Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
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Journal of the ACM (JACM)
Invitation to data reduction and problem kernelization
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Polylogarithmic-round interactive proofs for coNP collapse the exponential hierarchy
Theoretical Computer Science
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
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
Kernelization: New Upper and Lower Bound Techniques
Parameterized and Exact Computation
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
Solving MAX-r-SAT above a tight lower bound
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
On the Compressibility of $\mathcal{NP}$ Instances and Cryptographic Applications
SIAM Journal on Computing
Lower Bounds for Kernelizations and Other Preprocessing Procedures
Theory of Computing Systems
Kernelization of packing problems
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Parameterized Complexity
Kernelization of packing problems
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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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In this paper we use the notion of weak compositions to obtain polynomial kernelization lower-bounds for several natural parameterized problems. Let d ≥ 2 be some constant and let L1, L2 ⊆ {0, 1}* x N be two parameterized problems where the unparameterized version of L1 is NP-hard. Assuming coNP ⊈ NP/poly, our framework essentially states that composing t L1-instances each with parameter k, to an L2-instance with parameter k' ≤ t1/dkO(1), implies that L2 does not have a kernel of size O(kd−ε) for any ε 0. We show two examples of weak composition and derive polynomial kernelization lower bounds for d-Bipartite Regular Perfect Code and d-Dimensional Matching, parameterized by the solution size k. By reduction, using linear parameter transformations, we then derive the following lower-bounds for kernel sizes when the parameter is the solution size k (assuming coNP ⊈ NP/poly): • d-Set Packing, d-Set Cover, d-Exact Set Cover, Hitting Set with d-Bounded Occurrences, and Exact Hitting Set with d-Bounded Occurrences have no kernels of size O(kd−3−ε) for any ε 0. • Kd Packing and Induced K1,d Packing have no kernels of size O(kd−4−ε) for any ε 0. • d-Red-Blue Dominating Set and d-Steiner Tree have no kernels of sizes O(kd−3−ε) and O(kd−4−ε), respectively, for any ε 0. Our results give a negative answer to an open question raised by Dom, Lokshtanov, and Saurabh [ICALP2009] regarding the existence of uniform polynomial kernels for the problems above. All our lower bounds transfer automatically to compression lower bounds, a notion defined by Harnik and Naor [SICOMP2010] to study the compressibility of NP instances with cryptographic applications. We believe weak composition can be used to obtain polynomial kernelization lower bounds for other interesting parameterized problems. In the last part of the paper we strengthen previously known super-polynomial kernelization lower bounds to super-quasi-polynomial lower bounds, by showing that quasi-polynomial kernels for compositional NP-hard parameterized problems implies the collapse of the exponential hierarchy. These bounds hold even the kernelization algorithms are allowed to run in quasipolynomial time.