The Exponential Time Hypothesis

“Easy” for small values of the parameter

Fixed-Parameter Tractable: f(k)poly(n)

As hard as solving Clique

An O(nk) algorithm exists.

We are going to focus on problems that have O*(kk) algorithms,

!but are not expected to have O*(2o(k log k)) algorithms.

Input

QuestionIs there a clique that picks one vertex from each row, and one vertex from each column?

A graph over the vertex set [k] x [k].

Permutation Clique

Input

QuestionIs there a clique that picks one vertex from each row, and one vertex from each column?

A graph over the vertex set [k] x [k].

Permutation Clique

Unless ETH fails, there is no algorithm that solves Permutation Clique

in 2o(k log k) time.

Input

QuestionIs there a hitting set that picks one vertex from each row, and one vertex from each column?

A family of subsets over the universe [k] x [k].

Permutation Hitting Set

Permutation Hitting Set

Permutation Clique

Permutation Clique

Permutation Hitting Set

Permutation Clique

Permutation Clique

Permutation Hitting Set

Permutation Clique

Permutation Hitting Set

Permutation Hitting Set

Permutation Clique

Every Clique is in fact a hitting set too.

Permutation Hitting Set

Permutation Clique

Every Clique is in fact a hitting set too.

Permutation Hitting Set

Permutation Clique

Every Clique is in fact a hitting set too.

Permutation Hitting Set

Permutation Clique

Every Clique is in fact a hitting set too.

Input

QuestionIs there a hitting set that picks one vertex from each row, and one vertex from each column?

A family of subsets over the universe [k] x [k], such that every set has at most one element from every row.

Permutation Hitting Set With Thin Sets

Input

QuestionIs there a string of length d over A whose hamming distance from each xi is at most d?

n strings, x1, x2, …, xn of length L each over an alphabet A, and a budget d.

Closest String

x1

x2

…

xn

Input

QuestionIs there a string of length d over A whose hamming distance from each xi is at most d?

n strings, x1, x2, …, xn of length L each over an alphabet A, and a budget d.

Closest String

x1

x2

…

xn

Is there a hitting set that picks one vertex from

each row, and one vertex from each column?

A family of subsets over the universe [k] x [k], such that

every set has at most one element from every row.

Permutation Hitting Set With Thin Sets

Is there a hitting set that picks one vertex from

each row, and one vertex from each column?

A family of subsets over the universe [k] x [k], such that

every set has at most one element from every row.

Permutation Hitting Set With Thin Sets

1 3 2 ♠ ♠ 1

Is there a hitting set that picks one vertex from

each row, and one vertex from each column?

A family of subsets over the universe [k] x [k], such that

every set has at most one element from every row.

Permutation Hitting Set With Thin Sets

1 3 2 ♠ ♠ 14 ♠ 3 5 5 5

Is there a hitting set that picks one vertex from

each row, and one vertex from each column?

A family of subsets over the universe [k] x [k], such that

every set has at most one element from every row.

Permutation Hitting Set With Thin Sets

1 3 2 ♠ ♠ 14 ♠ 3 5 5 5♠ 6 5 4 3 ♠

Is there a hitting set that picks one vertex from

each row, and one vertex from each column?

A family of subsets over the universe [k] x [k], such that

every set has at most one element from every row.

Permutation Hitting Set With Thin Sets

1 3 2 ♠ ♠ 14 ♠ 3 5 5 5♠ 6 5 4 3 ♠♠ ♠ ♠ ♠ 1 2

Is there a hitting set that picks one vertex from

each row, and one vertex from each column?

A family of subsets over the universe [k] x [k], such that

every set has at most one element from every row.

Permutation Hitting Set With Thin Sets

1 3 2 ♠ ♠ 14 ♠ 3 5 5 5♠ 6 5 4 3 ♠

1 1 1 1 1 1

2 2 2 2 2 2

3 3 3 3 3 34 4 4 4 4 45 5 5 5 5 56 6 6 6 6 6

♠ ♠ ♠ ♠ 1 2

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.

Input

QuestionIs there a clique that picks one vertex from each row?

A graph over the vertex set [k] x [k].

[k]x[k] Clique

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

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.

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

A 2(k log k) algorithm.

A 2(k log k) algorithm.

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.

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.

3-Sat [N] Edge Clique Cover [k]

Reduce 3-SAT to Edge Clique Cover, and suppose n —> k* !

Run a algorithm. !

This should be a 2o(n) algorithm.

2o(2k�)

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

k =

�2N

log3 N

�

V1 V2 Vk…

k =

�2N

log3 N

�

V1 V2 Vk…

All possible 3-colorings of the Vi’s.

Add edges between compatible colorings…

k =

�2N

log3 N

�

V1 V2 Vk…

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.

3-SAT cannot be solved in 2o(n+m) time.

Exponential Time Hypothesis [ETH]