kernelization basics

KernelizationEvil is whatever distracts.

― Franz Kafka

Preprocess

Preprocess

(x, k) (x �, k �)

Preprocess

(x, k) (x �, k �)FAST

SMALL

SANE

Preprocess

(x, k) (x �, k �)|x|O(1) time

SMALL

SANE

Preprocess

(x, k) (x �, k �)|x|O(1) time

|x �| = f(k) k � k �and

SANE

Preprocess

(x, k) (x �, k �)|x|O(1) time

|x �| = f(k) k � k �and

(x, k) � (x �, k �)

Examples

Max Sat

Input

Max Sat

A CNF Formula over n variables with m clauses.

InputQuestion

Max Sat

Is there an assignment satisfying at least k clauses?


InputQuestion

Max Sat



What if k is at most m/2?

InputQuestion

Max Sat



Say YES.


InputQuestion

Max Sat



Say YES.

k > m/2?


InputQuestion

Max Sat



Say YES.

k > m/2?

The number of clauses is bounded by 2k.


InputQuestion

Max Sat



InputQuestion

Max Sat



variables

InputQuestion

Max Sat



Configuration of x

variables

InputQuestion

Max Sat



Configuration of x

If two variables x and y have the same “configuration”in the formula, then delete one of them.

variables

InputQuestion

Max Sat



Configuration of x

variables

InputQuestion

Max Sat



Configuration of x

We have at most variables left.3(2k)

variables

InputQuestion

Max Sat



GOAL

We have at most variables left.k

InputQuestionIs there an assignment satisfying at least k clauses?


Variables

Clauses

Max Sat



Max Sat

If we have at least k variables and we have a matching from Variables — Clauses

then we can say YES.

If we have at most k variables - nothing to do.



Max Sat

Else, we have at least k variables, but no matching from Variables — Clauses.

Hall’s Theorem

Hall’s Theorem

If, in a bipartite graph with parts A and B,there is no matching from A to B,

then there is a subset X of Asuch that

|N(X)| < |X|

Hall’s Theorem

Hall’s Theorem

Such an “obstructing set” can be computed in polynomial time.

Hall’s Theorem

Such an “obstructing set” can be computed in polynomial time.[inclusion-minimal]

Such an “obstructing set” can be computed in polynomial time.[inclusion-minimal]

Max Sat

V

C



Max Sat

V

C

removed |X| variables

removed |N(X)| clauses



Max Sat

Ask now if k - |N(X)| clauses can be satisfied.

V

C

removed |X| variables

removed |N(X)| clauses



Max Sat

V

C



Max Sat

V

C



We removed an inclusion-minimal violating set.

MAX SAT

V

C



Get rid of one vertex…

MAX SAT

V

C



The rest of it can be matched!

Get rid of one vertex…

MAX SAT



This implies a kernel with at most k variables and 2k clauses.

Vertex Cover

Input

Vertex Cover

A graph G = (V,E) with n vertices, m edges, and k.

Input

Question

Vertex Cover

Is there a subset of vertices S of size at most k that intersects all the edges?


Input

Question

Vertex Cover



What if a vertex has more than k neighbors?

Input

Question

Vertex Cover




Input

Question

Vertex Cover

We cannot afford to leave v out ofany vertex cover of size at most k.




Vertex Cover

If a vertex has more than k neighbors,

Input

QuestionIs there a subset of vertices S of size at most k that intersects all the edges?


Vertex Cover

delete it from the graph and reduce the budget by one.

If a vertex has more than k neighbors,

Input



Vertex Cover

When we have nothing more to do…

Input



Vertex Cover

every vertex has degree at most k.

When we have nothing more to do…

Input



Vertex Cover

If the graph has more than k2 edges,

Input



Vertex Cover

reject the instance.

If the graph has more than k2 edges,

Input



Vertex Cover

Otherwise:

Input



Vertex Cover

the number of edges is at most k2.

Otherwise:

Input



Vertex Cover


Otherwise:

Input



Vertices?

Vertex Cover


Otherwise:

Input



k2 edges can be involved in at most 2k2 vertices.Throw away isolated vertices.

Vertices?

Vertex Cover

Input



This implies a kernel with at most 2k2 vertices and k2 edges.

Vertex Cover

Input



This implies a kernel with at most 2k2 vertices and k2 edges.

[BUSS KERNELIZATION]

Feedback Arc Set on Tournaments

Input


A tournament on n vertices and an integer k.

Input

Question


Is there a subset of k arcs that can be reversed to make the tournament acyclic?


Input

Question




A tournament has a cycle if, and only if, it has a triangle.

Input

Question




A tournament has a cycle if, and only if, it has a triangle.

Delete a vertex if it does not belong to any triangle.

Input

Question




Delete a vertex if it does not belong to any triangle.

Input

Question




What about an edge that participates in more than k triangles?

Input

Question




What about an edge that participates in more than k triangles?

Reverse it and decrease the budget by one.

Input

Question



A tournament T on n vertices and an integer k.

Input

Question




At most k arcs in any solution.

Input

Question




At most k arcs in any solution.

Every vertex is in a triangle, and everyarc “sees” at most k triangles.

Input

Question




Input

Question




After reducing the graph, if there are more thank2 + 2k vertices,

reject the instance.

Input

Question




This implies a kernel with at most k2 + 2k vertices.

d-Hitting Set

Input

d-Hitting Set

A family F of d-sized sets over an universe U, and k.

Input

Question

d-Hitting Set

Does U have a subset S of size at most k that intersects every set in F?


Input

Question

d-Hitting Set



What if there are more than k sets with one element in common?

Input

Question

d-Hitting Set



☕�♤♧★☼

♕♖☕�♘♤ ⚖☕�✜✍♖

✂✍✎☕�♢


Input

Question

d-Hitting Set




☕�♤♧★☼

♕♖☕�♘♤ ⚖☕�✜✍♖

✂✍✎☕�♢

Input

Question

d-Hitting Set



What if there are more than k sets with one element in common, and every set is mutually disjoint otherwise?

Input

Question

d-Hitting Set



What if there are more than k sets with one element in common, and every set is mutually disjoint otherwise?

☕♤♧★☼

♕♖☕♘❀ ⚖☕✜✍♖

✂☞✎☕♢

Input

Question

d-Hitting Set



❀Sunflower Lemma

Input

Question

d-Hitting Set



Sunflower Lemma

{1,2,53,54,55} {1,2,68,69,67} {1,2,15,12,11} {1,2,23,24,29} {1,2,72,71,70}

Input

Question

d-Hitting Set



Sunflower Lemma

If a d-uniform family F has at least d!(k-1)d sets, then it has a subcollection of at least k sets that:

!have a common mutual intersection (a core)

are pairwise disjoint otherwise (petals).

Input

Question

d-Hitting Set



Sunflower Lemma [By the way…]

Sunflowers can also be computed in polynomial time.

Input

Question

d-Hitting Set



Intuition

As long as there is a sunflower involving at least k+1 petals, remove it and replace it with its core.

Input

Question

d-Hitting Set



{1,2,53,54,55} {1,2,68,69,67} {1,2,15,12,11} {1,2,23,24,29} {1,2,72,71,70}

Intuition

Input

Question

d-Hitting Set



{1,2}

Intuition

Input

Question

d-Hitting Set



Sunflower Lemma

The rule destroys d-uniformity. Fix this by iterating over d.

What if the core is empty? This means we have more than k disjoint sets,

so we can reject the instance.

Input

Question

d-Hitting Set



This implies a kernel with at most d!kd sets, and d(d!)kd elements.

Take Away…

Eliminate “high” degree

vertices

The problem is simple on

“bounded degree” graphs


vertices



May have one, but not the other,eg., Dominating Set,

Connected Vertex Cover.


vertices



May have one, but not the other,eg., Dominating Set,

Connected Vertex Cover.

High Degree Rulesmay be involved.

Eg: Feedback Vertex Set

Combinatorial facts (such as the Sunflower Lemma)and structure theorems (like Hall’s theorem)

can be often (surprisingly) useful.

The Big Picture

A parameterized problem is fixed-parameter tractable !

if and only if !

it has a kernelization algorithm.

A fundamental connection with FPT-land…

If you have a f(k)-sized kernel…

Obtain kernel and solve the problemusing some algorithm with running time g().

poly(n) + g(f(k))

If you have a f(k)-sized kernel…

If you have a f(k)p(n) FPT Algorithm…

We are done if n is at most f(k).


We are done if n is at most f(k).


If not, f(k)p(n) is at most n(p(n)), so the “FPT”algorithm can be run in polynomial time

to obtain a trivial kernel.

Do FPT problems always havepolynomial-sized kernels?

Do FPT problems always havepolynomial-sized kernels?

Remember to ask the question with an“above guarantee” parameterization

after you find a ‘cheat’ kernel.

kernelization basics

Technology

variables clauses

n variables

number of clauses

inputa cnf formula

variables x

nx clauses cmax satquestion

x variables v

variables configuration