# Kernelization Lower Bounds Through Colors and IDs

@article{Dom2014KernelizationLB,
title={Kernelization Lower Bounds Through Colors and IDs},
author={Michael Dom and Daniel Lokshtanov and Saket Saurabh},
journal={ACM Transactions on Algorithms (TALG)},
year={2014},
volume={11},
pages={1 - 20}
}
• Published 30 October 2014
• Mathematics, Computer Science
• ACM Transactions on Algorithms (TALG)
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 nonexistence of polynomial kernels for specific problems has been developed by Bodlaender et al. [2009] and Fortnow and…
135 Citations

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## References

SHOWING 1-10 OF 36 REFERENCES

### Cross-Composition: A New Technique for Kernelization Lower Bounds

• Computer Science, Mathematics
STACS
• 2011
It is shown that if an NP-hard problem cross-composes into a parameterized problem Q then Q does not admit a polynomial kernel unless thePolynomial hierarchy collapses, and its applicability is shown by proving kernelization lower bounds for a number of important graphs problems with structural (non-standard) parameterizations.

### Co-Nondeterminism in Compositions: A Kernelization Lower Bound for a Ramsey-Type Problem

This work presents the first example of how co-nondeterminism can help to make a composition algorithm, and studies the existence of polynomial kernels for a Ramsey-type problem where, given a graph G and an integer k, the question is whether G contains an independent set or a clique of size at least k.

### The Parameterized Complexity of the Unique Coverage Problem

• Mathematics
ISAAC
• 2007
This work considers the parameterized complexity of the UNIQUE COVERAGE problem: given a family of sets and a parameter k, whether there exists a subfamily that covers at least k elements exactly once, and shows that this problem is fixed-parameter tractable with respect to the parameter k.

### Kernels for the Dominating Set Problem on Graphs with an Excluded Minor

• Mathematics
Electron. Colloquium Comput. Complex.
• 2008
The results imply that there is a problem kernel of polynomial size for graphs with an excluded minor and a linear kernel for graphs that are K3,h-minor-free.

### Kernel bounds for disjoint cycles and disjoint paths

• Mathematics, Computer Science
Theor. Comput. Sci.
• 2009
Evidence is given that DisJoint Cycles and Disjoint Paths do not have polynomial kernels, unless NP ⊆ coNP/poly, and it is shown that the related Disj Joint Cycles Packing problem has a kernel of size O(k logk).

### On problems without polynomial kernels

• Computer Science
J. Comput. Syst. Sci.
• 2009

### Capacitated Domination and Covering: A Parameterized Perspective

• Computer Science, Mathematics
IWPEC
• 2008
Capacitated Vertex Cover is the first known "subset problem" which has turned out to be fixed parameter tractable when parameterized by solution size butW[1]-hard when parameterizing by treewidth.

### Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses

• Computer Science, Mathematics
STOC '10
• 2010
It is shown that if satisfiability for n-variable d-CNF formulas has a protocol of cost O(nd-ε) then coNP is in NP/poly, which implies that the polynomial-time hierarchy collapses to its third level.

### Kernel(s) for problems with no kernel: On out-trees with many leaves

• Mathematics
TALG
• 2012
This work gives the first polynomial kernel for Rooted k-Leaf-Out-Branching, a variant of k- leaf- out-branching where the root of the tree searched for is also a part of the input, and is the first ones separating Karp kernelization from Turing kernelization.