Document Open Access Logo

On the Hardness of Compressing Weights

Authors Bart M. P. Jansen , Shivesh K. Roy , Michał Włodarczyk



PDF
Thumbnail PDF

File

LIPIcs.MFCS.2021.64.pdf
  • Filesize: 1.25 MB
  • 21 pages

Document Identifiers

Author Details

Bart M. P. Jansen
  • Eindhoven University of Technology, The Netherlands
Shivesh K. Roy
  • Eindhoven University of Technology, The Netherlands
Michał Włodarczyk
  • Eindhoven University of Technology, The Netherlands

Cite AsGet BibTex

Bart M. P. Jansen, Shivesh K. Roy, and Michał Włodarczyk. On the Hardness of Compressing Weights. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 64:1-64:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.MFCS.2021.64

Abstract

We investigate computational problems involving large weights through the lens of kernelization, which is a framework of polynomial-time preprocessing aimed at compressing the instance size. Our main focus is the weighted Clique problem, where we are given an edge-weighted graph and the goal is to detect a clique of total weight equal to a prescribed value. We show that the weighted variant, parameterized by the number of vertices n, is significantly harder than the unweighted problem by presenting an 𝒪(n^{3 - ε}) lower bound on the size of the kernel, under the assumption that NP ̸ ⊆ coNP/poly. This lower bound is essentially tight: we show that we can reduce the problem to the case with weights bounded by 2^𝒪(n), which yields a randomized kernel of 𝒪(n³) bits. We generalize these results to the weighted d-Uniform Hyperclique problem, Subset Sum, and weighted variants of Boolean Constraint Satisfaction Problems (CSPs). We also study weighted minimization problems and show that weight compression is easier when we only want to {preserve the collection of} optimal solutions. Namely, we show that for node-weighted Vertex Cover on bipartite graphs it is possible to maintain the set of optimal solutions using integer weights from the range [1, n], but if we want to maintain the ordering of the weights of all inclusion-minimal solutions, then weights as large as 2^Ω(n) are necessary.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Problems, reductions and completeness
Keywords
  • kernelization
  • compression
  • edge-weighted clique
  • constraint satisfaction problems

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Amir Abboud, Shon Feller, and Oren Weimann. On the fine-grained complexity of parity problems. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), volume 168 of LIPIcs, pages 5:1-5:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.5.
  2. Amir Abboud, Kevin Lewi, and Ryan Williams. Losing weight by gaining edges. In Andreas S. Schulz and Dorothea Wagner, editors, Algorithms - ESA 2014 - 22th Annual European Symposium, Wroclaw, Poland, September 8-10, 2014. Proceedings, volume 8737 of Lecture Notes in Computer Science, pages 1-12. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-44777-2_1.
  3. Andreas Björklund. Determinant sums for undirected Hamiltonicity. SIAM J. Comput., 43(1):280-299, 2014. URL: https://doi.org/10.1137/110839229.
  4. Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM J. Discrete Math., 28(1):277-305, 2014. URL: https://doi.org/10.1137/120880240.
  5. Miroslav Chlebík and Janka Chlebíková. Crown reductions for the minimum weighted vertex cover problem. Discret. Appl. Math., 156(3):292-312, 2008. URL: https://doi.org/10.1016/j.dam.2007.03.026.
  6. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  7. Holger Dell and Dániel Marx. Kernelization of packing problems. In Yuval Rabani, editor, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 68-81. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.6.
  8. Holger Dell and Dieter van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM, 61(4):23:1-23:27, 2014. URL: https://doi.org/10.1145/2629620.
  9. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: https://doi.org/10.1007/978-1-4471-5559-1.
  10. Michael Etscheid, Stefan Kratsch, Matthias Mnich, and Heiko Röglin. Polynomial kernels for weighted problems. J. Comput. Syst. Sci., 84:1-10, 2017. URL: https://doi.org/10.1016/j.jcss.2016.06.004.
  11. Henning Fernau. Kernelization, Turing kernels. In Encyclopedia of Algorithms, pages 1043-1045. Springer, 2016. URL: https://doi.org/10.1007/978-1-4939-2864-4_528.
  12. András Frank and Éva Tardos. An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica, 7(1):49-65, 1987. Google Scholar
  13. Danny Harnik and Moni Naor. On the compressibility of NP instances and cryptographic applications. SIAM Journal on Computing, 39(5):1667-1713, 2010. URL: https://doi.org/10.1137/060668092.
  14. Alon Itai and Michael Rodeh. Finding a minimum circuit in a graph. SIAM J. Comput., 7(4):413-423, 1978. URL: https://doi.org/10.1137/0207033.
  15. Bart M. P. Jansen and Astrid Pieterse. Optimal sparsification for some binary CSPs using low-degree polynomials. TOCT, 11(4):28:1-28:26, 2019. URL: https://doi.org/10.1145/3349618.
  16. Bart M. P. Jansen, Shivesh K. Roy, and Michał Włodarczyk. On the hardness of compressing weights, 2021. URL: http://arxiv.org/abs/2107.02554.
  17. Bart M. P. Jansen and Michal Wlodarczyk. Optimal polynomial-time compression for Boolean Max CSP. In Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders, editors, 28th Annual European Symposium on Algorithms, ESA 2020, September 7-9, 2020, Pisa, Italy (Virtual Conference), volume 173 of LIPIcs, pages 63:1-63:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.63.
  18. Peter Jonsson and Andrei Krokhin. Maximum H-colourable subdigraphs and constraint optimization with arbitrary weights. Journal of Computer and System Sciences, 73(5):691-702, 2007. URL: https://doi.org/10.1016/j.jcss.2007.02.001.
  19. Richard 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, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  20. Richard M. Karp and Michael O. Rabin. Efficient randomized pattern-matching algorithms. IBM journal of research and development, 31(2):249-260, 1987. Google Scholar
  21. Dániel Marx and Michal Pilipczuk. Everything you always wanted to know about the parameterized complexity of subgraph isomorphism (but were afraid to ask), 2013. URL: http://arxiv.org/abs/1307.2187v3.
  22. Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. Matching is as easy as matrix inversion. Comb., 7(1):105-113, 1987. URL: https://doi.org/10.1007/BF02579206.
  23. Jesper Nederlof. Bipartite TSP in o(1.9999ⁿ) time, assuming quadratic time matrix multiplication. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 40-53. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384264.
  24. Jesper Nederlof, Erik Jan van Leeuwen, and Ruben van der Zwaan. Reducing a target interval to a few exact queries. In Branislav Rovan, Vladimiro Sassone, and Peter Widmayer, editors, Mathematical Foundations of Computer Science 2012 - 37th International Symposium, MFCS 2012, Bratislava, Slovakia, August 27-31, 2012. Proceedings, volume 7464 of Lecture Notes in Computer Science, pages 718-727. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-32589-2_62.
  25. G.L. Nemhauser and L.E.jun. Trotter. Vertex packings: structural properties and algorithms. Math. Program., 8:232-248, 1975. URL: https://doi.org/10.1007/BF01580444.
  26. Noam Nisan and Mario Szegedy. On the degree of Boolean functions as real polynomials. Computational Complexity, 4:301-313, 1994. URL: https://doi.org/10.1007/BF01263419.
  27. Virginia Vassilevska Williams. Hardness of easy problems: Basing hardness on popular conjectures such as the strong exponential time hypothesis (invited talk). In Thore Husfeldt and Iyad A. Kanj, editors, 10th International Symposium on Parameterized and Exact Computation, IPEC 2015, September 16-18, 2015, Patras, Greece, volume 43 of LIPIcs, pages 17-29. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.IPEC.2015.17.
  28. Virginia Vassilevska Williams and Ryan Williams. Finding, minimizing, and counting weighted subgraphs. In Michael Mitzenmacher, editor, Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, pages 455-464. ACM, 2009. URL: https://doi.org/10.1145/1536414.1536477.
  29. Virginia Vassilevska Williams and Ryan Williams. Finding, minimizing, and counting weighted subgraphs. SIAM J. Comput., 42(3):831-854, 2013. URL: https://doi.org/10.1137/09076619X.
  30. Virginia Vassilevska Williams and Ryan Williams. Subcubic equivalences between path, matrix, and triangle problems. J. ACM, 65(5):27:1-27:38, 2018. URL: https://doi.org/10.1145/3186893.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail