Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
skip to main content
research-article

On some FPT problems without polynomial Turing compressions

Published: 22 February 2022 Publication History

Abstract

An impressive hardness theory which can prove compression lower bounds for a large number of FPT problems has been established under the assumption that NP ⊈ coNP/poly. However, there are no problems in FPT for which the existence of polynomial Turing compressions under any widely believed complexity assumptions has been excluded. In this paper, we provide a technique which can be used to prove that some FPT problems have no small Turing compressions under the assumption that there exists a problem in NP which does not have small-sized circuits. These FPT problems, which include edge clique cover parameterized by the number of cliques, integer linear programming and a -choosability parameterized by some structural parameters, etc. have the property that they remain NP-hard under Cook reductions even if their parameter values are small. Moreover, a trade-off between the size of the Turing compression lower bound and the robustness of the complexity assumption is obtained. In particular, we demonstrate that these FPT problems have no polynomial Turing compressions unless every set in NP has quasi-polynomial-sized circuits, and have no 2 o ( k ) Turing compressions unless every set in NP has sub-exponential-sized circuits. Additionally, Turing kernelization lower bounds for these FPT problems are provided under some weaker complexity assumptions. Lastly, compression lower bounds for the above-mentioned FPT problems are proved under some complexity assumptions which are weaker than NP ⊈ coNP/poly, moreover, these results are proved under a method which is different from the previous hardness theory for compression lower bounds.

References

[1]
M. Cygan, F.V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, S. Saurabh, Parameterized Algorithms, Springer, 2015,.
[2]
R.G. Downey, M.R. Fellows, Fundamentals of Parameterized Complexity, Texts in Computer Science, Springer, 2013,.
[3]
N. Alon, G.Z. Gutin, E.J. Kim, S. Szeider, A. Yeo, Solving MAX-r-SAT above a tight lower bound, Algorithmica 61 (3) (2011) 638–655,.
[4]
M. Wahlström, Abusing the tutte matrix: an algebraic instance compression for the K-set-cycle problem, in: 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, February 27 - March 2, 2013, Kiel, Germany, 2013, pp. 341–352,.
[5]
H.L. Bodlaender, R.G. Downey, M.R. Fellows, D. Hermelin, On problems without polynomial kernels, J. Comput. Syst. Sci. 75 (8) (2009) 423–434,.
[6]
H. Dell, D. van Melkebeek, Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses, J. ACM 61 (4) (2014),.
[7]
A. Drucker, New limits to classical and quantum instance compression, SIAM J. Comput. 44 (5) (2015) 1443–1479,.
[8]
L. Fortnow, R. Santhanam, Infeasibility of instance compression and succinct PCPs for NP, J. Comput. Syst. Sci. 77 (1) (2011) 91–106,.
[9]
D. Binkele-Raible, H. Fernau, F.V. Fomin, D. Lokshtanov, S. Saurabh, Y. Villanger, Kernel(s) for problems with no kernel: on out-trees with many leaves, ACM Trans. Algorithms 8 (4) (2012),.
[10]
A.M. Ambalath, R. Balasundaram, H. Chintan Rao, V. Koppula, N. Misra, G. Philip, M.S. Ramanujan, On the kernelization complexity of colorful motifs, in: Parameterized and Exact Computation - 5th International Symposium, IPEC 2010, Chennai, India, December 13-15, 2010. Proceedings, 2010, pp. 14–25,.
[11]
H.L. Bodlaender, B.M.P. Jansen, S. Kratsch, Kernelization lower bounds by cross-composition, SIAM J. Discrete Math. 28 (1) (2014) 277–305,.
[12]
H. Donkers, B.M.P. Jansen, A Turing kernelization dichotomy for structural parameterizations of F-minor-free deletion, in: Graph-Theoretic Concepts in Computer Science - 45th International Workshop, WG 2019, Vall de Núria, Spain, June 19-21, 2019, Revised Papers, 2019, pp. 106–119,.
[13]
B.M.P. Jansen, Turing kernelization for finding long paths and cycles in restricted graph classes, J. Comput. Syst. Sci. 85 (2017) 18–37,.
[14]
B.M.P. Jansen, D. Marx, Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernels, in: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, 2015, pp. 616–629,.
[15]
B.M.P. Jansen, M. Pilipczuk, M. Wrochna, Turing kernelization for finding long paths in graphs excluding a topological minor, in: 12th International Symposium on Parameterized and Exact Computation, IPEC 2017, September 6-8, 2017, Vienna, Austria, 2017, pp. 23:1–23:13,.
[16]
A. Schäfer, C. Komusiewicz, H. Moser, R. Niedermeier, Parameterized computational complexity of finding small-diameter subgraphs, Optim. Lett. 6 (5) (2012) 883–891,.
[17]
S. Thomassé, N. Trotignon, K. Vuskovic, A polynomial Turing-kernel for weighted independent set in bull-free graphs, Algorithmica 77 (3) (2017) 619–641,.
[18]
D. Hermelin, S. Kratsch, K. Soltys, M. Wahlström, X. Wu, A completeness theory for polynomial (Turing) kernelization, Algorithmica 71 (3) (2015) 702–730,.
[19]
M. Cygan, F.V. Fomin, B.M.P. Jansen, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, S. Saurabh, Open problems for FPT school 2014, in: Dagstuhl Seminars or Workshop on Kernels, 2014, pp. 1–352. http://fptschool.mimuw.edu.pl/opl.pdf.
[20]
S. Kratsch, Recent developments in kernelization: a survey, Bull. Eur. Assoc. Theor. Comput. Sci. 113 (2014) http://eatcs.org/beatcs/index.php/beatcs/article/view/285.
[21]
N. Misra, V. Raman, S. Saurabh, Lower bounds on kernelization, Discrete Optim. 8 (1) (2011) 110–128,.
[22]
M. Cygan, M. Pilipczuk, M. Pilipczuk, Known algorithms for edge clique cover are probably optimal, SIAM J. Comput. 45 (1) (2016) 67–83,.
[23]
S. Chandran, D. Issac, A. Karrenbauer, On the parameterized complexity of biclique cover and partition, in: 11th International Symposium on Parameterized and Exact Computation, IPEC 2016, August 24-26, 2016, Aarhus, Denmark, 2016, pp. 11:1–11:13,.
[24]
V.B. Le, S. Peng, On the complete width and edge clique cover problems, J. Comb. Optim. 36 (2) (2018) 532–548,.
[25]
M. Cygan, S. Kratsch, M. Pilipczuk, M. Pilipczuk, M. Wahlström, Clique cover and graph separation: new incompressibility results, ACM Trans. Comput. Theory 6 (2) (2014),.
[26]
S. Arora, B. Barak, Computational Complexity - A Modern Approach, Cambridge University Press, 2009, http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521424264.
[27]
N. Pippenger, M.J. Fischer, Relations among complexity measures, J. ACM 26 (2) (1979) 361–381,.
[28]
Eisenbrand, F.; Hunkenschröder, C.; Klein, K.; Koutecký, M.; Levin, A.; Onn, S. (2019): An algorithmic theory of integer programming. CoRR http://arxiv.org/abs/1904.01361.
[29]
J. Gramm, J. Guo, F. Hüffner, R. Niedermeier, Data reduction and exact algorithms for clique cover, ACM J. Exp. Algorithmics 13 (2008),.
[30]
J. Orlin, Contentment in graph theory: covering graphs with cliques, Indag. Math. 80 (5) (1977) 406–424,.
[31]
I. Nor, D. Hermelin, S. Charlat, J. Engelstadter, M. Reuter, O. Duron, M. Sagot, Mod/resc parsimony inference: theory and application, Inf. Comput. 213 (2012) 23–32,.
[32]
M. Chang, L. Hung, T. Kloks, S. Peng, Block-graph width, Theor. Comput. Sci. 412 (23) (2011) 2496–2502,.
[33]
D. Marx, V. Mitsou, Double-exponential and triple-exponential bounds for choosability problems parameterized by treewidth, in: 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, 2016, pp. 28:1–28:15,.
[34]
M.R. Fellows, F.V. Fomin, D. Lokshtanov, F.A. Rosamond, S. Saurabh, S. Szeider, C. Thomassen, On the complexity of some colorful problems parameterized by treewidth, Inf. Comput. 209 (2) (2011) 143–153,.
[35]
S. Gutner, M. Tarsi, Some results on (a: b)-choosability, Discrete Math. 309 (8) (2009) 2260–2270,.
[36]
D. Knop, M. Pilipczuk, M. Wrochna, Tight complexity lower bounds for integer linear programming with few constraints, in: 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019, March 13-16, 2019, Berlin, Germany, 2019, pp. 44:1–44:15,.
[37]
R. Ganian, S. Ordyniak, The complexity landscape of decompositional parameters for ILP, Artif. Intell. 257 (2018) 61–71,.
[38]
M. Koutecký, A. Levin, S. Onn, A parameterized strongly polynomial algorithm for block structured integer programs, in: 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, 2018, pp. 85:1–85:14,.
[39]
R.M. Karp, Reducibility among combinatorial problems, in: Proceedings of a Symposium on the Complexity of Computer Computations, Held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA, 1972, pp. 85–103,.
[40]
G. Pan, M.Y. Vardi, Fixed-parameter hierarchies inside PSPACE, in: 21th IEEE Symposium on Logic in Computer Science (LICS 2006), 12-15 August 2006, Seattle, WA, USA, Proceedings, 2006, pp. 27–36,.
[41]
S.A. Cook, The complexity of theorem-proving procedures, in: M.A. Harrison, R.B. Banerji, J.D. Ullman (Eds.), Proceedings of the 3rd Annual ACM Symposium on Theory of Computing, May 3-5, 1971, Shaker Heights, Ohio, USA, ACM, 1971, pp. 151–158,.
[42]
R.M. Karp, R.J. Lipton, Some connections between nonuniform and uniform complexity classes, in: Proceedings of the 12th Annual ACM Symposium on Theory of Computing, April 28-30, 1980, Los Angeles, California, USA, 1980, pp. 302–309,.
[43]
S.R. Mahaney, Sparse complete sets for NP: solution of a conjecture of berman and hartmanis, in: 21st Annual Symposium on Foundations of Computer Science, Syracuse, New York, USA, 13-15 October 1980, IEEE Computer Society, 1980, pp. 54–60,.
[44]
H. Buhrman, J.M. Hitchcock, NP-hard sets are exponentially dense unless coNP ⊆ NP/poly, in: Proceedings of the 23rd Annual IEEE Conference on Computational Complexity, CCC 2008, 23-26 June 2008, College Park, Maryland, USA, 2008, pp. 1–7,.
[45]
J. Cai, S 2 p ⊆ ZP P NP, in: 42nd Annual Symposium on Foundations of Computer Science, FOCS 2001, 14-17 October 2001, las Vegas, Nevada, USA, 2001, pp. 620–629,.
[46]
H. Buhrman, S. Homer, Superpolynomial circuits, almost sparse oracles and the exponential hierarchy, in: Foundations of Software Technology and Theoretical Computer Science, 12th Conference, New Delhi, India, December 18-20, 1992, Proceedings, 1992, pp. 116–127,.
[47]
A. Pavan, A.L. Selman, S. Sengupta, N.V. Vinodchandran, Polylogarithmic-round interactive proofs for coNP collapse the exponential hierarchy, Theor. Comput. Sci. 385 (1–3) (2007) 167–178,.
[48]
R. Impagliazzo, V. Kabanets, A. Kolokolova, P. McKenzie, S. Romani, Does looking inside a circuit help?, in: 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017, August 21-25, 2017 - Aalborg, Denmark, 2017, pp. 1:1–1:13,.
[49]
R. Paturi, P. Pudlák, On the complexity of circuit satisfiability, in: Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, 2010, pp. 241–250,.
[50]
R. Impagliazzo, R. Paturi, F. Zane, Which problems have strongly exponential complexity?, J. Comput. Syst. Sci. 63 (4) (2001) 512–530,.
[51]
S. Homer, L. Longpré, On reductions of NP sets to sparse sets, J. Comput. Syst. Sci. 48 (2) (1994) 324–336,.
[52]
J. Chen, H. Fernau, I.A. Kanj, G. Xia, Parametric duality and kernelization: lower bounds and upper bounds on kernel size, SIAM J. Comput. 37 (4) (2007) 1077–1106,.
[53]
Y. Chen, J. Flum, M. Müller, Lower bounds for kernelizations and other preprocessing procedures, Theory Comput. Syst. 48 (4) (2011) 803–839,.
[54]
H. Fernau, T. Fluschnik, D. Hermelin, A. Krebs, H. Molter, R. Niedermeier, Diminishable parameterized problems and strict polynomial kernelization, in: Sailing Routes in the World of Computation - 14th Conference on Computability in Europe, CiE 2018, Kiel, Germany, July 30 - August 3, 2018, Proceedings, 2018, pp. 161–171,.

Cited By

View all

Index Terms

  1. On some FPT problems without polynomial Turing compressions
            Index terms have been assigned to the content through auto-classification.

            Recommendations

            Comments

            Information & Contributors

            Information

            Published In

            cover image Theoretical Computer Science
            Theoretical Computer Science  Volume 905, Issue C
            Feb 2022
            106 pages

            Publisher

            Elsevier Science Publishers Ltd.

            United Kingdom

            Publication History

            Published: 22 February 2022

            Author Tags

            1. Turing compression
            2. Turing kernelization
            3. Parameterized complexity
            4. Compression
            5. Kernelization

            Qualifiers

            • Research-article

            Contributors

            Other Metrics

            Bibliometrics & Citations

            Bibliometrics

            Article Metrics

            • Downloads (Last 12 months)0
            • Downloads (Last 6 weeks)0
            Reflects downloads up to 15 Oct 2024

            Other Metrics

            Citations

            Cited By

            View all

            View Options

            View options

            Get Access

            Login options

            Media

            Figures

            Other

            Tables

            Share

            Share

            Share this Publication link

            Share on social media