Two Edge Modification Problems without Polynomial Kernels

Kratsch, Stefan; Wahlström, Magnus

doi:10.1007/978-3-642-11269-0_22

Cited by 41 publications

(42 citation statements)

References 18 publications

(14 reference statements)

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“…Polynomial parameter transformations have since been used extensively, e.g. in [18,23]. Recently, Bodlaender et al [6] extended the kernelization lower bounds machinery in a new direction by introducing the notion of so-called cross composition.…”

Section: Introductionmentioning

confidence: 99%

Weak Compositions and Their Applications to Polynomial Lower Bounds for Kernelization

Hermelin¹,

Wu²

2012

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms

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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 L 1 , L 2 ⊆ {0, 1} * × N be two parameterized problems where the unparameterized version of L 1 is NP-hard. Assuming coNP ⊆ NP/poly, our framework essentially states that composing t L 1 -instances each with parameter k, to an L 2 -instance with parame-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):Cover, Hitting Set with d-Bounded Occurrences, and Exact Hitting Set with dBounded Occurrences have no kernels of size O(k d−3−ε ) for any ε > 0.• K d Packing and Induced K 1,d Packing have no kernels of size O(k d−4−ε ) for any ε > 0.• Our results give a negative answer to an open question raised by Dom, Lokshtanov, and * The full version of this paper can be found at [21].† Contact: Max Plank Institute for Informatics, Germany, Email: hermelin@mpi-inf.mpg.de.‡ Contact: University of Wisconsin-Madison, USA, Email: xiwu@cs.wisc.edu.Work done during the research visit at Max Planck Institute for Informatics, Germany. Research partially supported by NSF grant 1017597, NSFC grant 60973026, the Shanghai Leading Academic Discipline Project (no. B114), and the Shanghai Committee of Science and Technology (nos. 08DZ2271800 and 09DZ2272800).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.

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Section: Introductionmentioning

confidence: 99%

Weak Compositions and Their Applications to Polynomial Lower Bounds for Kernelization

Hermelin¹,

Wu²

2012

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In this case, the problem is a natural generalization of c-hitting set. While chitting set has a k O(c) kernel [AK07], a polynomial sized kernel is unlikely for min ones c-sat even for c = 3, as a special case of min ones 3-sat (not-1-in-3 SAT) is unlikely to have a polynomial sized kernel [KW09]. See [KWar] for a classification of the types of bounded variable constraints for which polynomial sized kernel is possible.…”

Section: Discussionmentioning

confidence: 99%

Solving minones-2-sat as Fast as vertex cover

Misra

Narayanaswamy

Raman

et al. 2010

Mathematical Foundations of Computer Science 2010

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Abstract. The problem of finding a satisfying assignment for a 2-SAT formula that minimizes the number of variables that are set to 1 (min ones 2-sat) is NP-complete. It generalizes the well-studied problem of finding the smallest vertex cover of a graph, which can be modeled using a 2-SAT formula with no negative literals. The natural parameterized version of the problem asks for a satisfying assignment of weight at most k.In this paper, we present a polynomial-time reduction from min ones 2-sat to vertex cover without increasing the parameter and ensuring that the number of vertices in the reduced instance is equal to the number of variables of the input formula. Consequently, we conclude that this problem also has a simple 2-approximation algorithm and a 2k variables kernel subsuming these results known earlier. Further, the problem admits algorithms for the parameterized and optimization versions whose runtimes will always match the runtimes of the best-known algorithms for the corresponding versions of vertex cover.

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“…The above H-free edge modification problems are FPT for every fixed H following a general result of the first author [6]. In IWPEC'06 [2], the same author raised the issue of determining the existence of polynomial kernels for H-Free Edge Deletion in terms of the structure of H. Kratsch and Wahlström [11] constructed the first H for which neither H-Free Edge Deletion nor H-Free Edge Editing admits polynomial kernels, and Guillemot et al [10] established the nonexistence of polynomial kernels for H-Free Edge Deletion when H is a path P l with l ≥ 13 or a cycle C l with l ≥ 12, provided that coNP ⊆ NP/poly. On the other hand, Gramm et al [9] obtained polynomial kernels for P 3 -Free Edge Deletion, Completion and Editing, and Guillemot et al [10] presented polynomial kernels for P 4 -Free Edge Deletion, Completion and Editing.…”

Section: H-free Edge Deletionmentioning

confidence: 99%

“…Propagational functions f (x, y, z) generalize function Not-1-in-3 of Kratsch and Wahlström [11], and capture the relation that "whatever happens to x must happen to either y or z", which is of great use when we deal with edge modification problems because of propagations of edge deletions/additions. The following example of C 4 -Free Edge Deletion explains such a connection.…”

Section: Satisfiability Of Propagational Formulasmentioning

confidence: 99%

Incompressibility of H-Free Edge Modification

Cai

2013

Parameterized and Exact Computation

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Abstract. Given a fixed graph H, the H-Free Edge Deletion (resp., Completion, Editing) problems ask whether it is possible to delete from (resp., add to, delete from or add to) the input graph at most k edges so that the resulting graph is H-free, i.e., contains no induced subgraph isomorphic to H. These H-free edge modification problems are well known to be FPT for every fixed H. In this paper, we study the nonexistence of polynomial kernels for them in terms of the structure of H, and completely characterize their nonexistence for H being paths, cycles or 3-connected graphs. As a very effective tool, we have introduced a constrained satisfiability problem Propagational Satisfiability to cope with the propagation of edge additions/deletions, and we expect the problem to be useful in studying the nonexistence of polynomial kernels.

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Two Edge Modification Problems without Polynomial Kernels

Cited by 41 publications

References 18 publications

Weak Compositions and Their Applications to Polynomial Lower Bounds for Kernelization

Weak Compositions and Their Applications to Polynomial Lower Bounds for Kernelization

Solving minones-2-sat as Fast as vertex cover

Incompressibility of H-Free Edge Modification

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