On the completeness of a generalized matching problem
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Theoretical Computer Science - Parameterized and exact computation
Infeasibility of instance compression and succinct PCPs for NP
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Faster Fixed-Parameter Tractable Algorithms for Matching and Packing Problems
Algorithmica - Parameterized and Exact Algorithms
A more effective linear kernelization for cluster editing
Theoretical Computer Science
A Problem Kernelization for Graph Packing
SOFSEM '09 Proceedings of the 35th Conference on Current Trends in Theory and Practice of Computer Science
Incompressibility through Colors and IDs
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
On problems without polynomial kernels
Journal of Computer and System Sciences
Two Edge Modification Problems without Polynomial Kernels
Parameterized and Exact Computation
A 4k2 kernel for feedback vertex set
ACM Transactions on Algorithms (TALG)
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses
Proceedings of the forty-second ACM symposium on Theory of computing
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Preprocessing of min ones problems: a dichotomy
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Parameterized complexity and kernelizability of max ones and exact ones problems
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
Lower Bounds for Kernelizations and Other Preprocessing Procedures
Theory of Computing Systems
A linear kernel for a planar connected dominating set
Theoretical Computer Science
Preprocessing for treewidth: a combinatorial analysis through kernelization
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Weak compositions and their applications to polynomial lower bounds for kernelization
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Finding k disjoint triangles in an arbitrary graph
WG'04 Proceedings of the 30th international conference on Graph-Theoretic Concepts in Computer Science
Linear kernels in linear time, or how to save k colors in O(n2) steps
WG'04 Proceedings of the 30th international conference on Graph-Theoretic Concepts in Computer Science
Research paper: Combinatorial and computational aspects of graph packing and graph decomposition
Computer Science Review
Discrete Optimization
Weak compositions and their applications to polynomial lower bounds for kernelization
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Kernelization --- preprocessing with a guarantee
The Multivariate Algorithmic Revolution and Beyond
What's next? future directions in parameterized complexity
The Multivariate Algorithmic Revolution and Beyond
Kernelization hardness of connectivity problems in d-degenerate graphs
Discrete Applied Mathematics
Clique cover and graph separation: new incompressibility results
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Kernel lower bounds using co-nondeterminism: finding induced hereditary subgraphs
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Parameterized complexity of induced h-matching on claw-free graphs
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Kernel bounds for path and cycle problems
Theoretical Computer Science
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Kernelization algorithms are polynomial-time reductions from a problem to itself that guarantee their output to have a size not exceeding some bound. For example, d-Set Matching for integers d ≥ 3 is the problem of finding a matching of size at least k in a given d-uniform hypergraph and has kernels with O(kd) edges. Recently, Bodlaender et al. [ICALP 2008], Fortnow and Santhanam [STOC 2008], Dell and Van Melkebeek [STOC 2010] developed a framework for proving lower bounds on the kernel size for certain problems, under the complexity-theoretic hypothesis that coNP is not contained in NP/poly. Under the same hypothesis, we show lower bounds for the kernelization of d-Set Matching and other packing problems. Our bounds are tight for d-Set Matching: It does not have kernels with O(kd−ε) edges for any ε 0 unless the hypothesis fails. By reduction, this transfers to a bound of O(kd−1−ε) for the problem of finding k vertex-disjoint cliques of size d in standard graphs. It is natural to ask for tight bounds on the kernel sizes of such graph packing problems. We make first progress in that direction by showing non-trivial kernels with O(k2.5) edges for the problem of finding k vertex-disjoint paths of three edges each. This does not quite match the best lower bound of O(k2−ε) that we can prove. Most of our lower bound proofs follow a general scheme that we discover: To exclude kernels of size O(kd−ε) for a problem in d-uniform hypergraphs, one should reduce from a carefully chosen d-partite problem that is still NP-hard. As an illustration, we apply this scheme to the vertex cover problem, which allows us to replace the number-theoretical construction by Dell and Van Melkebeek [STOC 2010] with shorter elementary arguments.