1. Implications of the ETH
Research Exam
Anant Dhayal
Advised by Prof. Russell Impagliazzo

2. Overview The Sparsification Lemma and ETH
The Dichotomy Theorem & Lower Bounds for NP-complete CSP’s
Strong ETH & Open questions

3. The Exponential Time Hypothesis

4. CNF-SAT : “The” NP-complete problem
Conjunctive Normal Form (CNF):
Variables → 𝑥 1 , 𝑥 2 , …, 𝑥 𝑛
Literals → 𝑥 1 , 𝑥 1 , 𝑥 2 , 𝑥 2 , …, 𝑥 𝑛 , 𝑥 𝑛
Clause →(𝑥 𝑥 𝑥 3 )
CNF formula →(𝑥 𝑥 𝑥 3 )⋀ (𝑥 𝑥 𝑥 7 )
CNF-SAT
Input: CNF formula
Output : Yes, if satisfiable No, otherwise
𝑘-SAT : Number of literals in a clause is at most 𝑘
Assumptions : 𝑛= #variables 𝑚= #clauses
Sparse formulae : 𝑚∈𝑂(𝑛)

5. 𝑘-SAT Classical complexity theory: 𝑘<3 Polynomial time 𝑘≥3
CNF-SAT ≤ 𝑝 𝑘-SAT
Clause width reduction
(𝑥 1 ˅ 𝑥 2 ˅ 𝑥 3 ˅ 𝑥 4 ) → (𝑥 1 ˅ 𝑥 2 ˅ 𝑦) ˄ ( 𝑦 ˅ 𝑥 3 ˅ 𝑥 4 )

6. 𝑘-SAT Classical complexity theory: 𝑘<3 Polynomial time 𝑘≥3
NP-complete

7. P≠NP: Super-polynomial time
𝑘-SAT
Classical complexity theory:
𝑘<3
Polynomial time
𝑘≥3
P≠NP: Super-polynomial time

8. Sub-exponential or not ??
𝑘-SAT
Fine-grained complexity theory:
𝑘<3
Polynomial time
𝑘≥3
Sub-exponential or not ??
Does a sub-exponential time algorithm for 3-SAT also imply a sub-exponential time algorithm for 13-SAT?

9. SE: Sub-exponential Π,𝑝 ∈𝑆𝐸:
∀𝜖>0 there is an algorithm that solves Π and runs in time 2 𝜖 𝑝 (𝑥) for any input 𝑥.
zitidaxiao

10. 𝑘-SAT Fine-grained complexity theory: 𝑘≥3 Sub-exponential or not ??
Does a sub-exponential time algorithm for 3-SAT also imply a sub-exponential time algorithm for 13-SAT?
Does clause width reduction answer this for parameter 𝑚?
Yes – It doubles the number of clauses!

11. We need reductions which preserve sub-exponential time complexity
𝑘-SAT
Fine-grained complexity theory:
𝑘≥3
Sub-exponential or not ??
Does a sub-exponential time algorithm for 3-SAT also imply a sub-exponential time algorithm for 13-SAT?
Does clause width reduction answer this for parameter 𝑛?
NO – It adds one new variable per clause, and there are 𝑂( 𝑛 𝑘 ) clauses
We need reductions which preserve sub-exponential time complexity

12. Sub-Exponetial Reduction Family (SERF)
Inclusion in SE:
Π 1 , 𝑝 1 ≤ 𝑆𝐸𝑅𝐹 Π 2 , 𝑝 2
Π 2 , 𝑝 2 ∈SE→ Π 1 , 𝑝 1 ∈SE
Transitivity:
Π 1 , 𝑝 1 ≤ 𝑆𝐸𝑅𝐹 Π 2 , 𝑝 2 ≤ 𝑆𝐸𝑅𝐹 Π 3 , 𝑝 → Π 1 , 𝑝 1 ≤ 𝑆𝐸𝑅𝐹 Π 3 , 𝑝 3

13. SERF Reductions
𝛱 1 , 𝑝 1 ≤ 𝑆𝐸𝑅𝐹 ( 𝛱 2 , 𝑝 2 ), if there exists a reduction which
Takes Π 1 instance 𝑥 as input
Runs in sub−exponetial time*
Makes bounded∗∗ oracle queries to Π 2
𝑂𝑢𝑡𝑝𝑢𝑡=1 ⟷ Π 1 𝑥 =1
* (in the family) there exists a reduction for ∀𝜖>0
**every query 𝑞 should satisfy, 𝑝 2 𝑞 ∈𝑂( 𝑝 1 (𝑥))

14. Sparsification Lemma [IPZ98]
(k-SAT,n) ≤ 𝑆𝐸𝑅𝐹 (sparse-k-SAT,n)
If (𝑘-SAT,m) ∈ SE → Run sparsification
→ We get m∈𝑂(𝑛)
→ Run SE algorithm for m → We get (𝑘-SAT,n)∈ SE
𝑛 plays the same role as 𝑚

15. 𝑘-SAT Fine-grained complexity theory:
For 𝑘≥3 start with 𝑘-SAT formula over 𝑛 variables
Sparsification gives 𝑘-SAT formulae over 𝑛 variables
[𝑂(𝑛) clauses]
Clause width reduction adds one new variable per clause, thus gives 3-SAT formulae over 𝑂(𝑛) variables OR
Sub-exponential time
Exponential time

16. 𝑘-SAT Fine-grained complexity theory: 𝑘<3 Polynomial time 𝑘≥3
Sub-exponential OR
Exponential time
All 𝑘≥3 play the same role

17. 𝑘-SAT Fine-grained complexity theory: Exponential Time Hypothesis:
𝑘<3
Polynomial time
Exponential Time Hypothesis:
There exists 𝜖>0 such that 𝑘-SAT for 𝑘≥3 cannot be solved in time O( 2 𝜖𝑛 ). In other words, 3-SAT ∉ SE.

18. ETH Based Lower Bounds
ETH implies a 2 𝑛 Ω(1) lower-bound for all NP- complete problems.
NP-hardness of many graph problems including Vertex Cover, Independent Set and Clique was established by reducing from 3-SAT. These reductions produce graphs with |𝑉|=𝑂(𝑚).
Since m=𝑂( 𝑛 3 ) ETH implies a lower bound of 2 Ω( |𝑉| 1/3 ) .
Role of sparsification: Starting with sparse-3-SAT produces graphs with 𝑉 =𝑂 𝑛 . Thus we get a better lower-bound of 2 Ω(|𝑉|) .

19. Constraint Languages & CSP’s

20. 3-Coloring as a Set of Constraints
Input: 𝐺(𝑉,𝐸)
Goal: To find a 𝜙:𝑉→{0,1,2} such that any 𝑢,𝑣 ∈𝐸 satisfies the following constraint 𝑢 1 2 𝑣
𝑒𝑑𝑔𝑒
𝑐𝑜𝑙𝑜𝑟 𝑡𝑢𝑝𝑙𝑒𝑠 𝑡ℎ𝑎𝑡
𝑐𝑎𝑛 𝑏𝑒 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑
White columns: valid assignments Red columns: invalid assignments

21. Binary relation over 𝐷={0,1,2}
Constraint Language
𝑘-ary relation over domain 𝐷:
Set of all finitary relations over 𝐷:
Constraint language over 𝐷:
𝜌⊆ 𝐷 𝑘
Binary relation over 𝐷={0,1,2} 1 2
𝑅 𝐷
Γ⊆ 𝑅 𝐷

22. 𝜙:𝑉→𝐷 such that 𝜙(𝑣 1 ),…, 𝜙(𝑣 𝑘 ) ∈𝜌
Constraints
𝑘-ary constraint 𝐶 over domain D and variable set V:
Satisfying assignment of 𝐶:
𝜌 𝑣 1 ,…, 𝑣 𝑘 : interpreted as 𝑣 1 ,…, 𝑣 𝑘 ∈𝜌, where 𝜌⊆ 𝐷 𝑘 and 𝑣 1 , …, 𝑣 𝑘 ∈ 𝑉 𝑘 𝑢 1 2 𝑣
𝜙:𝑉→𝐷 such that 𝜙(𝑣 1 ),…, 𝜙(𝑣 𝑘 ) ∈𝜌

23. Constraint Satisfaction Problem (CSP)
𝐶𝑆𝑃(Γ)
Input: 𝑉,ℭ
𝑉={ 𝑣 1 , 𝑣 2 ,…, 𝑣 𝑛 } is a set of variables.
ℭ is a set of constraints over Γ and 𝑉.
Output: Yes, if ℭ is satisfiable No, otherwise.

24. 3-SAT as a CSP Domain: 𝐵={0,1}.
Constraint language: 𝐵 3 \{ 𝑎,𝑏,𝑐 | 𝑎,𝑏,𝑐∈𝐵}.
Clauses to constraints:
( 𝑥 𝑥 𝑥 3 )
𝑥 1 1
𝑥 2
𝑥 3
𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠
𝑡𝑢𝑝𝑙𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛

25. 3-SAT as a CSP Domain: 𝐵={0,1}.
Constraint language: 𝐵 3 \{ 𝑎,𝑏,𝑐 | 𝑎,𝑏,𝑐∈𝐵}.
Clauses to constraints:
( 𝑥 𝑥 𝑥 3 )
𝑥 1 1
𝑥 2
𝑥 3
𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠
𝑡𝑢𝑝𝑙𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛

26. 3-SAT as a CSP Domain: 𝐵={0,1}.
Constraint language: 𝐵 3 \{ 𝑎,𝑏,𝑐 | 𝑎,𝑏,𝑐∈𝐵}.
Clauses to constraints:
( 𝑥 𝑥 𝑥 3 )
𝑥 1 1
𝑥 2
𝑥 3
𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠
𝑡𝑢𝑝𝑙𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛

27. Power of 3-SAT Constraints
Can express all ternary Boolean relations
Example:
𝑥 1 ⋁ 𝑥 2 ⋁ 𝑥 ⋀ 𝑥 1 ⋁ 𝑥 2 ⋁ 𝑥 3
𝑥 1 1
𝑥 2
𝑥 3
𝑥 1 1
𝑥 2
𝑥 3
𝑥 1 1
𝑥 2
𝑥 3

28. Power of 3-SAT Constraints
Can express all finitary Boolean relations
Example:
( 𝑥 𝑥 𝑥 𝑥 4 )≡ ∃𝑦 ( 𝑥 𝑥 𝑦) ⋀ ( 𝑦 𝑥 𝑥 4 )
𝑥 1 1
𝑥 2 𝑦
𝑥 1 … 1
𝑥 2
𝑥 3
𝑥 4 𝑦 1
𝑥 3
𝑥 4

29. Clone of a Constraint Language
= 𝐷 : the relation 𝑑,𝑑 𝑑∈ 𝐷}.
pp-formula: first-order formula involving only conjunctions (⋀) and existential quantification ∃ .
Relational Clone of Γ: Set of all finitary relations expressed by pp-formulae using relations from the set Γ∪ = 𝐷 . Denoted by Γ . 𝑥 1 𝑦
sparse-𝐶𝑆𝑃 𝛤 ≡ 𝑆𝐸𝑅𝐹 sparse-𝐶𝑆𝑃(⟨Γ⟩)

30. Universal Algebra

31. Universal Algebra 𝑘-ary operation over 𝐷:
Set of all finitary operations over 𝐷:
Algebra over 𝐷:
𝑓: 𝐷 𝑘 →𝐷
𝑂 𝐷
𝐹⊆ 𝑂 𝐷

32. Composition of Operations
Composition of an 𝑛-ary operation 𝑓 with 𝑛 𝑘-ary operations { 𝑔 1 ,…, 𝑔 𝑛 } is the 𝑘-ary operation ℎ, defined by:
ℎ 𝑑 1 ,…, 𝑑 𝑘 = 𝑓( 𝑔 1 𝑑 1 , …, 𝑑 𝑘 ,…, 𝑔 𝑛 ( 𝑑 1 , …, 𝑑 𝑘 )) 𝑓
𝑔 𝑔 2 … 𝑔 𝑛
𝑑 𝑑 … 𝑑 𝑘

33. Term Operations of an Algebra
𝑝𝑟𝑜 𝑗 𝐷 : The set of all finitary projection operations over 𝐷.
Term operations of 𝐹 : Set of all finitary operations expressed by compositions using operations from the set 𝐹∪ 𝑝𝑟𝑜𝑗 𝐷 . Denote by 𝑇𝑒𝑟𝑚(𝐹).
Helps in generalizing the definition of compositions 𝑔
𝑑 𝑑 𝑑 3 … 𝑑 𝑘

34. The Galois Correspondence

35. Polymorphisms
𝑓∈ 𝑂 𝐷 is a polymorphism of 𝜌∈ 𝑅 𝐷 if ∀𝑑 1 ,…, 𝑑 𝑛 ∈𝜌, 𝑓 𝑑 1 ,…, 𝑑 𝑛 ∈𝜌.
We also say “𝑓 preserves 𝜌” or “𝜌 is invariant under 𝑓”. 1 ˄ ( 1 ) = ˅ ( 1 ) =

36. Polymorphisms 𝑃𝑜𝑙 Γ : Constraint Languages → Algebras
{𝑓∈ 𝑂 𝐷 | f preserves each relation in Γ}
𝐼𝑛𝑣 F : Algebras → Constraint Languages
{𝜌∈ 𝑅 𝐷 | 𝜌 is invariant under each operation in 𝐹}

37. Clones vs Terms 𝑂 𝐷 𝑅 𝐷 For any 𝐹, 𝐼𝑛𝑣(𝐹) is a relational clone.
𝑃𝑜𝑙 𝐼𝑛𝑣 𝐹 =𝑇𝑒𝑟𝑚(𝐹). Γ
𝑇𝑒𝑟𝑚(𝐹) Γ 𝐹
𝑅 𝐷
𝑂 𝐷

38. Clones vs Terms 𝑂 𝐷 𝑅 𝐷 For any Γ, 𝑃𝑜𝑙 𝛤 is a set of term operations.
𝐼𝑛𝑣(𝑃𝑜𝑙(Γ))=⟨Γ⟩. Γ
𝑇𝑒𝑟𝑚(𝐹) Γ 𝐹
𝑅 𝐷
𝑂 𝐷

39. Recap
sparse-𝐶𝑆𝑃(𝛤)
sparse-𝐶𝑆𝑃( 𝛤 )
sparse-𝐶𝑆𝑃(𝐼𝑛𝑣(𝐹))
where 𝐹=𝑃𝑜𝑙(Γ)

40. Idempotent Algebra & the Dichotomy Theorem

41. Idempotent Algebra
Idempotent operation: An operation 𝑓∈ 𝑂 𝑑 is idempotent if ∀𝑑∈𝐷, 𝑓 𝑑,…,𝑑 =𝑑.
Idempotent algebra: An algebra whose term operations are idempotent.
Every Algebra 𝐹 has an associated idempotent algebra 𝐼 s.t.
sparse-𝐶𝑆𝑃(𝐼𝑛𝑣(𝐹)) ≡ 𝑆𝐸𝑅𝐹 sparse-𝐶𝑆𝑃(𝐼𝑛𝑣(𝐼))

42. Factors of Algebra Idempotent Algebra 𝑯 is a Homomomorphic Image of 𝑺
sparse-CSP(𝚪)
Factors of Algebra
𝑰: Idempotent algebra over domain 𝑫
Idempotent Algebra
sparse-CSP(𝑰𝒏𝒗 𝑰 )
𝑺: Subalgebra of 𝑰 over domain 𝑪
For any 𝑪⊆𝑫 which is preserved by all 𝒇∈𝑰:
𝑺=𝑰 ​ 𝑪 (operations of 𝑰 restricted to domain 𝑪) is a subalgebra
sparse-CSP(𝑰𝒏𝒗 𝑺 )
𝑯: Homomorphic
Image of 𝑺
over domain 𝑩
𝑯 is a Homomomorphic Image of 𝑺
𝐿𝑒𝑡 𝑆= 𝑓 𝑖 𝑖∈𝐼} and 𝐻={ 𝑔 𝑖 | 𝑖∈𝐼}
there exists a surjective 𝝓:𝑪→𝑩
s.t. ∀𝑖 𝜙 𝑓 𝑖 𝑑 1 ,…, 𝑑 𝑘 = 𝑔 𝑖 (𝜙 𝑑 1 ,…,𝜙( 𝑑 𝑘 ))
sparse-CSP(𝑰𝒏𝒗 𝑯 )

43. The Dichotomy Theorem & Lower Bounds
Idempotent algebra is NP-complete if it has a non-trivial factor which contains only projections. Otherwise it is tractable.
sparse-CSP(𝚪)
sparse-CSP(𝑰𝒏𝒗 𝑯 )
NP-complete Constraint Language
sparse-𝟑-SAT
Factors of associated Idempotent Algebra

44. Strong ETH & Open Questions

45. SETH [IP99] 𝑠 𝑘 ≤ 1− 𝑐 𝑘 𝑠 ∞ , where 𝑠 ∞ = lim 𝑘 →∞ 𝑠 𝑘 SETH: 𝑠 ∞ = 1
Let 𝑠 𝑘 =𝑖𝑛𝑓 𝛿 ∃ 2 𝛿𝑛 time 𝑘-SAT algorithm}.
ETH ≡ 𝑠 3 >0→ s 𝑘 𝑘≥3 increases infinitely often.
Best known algorithm for 𝑘-SAT runs in time 2 1− 𝑑 𝑘 𝑛 . [PPSZ98]
[IP99]
𝑠 𝑘 ≤ 1− 𝑐 𝑘 𝑠 ∞ , where 𝑠 ∞ = lim 𝑘 →∞ 𝑠 𝑘
Harder -> easier
SETH: 𝑠 ∞ = 1
It also implies that CNF-SAT can’t be solved in time 2 1−𝜖 𝑛 for 𝜖>0.

46. Open Questions 𝑘-Coloring ETH implies a lower bound of 2 Ω( 𝑉 ) .
Best known algorithm requires 𝑂( 2 |𝑉| ) time.
Is there a reduction from CNF-SAT to 𝑘-Coloring which gives a Ω 2 𝑉 lower bound based on SETH?
Does ETH imply SETH?

47. Thank You!

48. References
[IPZ98] R. Impagliazzo, R. Paturi, and F. Zane. Which problems have strongly exponential complexity? In Foundations of Computer Science, Proceedings. 39th Annual Symposium on, pages 653–662. IEEE, 1998.
[B17] Andrei A. Bulatov. A dichotomy theorem for nonuniform csps. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages , 2017.
[Z17] Dmitriy Zhuk. A proof of CSP dichotomy conjecture. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages , 2017.
[PPSZ98] Ramamohan Paturi, Pavel Pudlak, Michael E. Saks, and Francis Zane. An improved exponential-time algorithm for k-sat. J. ACM, 52(3): , May 2005.
[IP99] Russell Impagliazzo and Ramamohan Paturi. Complexity of k-sat. In Proceedings of the14th Annual IEEE Conference on Computational Complexity, Atlanta, Georgia, USA, May 4-6, 1999, pages , 1999.
[BH06] Andreas Bjorklund and Thore Husfeldt. Inclusion{exclusion algorithms for counting set partitions. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS '06, pages , Washington, DC, USA, IEEE Computer Society.
[K06] Mikko Koivisto. An o*(2^n) algorithm for graph coloring and other partitioning problems via inclusion{exclusion. In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), October 2006, Berkeley, California, USA, Proceedings, pages , 2006.