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Satisfiability Allows No Nontrivial Sparsification unless the Polynomial-Time Hierarchy Collapses

Holger Dell, Dieter Van Melkebeek
2014 Journal of the ACM  
For any integer d ≥ 3 and positive real ǫ we show that if satisfiability for n-variable d-CNF formulas has a protocol of cost O(n d−ǫ ) then coNP is in NP/poly, which implies that the polynomial-time hierarchy  ...  The case d = 2 implies that no NP-hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k 2−ǫ ) edges unless coNP is in NP/poly, where k  ...  We would like to thank the following people for discussions, comments, pointers to the literature, and guidance: Matt Anderson, Albert Atserias, Kord Eickmeyer, Martin  ... 
doi:10.1145/2629620 fatcat:p5rgbdgfujderlax7sxyokquay

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

Holger Dell, Dieter van Melkebeek
2010 Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10  
For any integer d ≥ 3 and positive real ǫ we show that if satisfiability for n-variable d-CNF formulas has a protocol of cost O(n d−ǫ ) then coNP is in NP/poly, which implies that the polynomial-time hierarchy  ...  The case d = 2 implies that no NP-hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k 2−ǫ ) edges unless coNP is in NP/poly, where k  ...  We would like to thank the following people for discussions, comments, pointers to the literature, and guidance: Matt Anderson, Albert Atserias, Kord Eickmeyer, Martin  ... 
doi:10.1145/1806689.1806725 dblp:conf/stoc/DellM10 fatcat:qzmziun5kjdi7dlgmexpan6tne

Best-case and Worst-case Sparsifiability of Boolean CSPs [article]

Hubie Chen, Bart M. P. Jansen, Astrid Pieterse
2018 arXiv   pre-print
We continue the investigation of polynomial-time sparsification for NP-complete Boolean Constraint Satisfaction Problems (CSPs).  ...  The first result essentially shows that for any arity k, the only constraint type for which no nontrivial sparsification is possible has exactly one falsifying assignment, and corresponds to logical OR  ...  Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J.  ... 
arXiv:1809.06171v1 fatcat:al6bgjkptrfp5gjpaerc5crv6i

Kernelization Lower Bounds [chapter]

Rodney G. Downey, Michael R. Fellows
2013 Texts in Computer Science  
Composition algorithms Simple criterion for ruling out polynomial kernels Simple to understand Not always easy to apply!  ...  Philip 24/43 Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses Dell and van Melkebeek STOC 2010, JACM 2014 Pre-FSTTCS 2014 Workshop Kernelization  ...  Remark The condition coNP ⊆ NP/poly is considered unlikely, because it implies a collapse in the Polynomial Hierarchy. Corollary k-Path does not have a polynomial-size kernel, unless coNP ⊆ NP/poly.  ... 
doi:10.1007/978-1-4471-5559-1_30 fatcat:k35c6pmoaregboa6whaa4pxusq

Sparsification Upper and Lower Bounds for Graph Problems and Not-All-Equal SAT

Bart M. P. Jansen, Astrid Pieterse
2016 Algorithmica  
of an arbitrary problem, with bitsize O(n 2−ε ) for ε > 0, unless NP ⊆ coNP/poly and the polynomial-time hierarchy collapses.  ...  For the problems 4-Coloring, (Directed) Hamiltonian Cycle, and (Connected) Dominating Set, we prove that there is no polynomial-time algorithm that reduces any n-vertex input to an equivalent instance,  ...  sparsification of d-nae-sat.  ... 
doi:10.1007/s00453-016-0189-9 fatcat:6qyiuz6p5vb2jovwn7dbufvzo4

Sparsification Upper and Lower Bounds for Graph Problems and Not-All-Equal SAT [article]

Bart M.P. Jansen, Astrid Pieterse
2015 arXiv   pre-print
of an arbitrary problem, with bitsize O(n^2-e) for e > 0, unless NP is in coNP/poly and the polynomial-time hierarchy collapses.  ...  For the problems 4-Coloring, (Directed) Hamiltonian Cycle, and (Connected) Dominating Set, we prove that there is no polynomial-time algorithm that reduces any n-vertex input to an equivalent instance,  ...  v) ⊆ [4] of allowed colors.  ... 
arXiv:1509.07437v1 fatcat:mc3xjtofarbd7gcla7vle4cyhe

On Sparsification for Computing Treewidth [article]

Bart M. P. Jansen
2013 arXiv   pre-print
of an arbitrary problem, with O(n^2-e) bits, then NP is in coNP/poly and the polynomial hierarchy collapses to its third level.  ...  for any e > 0, unless NP is in coNP/poly.  ...  equivalent instances of an arbitrary problem, with O(n 2− ) bits, for some > 0, then NP ⊆ coNP/poly and the polynomial hierarchy collapses [27] .  ... 
arXiv:1308.3665v1 fatcat:ensn3iaj75dqff2mfiabif6due

Co-nondeterminism in compositions: A kernelization lower bound for a Ramsey-type problem [article]

Stefan Kratsch
2011 arXiv   pre-print
(JCSS 2009) showed that, unless NP \subseteq coNP/poly, the existence of a deterministic polynomial-time composition algorithm, i.e., an algorithm which outputs an instance of bounded parameter value which  ...  Thus we show that, unless NP \subseteq coNP/poly, the problem does not admit a kernelization with polynomial size guarantee.  ...  Acknowledgement The author is grateful to Pascal Schweitzer for providing Lemma 5 and for many useful comments on the paper.  ... 
arXiv:1107.3704v2 fatcat:qns7jo4gfjdwrekuylmhc7iwju

The Landscape of Communication Complexity Classes

Mika Göös, Toniann Pitassi, Thomas Watson
2018 Computational Complexity  
the full version), and no nontrivial lower bounds against it are known for any explicit function.  ...  One of the most notorious open problems in communication complexity is to prove lower bounds against the analogue of the polynomial hierarchy (PH) for any explicit two-party function.  ...  However, it is unlikely that US ⊆ coDP: [10] showed that this inclusion cannot hold in the classical time-bounded setting unless the polynomial hierarchy collapses.  ... 
doi:10.1007/s00037-018-0166-6 fatcat:xogeslxvpfe63ehwznalvplw4e

Introducing lop-kernels: a framework for kernelization lower bounds [article]

Júlio Araújo, Marin Bougeret, Victor A. Campos, Ignasi Sau
2021 arXiv   pre-print
Finally, we prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover of the input graph, even on bipartite graphs, unless NP⊆ coNP / poly.  ...  , the paw, or the complete graphs, by making use of the Erdös-Hajnal property.  ...  Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. Journal of the ACM, 61(4):23:1–23:27, 2014. doi:10.1145/2629620. 29 Reinhard Diestel.  ... 
arXiv:2102.02484v3 fatcat:fzq4726v7feydmutjof6zvkik4

Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs [article]

Marcin Pilipczuk, Michał Pilipczuk, Piotr Sankowski, Erik Jan van Leeuwen
2017 arXiv   pre-print
In the language of parameterized complexity, these results imply the first polynomial kernels for Steiner Tree and Steiner Forest on planar and bounded-genus graphs (parameterized by the size of the tree  ...  and a terminal set S ⊆ V(G), one can in polynomial time find a set F ⊆ E(G) that contains an optimal (edge) multiway cut C separating S and that has size polynomial in |C|.  ...  [5] can be strengthened to also refute a subexponential-time algorithm.  ... 
arXiv:1306.6593v4 fatcat:oddhcwecevg5xhyo5mlqkzl3gm

Nondeterministic Direct Product Reductions and the Success Probability of SAT Solvers

Andrew Drucker
2013 2013 IEEE 54th Annual Symposium on Foundations of Computer Science  
In these theorems one assumes that a target function f is mildly hard against nondeterministic circuits, and concludes that the direct product f ⊗t is extremely hard against (only polynomially smaller)  ...  As an application, we show that if NP is not in coNP/poly then, for every PPT algorithm attempting to produce satisfying assigments to Boolean formulas, there are infinitely many instances where the algorithm's  ...  The hypothesis that the Polynomial Hierarchy does not collapse to Σ p 3 can be equivalently restated as one such assumption: namely, that there is no polynomial-time procedure that reduces a 4-round game  ... 
doi:10.1109/focs.2013.84 dblp:conf/focs/Drucker13 fatcat:vcqxpllt3vedbacdlbsc5mwjdi

Sparsification Upper and Lower Bounds for Graphs Problems and Not-All-Equal SAT *

Bart Jansen, Astrid Pieterse
unpublished
of an arbitrary problem, with bitsize O(n 2−ε) for ε > 0, unless NP ⊆ coNP/poly and the polynomial-time hierarchy collapses.  ...  For the problems 4-Coloring, (Directed) Hamiltonian Cycle, and (Connected) Dominating Set, we prove that there is no polynomial-time algorithm that reduces any n-vertex input to an equivalent instance,  ...  be encoded in O(n 2−ε ) bits, then NP ⊆ coNP/poly and the polynomial-time hierarchy collapses.  ... 
fatcat:352dbfeblvdtddiii7iizqkvoe

Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs

Marcin Pilipczuk, Michal Pilipczuk, Piotr Sankowski, Erik Jan van Leeuwen
2014 2014 IEEE 55th Annual Symposium on Foundations of Computer Science  
van (2014) Network sparsification for Steiner problems on planar and bounded-genus graphs.  ...  [5] can be strengthened to also refute a subexponential-time algorithm.  ...  Acknowledgements We thank Daniel Lokshtanov and Saket Saurabh for showing us the application of Baker's approach to Planar Multiway Cut (Theorem 13.1) and for allowing us to include the proof in this paper  ... 
doi:10.1109/focs.2014.37 dblp:conf/focs/PilipczukPSL14 fatcat:33nagim3pvejpkiakhdive3zsy

The Complexity of Propositional Proofs

Nathan Segerlind
2007 Bulletin of Symbolic Logic  
Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms.  ...  AbstractPropositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies.  ...  The author wishes to thank Paul Beame for answering some questions on [33] and [2] , and also for providing Figure 8 .  ... 
doi:10.2178/bsl/1203350879 fatcat:mzffx4yqzfcg7ell47v452f2eu
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