Incompressibility through Colors and IDs

In parameterized complexity each problem instance comes with a parameter

k

, and a parameterized problem is said to admit a

polynomial kernel

if there are polynomial time preprocessing rules that reduce the input instance to an instance with size polynomial in

k

. Many problems have been shown to admit polynomial kernels, but it is only recently that a framework for showing the non-existence of polynomial kernels has been developed by Bodlaender et al. [4] and Fortnow and Santhanam [9]. In this paper we show how to combine these results with combinatorial reductions which use colors and IDs in order to prove kernelization lower bounds for a variety of basic problems:

We show that the

Steiner Tree

problem parameterized by the number of terminals and solution size

k

, and the

Connected Vertex Cover

and

Capacitated Vertex Cover

problems do not admit a polynomial kernel. The two latter results are surprising because the closely related

Vertex Cover

problem admits a kernel of size 2

k

.

Alon and Gutner obtain a

k

poly

(

h

)

kernel for

Dominating Set in

H

-Minor Free Graphs

parameterized by

h

 = |

H

| and solution size

k

and ask whether kernels of smaller size exist [2]. We partially resolve this question by showing that

Dominating Set in

H

-Minor Free Graphs

does not admit a kernel with size polynomial in

k

 + 

h

.

Harnik and Naor obtain a “compression algorithm” for the

Sparse Subset Sum

problem [13]. We show that their algorithm is essentially optimal since the instances cannot be compressed further.

Hitting Set

and

Set Cover

admit a kernel of size

k

O

(

d

)

when parameterized by solution size

k

and maximum set size

d

. We show that neither of them, along with the

Unique Coverage

and

Bounded Rank Disjoint Sets

problems, admits a polynomial kernel.

All results are under the assumption that the polynomial hierarchy does not collapse to the third level. The existence of polynomial kernels for several of the problems mentioned above were open problems explicitly stated in the literature [2,3,11,12,14]. Many of our results also rule out the existence of compression algorithms, a notion similar to kernelization defined by Harnik and Naor [13], for the problems in question.