Exponential Time Paradigms Through the Polynomial Time Lens

Authors Andrew Drucker, Jesper Nederlof, Rahul Santhanam

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Andrew Drucker
Jesper Nederlof
Rahul Santhanam

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Andrew Drucker, Jesper Nederlof, and Rahul Santhanam. Exponential Time Paradigms Through the Polynomial Time Lens. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 36:1-36:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


We propose a general approach to modelling algorithmic paradigms for the exact solution of NP-hard problems. Our approach is based on polynomial time reductions to succinct versions of problems solvable in polynomial time. We use this viewpoint to explore and compare the power of paradigms such as branching and dynamic programming, and to shed light on the true complexity of various problems. As one instantiation, we model branching using the notion of witness compression, i.e., reducibility to the circuit satisfiability problem parameterized by the number of variables of the circuit. We show this is equivalent to the previously studied notion of `OPP-algorithms', and provide a technique for proving conditional lower bounds for witness compressions via a constructive variant of AND-composition, which is a notion previously studied in theory of preprocessing. In the context of parameterized complexity we use this to show that problems such as Pathwidth and Treewidth and Independent Set parameterized by pathwidth do not have witness compression, assuming NP subseteq coNP/poly. Since these problems admit fast fixed parameter tractable algorithms via dynamic programming, this shows that dynamic programming can be stronger than branching, under a standard complexity hypothesis. Our approach has applications outside parameterized complexity as well: for example, we show if a polynomial time algorithm outputs a maximum independent set of a given planar graph on n vertices with probability exp(-n^{1-epsilon}) for some epsilon>0, then NP subseteq coNP/poly. This negative result dims the prospects for one very natural approach to sub-exponential time algorithms for problems on planar graphs. As two other illustrations (more exploratory) of our approach, we model algorithms based on inclusion-exclusion or group algebras via the notion of "parity compression", and we model a subclass of dynamic programming algorithms with the notion of "disjunctive dynamic programming". These models give us a way to naturally classify various parameterized problems with FPT algorithms. In the case of the dynamic programming model, we show that Independent Set parameterized by pathwidth is complete for this model.
  • exponential time paradigms
  • branching
  • dynamic programming
  • lower bounds


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  1. Michael Alekhnovich, Allan Borodin, Joshua Buresh-Oppenheim, Russell Impagliazzo, Avner Magen, and Toniann Pitassi. Toward a model for backtracking and dynamic programming. In 20th Annual IEEE Conference on Computational Complexity (CCC 2005), 11-15 June 2005, San Jose, CA, USA, pages 308-322. IEEE Computer Society, 2005. URL: http://dx.doi.org/10.1109/CCC.2005.32.
  2. Eric Allender, Shiteng Chen, Tiancheng Lou, Periklis A. Papakonstantinou, and Bangsheng Tang. Width-parametrized SAT: time-space tradeoffs. Theory of Computing, 10:297-339, 2014. URL: http://dx.doi.org/10.4086/toc.2014.v010a012.
  3. Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. On problems without polynomial kernels. J. Comput. Syst. Sci., 75(8):423-434, 2009. URL: http://dx.doi.org/10.1016/j.jcss.2009.04.001.
  4. Liming Cai and Jianer Chen. On the amount of nondeterminism and the power of verifying. SIAM Journal on Computing, 26(3):733-750, 1997. URL: http://dx.doi.org/10.1137/S0097539793258295.
  5. Michele Conforti, Gérard Cornuéjols, and Giacomo Zambelli. Extended formulations in combinatorial optimization. 4OR, 8(1):1-48, 2010. URL: http://dx.doi.org/10.1007/s10288-010-0122-z.
  6. Marek Cygan, Fedor Fomin, Bart M.P. Jansen, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, and Saket Saurabh Michal Pilipczuk. Open problems for fpt school 2014. Google Scholar
  7. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
  8. Evgeny Dantsin and Edward A. Hirsch. Satisfiability certificates verifiable in subexponential time. In Karem A. Sakallah and Laurent Simon, editors, Theory and Applications of Satisfiability Testing - SAT 2011 - 14th International Conference, SAT 2011, Ann Arbor, MI, USA, June 19-22, 2011. Proceedings, volume 6695 of Lecture Notes in Computer Science, pages 19-32. Springer, 2011. URL: http://dx.doi.org/10.1007/978-3-642-21581-0_4.
  9. 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: http://dx.doi.org/10.1145/2629620.
  10. Andrew Drucker. Nondeterministic direct product reductions and the success probability of SAT solvers. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 736-745. IEEE Computer Society, 2013. URL: http://dx.doi.org/10.1109/FOCS.2013.84.
  11. Andrew Drucker. New limits to classical and quantum instance compression. SIAM J. Comput., 44(5):1443-1479, 2015. URL: http://dx.doi.org/10.1137/130927115.
  12. David Eppstein. Quasiconvex analysis of multivariate recurrence equations for backtracking algorithms. ACM Trans. Algorithms, 2(4):492-509, October 2006. URL: http://dx.doi.org/10.1145/1198513.1198515.
  13. Lance Fortnow and Rahul Santhanam. Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci., 77(1):91-106, 2011. URL: http://dx.doi.org/10.1016/j.jcss.2010.06.007.
  14. Oded Goldreich. Computational complexity - a conceptual perspective. Cambridge University Press, 2008. Google Scholar
  15. Yuri Gurevich and Saharon Shelah. Expected computation time for hamiltonian path problem. SIAM J. Comput., 16(3):486-502, 1987. URL: http://dx.doi.org/10.1137/0216034.
  16. Paul Helman. A common schema for dynamic programming and branch and bound algorithms. Journal of the ACM, 36(1):97-128, 1989. Google Scholar
  17. Stefan Kratsch. Recent developments in kernelization: A survey. Bulletin of the EATCS, 113, 2014. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/285.
  18. Richard J. Lipton and Robert Endre Tarjan. Applications of a planar separator theorem. SIAM J. Comput., 9(3):615-627, 1980. URL: http://dx.doi.org/10.1137/0209046.
  19. Daniel Lokshtanov, Matthias Mnich, and Saket Saurabh. Planar k-path in subexponential time and polynomial space. In Petr Kolman and Jan Kratochvíl, editors, Graph-Theoretic Concepts in Computer Science - 37th International Workshop, WG 2011, Teplá Monastery, Czech Republic, June 21-24, 2011. Revised Papers, volume 6986 of Lecture Notes in Computer Science, pages 262-270. Springer, 2011. URL: http://dx.doi.org/10.1007/978-3-642-25870-1_24.
  20. Dániel Marx. What’s next? reductions other than kernelization. Talk at WorKer 2010: Workshop on Kernelization, Leiden, The Netherlands, 2010. Google Scholar
  21. Dániel Marx. What’s next? future directions in parameterized complexity. In Hans L. Bodlaender, Rod Downey, Fedor V. Fomin, and Dániel Marx, editors, The Multivariate Algorithmic Revolution and Beyond - Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday, volume 7370 of Lecture Notes in Computer Science, pages 469-496. Springer, 2012. URL: http://dx.doi.org/10.1007/978-3-642-30891-8_20.
  22. Jesper Nederlof. Saving space by algebraization. Talks at seminars in Utrecht (January 8) and UiB (Februari 11), UiB ICT Research school (May 18), STOC, Dagstuhl `Exact Complexity of NP-hard Problems', 2010. Google Scholar
  23. Jesper Nederlof. Space and Time Efficient Structural Improvements of Dynamic Programming Algorithms. PhD thesis, University of Bergen, 2011. URL: http://www.win.tue.nl/~jnederlo/PhDThesis.pdf.
  24. Ramamohan Paturi and Pavel Pudlák. On the complexity of circuit satisfiability. In Leonard J. Schulman, editor, Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 241-250. ACM, 2010. URL: http://dx.doi.org/10.1145/1806689.1806724.
  25. Michal Pilipczuk and Marcin Wrochna. On space efficiency of algorithms working on structural decompositions of graphs. In Nicolas Ollinger and Heribert Vollmer, editors, 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016, February 17-20, 2016, Orléans, France, volume 47 of LIPIcs, pages 57:1-57:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2016.57.
  26. Rahul Santhanam. On separators, segregators and time versus space. In Computational Complexity, 16th Annual IEEE Conference on, 2001., pages 286-294, 2001. URL: http://dx.doi.org/10.1109/CCC.2001.933895.
  27. Uwe Schöning. A probabilistic algorithm for k-sat and constraint satisfaction problems. In 40th Annual Symposium on Foundations of Computer Science, FOCS'99, 17-18 October, 1999, New York, NY, USA, pages 410-414. IEEE Computer Society, 1999. URL: http://dx.doi.org/10.1109/SFFCS.1999.814612.
  28. Seinosuke Toda. PP is as hard as the polynomial-time hierarchy. SIAM J. Comput., 20(5):865-877, 1991. URL: http://dx.doi.org/10.1137/0220053.
  29. Leslie G. Valiant and Vijay V. Vazirani. NP is as easy as detecting unique solutions. Theor. Comput. Sci., 47(3):85-93, 1986. URL: http://dx.doi.org/10.1016/0304-3975(86)90135-0.
  30. Magnus Wahlström. Abusing the tutte matrix: An algebraic instance compression for the K-set-cycle problem. In Natacha Portier and Thomas Wilke, editors, 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, February 27 - March 2, 2013, Kiel, Germany, volume 20 of LIPIcs, pages 341-352. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2013. URL: http://www.dagstuhl.de/dagpub/978-3-939897-50-7, URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2013.341.
  31. Ryan Williams. Inductive time-space lower bounds for sat and related problems. computational complexity, 15(4):433-470, 2006. Google Scholar
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