Fixed-Parameter Intractability

Fixed-Parameter Intractability

Vertex Cover

Feedback Vertex Set

Odd Cycle Transversal

Max Sat

Point Line Cover

Feedback Arc Set on Tournaments

Closest String

3-Hitting Set

d-Clustering

nf(k)

Vertex Cover

Feedback Vertex Set

Odd Cycle Transversal

Max Sat

Point Line Cover

Feedback Arc Set on Tournaments

Closest String

3-Hitting Set

d-Clustering

nf(k)O(1)

Vertex Cover

Feedback Vertex Set

Odd Cycle Transversal

Max Sat

Point Line Cover

Feedback Arc Set on Tournaments

Closest String

3-Hitting Set

d-Clustering

Some problems tend to be harder than others.

FPT

FPT“Hard”

Your Problem

CliqueYour Problem

CliqueYour Problem

solveClique{ blah blah

!}

Independent Set Clique

Independent Set Clique

Independent Set Clique

SolveClique(G,k) { Return IndependentSet(Gc,k ); }

solveClique{ blah blah

!}

(x, k) (x �, k �)

Instance of “your problem”

A known “hard” problem

solveClique{ blah blah

!}

(x, k) (x �, k �)

Instance of “your problem”

A known “hard” problem

The reduction itself must run in FPT time.

solveClique{ blah blah

!}

(x, k) (x �, k �)

Instance of “your problem”

A known “hard” problem

The reduction itself must run in FPT time.

solveClique{ blah blah

!}

The time taken to solve the generated instance will bef(k �)p(|x �|)

(x, k) (x �, k �)

Instance of “your problem”

A known “hard” problem

The reduction itself must run in FPT time.

solveClique{ blah blah

!}

So the parameter of the output is constrained to being a function of the original parameter.

SolveIndSet(G,k) { Return VertexCover(G,n-k ); }

SolveIndSet(G,k) { Return VertexCover(G,n-k ); }

NP-hardness reductions are not necessarily FPT reductions.

Given a graph G and a number k: !

return YES if k ≤ log n and G has a clique on k vertices.

Instance of CLIQUE, (G,k)

Instance of log-CLIQUE, (G,k)

Instance of CLIQUE, (G,k)

Instance of log-CLIQUE, (G,k)

Instance of CLIQUE, (G,k)

Instance of log-CLIQUE, (G,k)

Add 2k isolated vertices.

FPT reductions are not always NP-hardness reductions.

Multi-Colored Clique

Multi-Colored Clique

Multi-Colored Clique

Multi-Colored Clique

Multi-Colored Clique

Clique

Multi-Colored Clique

Multi-Colored Clique