The document discusses several problems related to the exponential time hypothesis (ETH). It states that unless ETH fails, there is no algorithm that can solve problems like Permutation Clique, Permutation Hitting Set, or Closest String in less than 2o(k log k) time, where k is a parameter of the problem. It also discusses how 3-Colorability can be reduced to the [k]x[k] Clique problem, and that unless ETH fails, [k]x[k] Clique does not have a 2o(k log k) time algorithm.

1. The Exponential Time
Hypothesis

2. “Easy” for small
values of the parameter
An O(nk) algorithm exists.
Fixed-Parameter
Tractable: f(k)poly(n)
As hard as solving
Clique

3. We are going to focus on problems that have
O*(kk) algorithms,
!
but are not expected to have O*(2o(k log k)) algorithms.

4. Input
A graph over the vertex set [k] x [k].
Is there a clique that picks one vertex from each row,
Question
and one vertex from each column?
Permutation Clique

5. Input
A graph over the vertex set [k] x [k].
Is there a clique that picks one vertex from each row,
Question
and one vertex from each column?
Permutation Clique

6. Unless ETH fails,
there is no algorithm that solves Permutation Clique
in 2o(k log k) time.

7. Input
A family of subsets over the universe [k] x [k].
Is there a hitting set that picks one vertex from
each row, and one vertex from each column?
Permutation Hitting Set
Question

8. Permutation Clique
Permutation Hitting Set
Permutation Clique

9. Permutation Clique
Permutation Hitting Set
Permutation Clique

10. Permutation Clique
Permutation Hitting Set
Permutation Hitting Set

11. Permutation Clique
Permutation Hitting Set
Every Clique is in fact a hitting set too.

12. Permutation Clique
Permutation Hitting Set
Every Clique is in fact a hitting set too.

13. Permutation Clique
Permutation Hitting Set
Every Clique is in fact a hitting set too.

14. Permutation Clique
Permutation Hitting Set
Every Clique is in fact a hitting set too.

15. Input
A family of subsets over the universe [k] x [k], such that
every set has at most one element from every row.
Is there a hitting set that picks one vertex from
each row, and one vertex from each column?
Question
Permutation Hitting Set With Thin Sets

16. Input
n strings, x1, x2, …, xn of length L each over an alphabet A,
and a budget d.
Is there a string of length d over A whose hamming
distance from each xi is at most d?
x1
…
x2
xn
Closest String
Question

17. Input
n strings, x1, x2, …, xn of length L each over an alphabet A,
and a budget d.
Is there a string of length d over A whose hamming
distance from each xi is at most d?
x1
…
x2
xn
Closest String
Question

18. A family of subsets over the
universe [k] x [k], such that
every set has at most one element
from every row.
Is there a hitting set that
picks one vertex from
each row, and one vertex
from each column?
Permutation Hitting
Set With Thin Sets

19. A family of subsets over the
universe [k] x [k], such that
every set has at most one element
from every row.
Is there a hitting set that
picks one vertex from
each row, and one vertex
from each column?
132♠♠ 1
Permutation Hitting
Set With Thin Sets

20. A family of subsets over the
universe [k] x [k], such that
every set has at most one element
from every row.
Is there a hitting set that
picks one vertex from
each row, and one vertex
from each column?
132♠♠ 1
4♠3555
Permutation Hitting
Set With Thin Sets

21. A family of subsets over the
universe [k] x [k], such that
every set has at most one element
from every row.
Is there a hitting set that
picks one vertex from
each row, and one vertex
from each column?
132♠♠ 1
4♠3555
♠6543♠
Permutation Hitting
Set With Thin Sets

22. A family of subsets over the
universe [k] x [k], such that
every set has at most one element
from every row.
Is there a hitting set that
picks one vertex from
each row, and one vertex
from each column?
132♠♠ 1
4♠3555
♠6543♠
♠♠♠♠12
Permutation Hitting
Set With Thin Sets

23. A family of subsets over the
universe [k] x [k], such that
every set has at most one element
from every row.
111111
Is there a hitting set that
picks one vertex from
each row, and one vertex
from each column?
333333
444444
555555
666666
222222
132♠♠ 1
4♠3555
♠6543♠
♠♠♠♠12
Permutation Hitting
Set With Thin Sets

24. Permutation Hitting Set with Thin Sets
is unlikely to admit a 2o(k log k) algorithm.
Closest String
is unlikely to admit a 2o(d log d) algorithm.
Closest String
is unlikely to admit a 2o(d log |A|) algorithm.

25. Input
A graph over the vertex set [k] x [k].
Is there a clique that picks one vertex from each row? Question
[k]x[k] Clique

27. Unless ETH fails,
there is no algorithm that solves 3-Colorability in 2o(n) time.

28. Unless ETH fails,
there is no algorithm that solves 3-Colorability in 2o(n) time.
Unless ETH fails,
there is no algorithm that solves [k]x[k] Clique in 2o(k log k) time.

29. No 2o(k) algorithm.
A 2(k log k) algorithm.

30. A 2(k log k) algorithm.

31. A 2(k log k) algorithm.

32. Unless ETH fails,
there is no algorithm that solves 3-Colorability in 2o(n) time.
Unless ETH fails,
there is no algorithm that solves [k]x[k] Clique in 2o(k log k) time.

33. 3-Colorability [N]
[k]x[k] Clique
Reduce 3-COL to [k]x[k] Clique, and suppose n —> k*
!
Run a 2o(k* log k*) algorithm.
!
This should be a 2o(n) algorithm.

34. 3-Sat [N]
Edge Clique Cover [k]
Reduce 3-SAT to Edge Clique Cover, and suppose n —> k*
!
Run a 2
o(2k )
algorithm.
!
This should be a 2o(n) algorithm.

35. 3-Colorability for a graph with N vertices
reduces to [k]x[k] Clique with k = O(N/log N).

36. 2N
k=
log3 N
V1
V2
…
Vk

37. 2N
k=
log3 N
V1
V2
…
All possible 3-colorings of the Vi’s.
Vk

38. 2N
k=
log3 N
V1
V2
…
Vk
Add edges between compatible colorings…

39. Clique does not admit a f(k)no(k) algorithm
unless ETH fails, for any computable function f.
Specifically, if W[1] = FPT, then ETH fails.

40. Exponential Time Hypothesis [ETH]
3-SAT cannot be solved in 2o(n+m) time.