂(𝑚⋅ 𝑛 2−𝜀 ) algorithm for All Pairs Maxflow
[KT’17]
Dynamic graph algorithms, ...

7. Fine-Grained Complexity
OV, 𝑛 2
APSP, 𝑛 3 ⟹
Problem All Pairs Shortest Paths:
Given graph 𝐺 with distance (weight) per edge
compute distance between any two vertices
SAT, 2 𝑛
SETH
APSP-Hypothesis:
∀𝜀>0: APSP has no 𝑂( 𝑛 3−𝜀 ) algorithm
Example Implications:
[VWW’10,AGVW’15]
No 𝑂( 𝑛 3−𝜀 ) algorithm for: Negative Triangle, Shortest Cycle, Radius, ...

8. Fine-Grained Complexity
OV, 𝑛 2
APSP, 𝑛 3
3SUM, 𝑛 2 ⟹
Problem 3SUM:
Given set 𝑍 of 𝑛 integers, are there 𝑎,𝑏,𝑐∈𝑍 with 𝑎+𝑏=𝑐?
SAT, 2 𝑛
SETH
3SUM-Hypothesis:
∀𝜀>0: 3SUM has no 𝑂( 𝑛 2−𝜀 ) algorithm
Example Implications:
[GO’95,P’10,AVWW’14]
No 𝑂( 𝑛 2−𝜀 ) algorithm for: Colinearity, Conv3SUM, ...
No 𝑂( 𝑛 3−𝜀 ) algorithm for ExactWeightTriangle
[P’10,VW’09]
Dynamic Problems, Listing Triangles, ...

9. Fine-Grained Complexity
OV, 𝑛 2
APSP, 𝑛 3
3SUM, 𝑛 2 ⟹
Success of fine-grained complexity:
SAT, 2 𝑛
These hypotheses explain why we cannot improve running times of fastest algorithms for many problems
SETH
Yield reasons to stop searching for faster algorithms
Non-matching lower bounds suggest faster algorithms
→ same as for NP-hardness, but not as elegant…..
Why are these hypotheses worth introducing, besides mentioned implications?

10. Reason 1: Decades of Effort
OV, 𝑛 2
Many people tried ⟹
Fastest known algorithms for 𝑑-SAT:
𝑂( 2 1−𝑑/𝑘 𝑛 ) for some 𝑐>0
[PPSZ’98]
SAT, 2 𝑛
SETH
Not for lack of trying:
[MS’85,S’92,K’99,PPZ’97,S’99,DGHKKPRS’02,
HSSW’02,BS’03,MS’11,CSTT’13,H’14,…]
OV is at least as hard as 𝑑-SAT

11. Reason 2: Restricted Algorithms
OV, 𝑛 2
Hypotheses hold for algorithms of some specific form ⟹
𝑎∨𝑏
𝑐∨¬𝑏
𝑎∨𝑐
SAT, 2 𝑛
Standard framework for refuting a 𝑑-SAT instance: Resolution
SETH
SETH holds for regular resolution and tree-like resolution
[BI’13,BT’16]
A weak version of SETH holds for `witness compression’ algorithms
[PP’10,D’13]
OVH holds for formulas and branching programs
[KW’18+]

12. Reason 3: Circuit Lower Bounds
OV, 𝑛 2
Falsifying these hypotheses is difficult,
since it would imply difficult-to-prove statements ⟹
If SETH fails then:
[W’13,JMV’15]
SAT, 2 𝑛
SETH
NEXP is not contained in linear-size VSP-circuits
is very likely, but seems difficult to prove

13. Reason 4: Implications of falsifying hypotheses
OV, 𝑛 2
Falsifying these hypotheses implies other `amazing’ algorithms ⟹
If SETH fails then there are 𝑂 ∗ 2−𝜀 𝑛 -time algorithms for:
Hitting Set, Set Splitting, NAE-SAT,
SAT on restricted circuit classes…
SAT, 2 𝑛
[CDLMNOPSW‘16]
SETH
add TC 0
If OVH fails then:
[GIKW‘17]
First-order model checking with 𝑘≥3 quantifiers is
in time 𝑂( 𝑚 𝑘−1−𝜀 ) on structures of total size 𝑚.
e.g. ∃𝑢∃𝑣∃𝑤:𝐸 𝑢,𝑣 ∧𝐸 𝑣,𝑤 ∧𝐸(𝑢,𝑤)
add a weighted, dense problem
„does 𝐺 contain a triangle?“
∀𝑢∀𝑣∃𝑤:𝐸 𝑢,𝑤 ∧𝐸 𝑤,𝑣
„does 𝐺 have diameter=2?“

14. Our Contributions ⟹ OV, 𝑛 2
[ABDN‘18]
OV, 𝑛 2
We find more consequences of falsifying SETH/OVH ⟹
If SETH fails then:
SAT, 2 𝑛
there are 𝑂 ∗ 2−𝜀 𝑛 -time algorithms for sparse- TC 0 -SAT
SETH
If OVH fails then:
there are 𝑂 ∗ 2−𝜀 𝑛 -time algorithms for sparse- TC 1 -SAT
there are 𝑂 𝑛 1−𝜀 𝑘 -time algorithms for weighted 𝑘-Clique (even in hypergraphs)

15. Circuit-SAT

16. Circuit-SAT ⟺ ⟺ ⟹ ∧ 𝒞 = class of circuits 𝐶 𝒞-SAT:
Given a circuit 𝐶∈𝒞 on 𝑛 variables,
is 𝐶 satisfiable? ∨ ∨
𝜔(𝒞) = inf{ 𝑤 | 𝒞-SAT is in time 𝑂 ∗ ( 2 𝑤⋅𝑛 ) }
SETH:
lim 𝑑→∞ 𝜔 𝑑−CNF =1
𝑥 1
𝑥 2 …
𝑥 𝑛
sparsification lemma ⟺
lim 𝑑,𝑐→∞ 𝜔 𝑐−sparse 𝑑−CNF =1
[IPZ’01] ⟺
lim 𝑐,𝛿→∞ 𝜔 𝑐−sparse depth−d ∨,∧,¬,𝑇𝐻𝑅 −circuit =1
𝑐−sparse depth−δ ∨,∧,¬,𝑇𝐻𝑅 −circuit
..sparse TC 0
[ABDN‘18]
OVH fails: ⟹
lim 𝑐,𝛿→∞ 𝜔 𝑐−sparse depth− 𝛿 log 𝑛 ∨,∧,¬,𝑇𝐻𝑅 −circuit <1
[ABDN‘18]
.. sparse TC 1
Proof techniques: Refined Cook-Levin; Depth reduction by Valiant

17. Weighted k-Clique

18. Weighted k-Clique ⟹ ⟹ OV, 𝑛 2 APSP, 𝑛 3 SAT, 2 𝑛 Neg-𝑘-Clique, 𝑛 𝑘1 -2 4 ⟹ ⟹ -1 3 -2
[VWW’10] 2 3 1 -1
SAT, 2 𝑛
Neg-𝑘-Clique, 𝑛 𝑘
SETH -5

19. Weighted k-Clique ⟹ ⟹ OV, 𝑛 2 APSP, 𝑛 3 SAT, 2 𝑛 Neg-𝑘-Clique, 𝑛 𝑘1 -2 4 ⟹ ⟹ -1 3 -2
[VWW’10] 2 3 1 -1
SAT, 2 𝑛
Neg-𝑘-Clique, 𝑛 𝑘
SETH -5
Problem Negative-𝒌-Clique:
Given edge-weighted graph 𝐺
is there a k-Clique with negative total edge-weight?
Neg-𝒌-Clique-Hypothesis:
∀𝜀>0,𝑘≥3: Neg-𝑘-Clique has no 𝑂( 𝑛 𝑘−𝜀 ) algorithm

20. ⟹ Weighted k-Clique ⟹ ⟹ OV, 𝑛 2 APSP, 𝑛 3 SAT, 2 𝑛 Neg-𝑘-Clique, 𝑛 𝑘1 -2 ⟹ 4 ⟹ ⟹ -1 3 -2
[VWW’10] 2
[ABDN‘18] 3 1 -1
SAT, 2 𝑛
Neg-𝑘-Clique, 𝑛 𝑘
SETH -5
relates two fine-grained hypotheses
falsifying OV implies amazing algorithm for Neg-𝑘-Clique
Proof technique: chain of tight reductions
Neg-𝑘-Clique -> Exact-𝑘-Clique-> Clique in Hypergraphs -> OV

21. Concluding Remarks
Fine-grained complexity is a currently very popular subject
Hypotheses are very productive (often lead to faster algorithms if no connection to hypotheses can be made)
Two of these hypotheses are
SETH ( 𝑂 ∗ (2 𝑛 ) is best for 𝑑-CNF-SAT) and
OVH ( 𝑂 𝑛 2 poly 𝑑 is best for OV)
This work*:
If SETH fails then:
there are 𝑂 ∗ 2−𝜀 𝑛 -time algorithms for sparse- TC 0 -SAT
If OVH fails then:
there are 𝑂 ∗ 2−𝜀 𝑛 -time algorithms for sparse- TC 1 -SAT
there are 𝑂 ∗ 𝑛 1−𝜀 𝑘 -time algorithms for weighted 𝑘-Clique
*Amir Abboud and Karl Bringmann and Holger Dell and JN: More Consequences of Falsifying SETH and the Orthogonal Vectors Conjecture. In the conference proceedings of STOC 2018.