Abstract
This article analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP ⊈ coNP/poly, no efficient preprocessing algorithm can reduce n-variable instances of cnf-sat with d literals per clause to equivalent instances with O(nd−ε ) bits for any ε > 0. For the Not-All-Equal sat problem, a compression to size Õ(nd−1) exists. We put these results in a common framework by analyzing the compressibility of CSPs with a binary domain. We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several settings. Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role. For example, for Exact Satisfiability with unbounded clause length it is possible to efficiently reduce the number of constraints to n+1, yet no polynomial-time algorithm can reduce to an equivalent instance with O(n2−ε ) bits for any ε > 0, unless NP ⊆ coNP/poly.
- David A. Mix Barrington. 1992. Some problems involving Razborov-Smolensky polynomials. In Proceedings of the London Mathematical Society Symposium on Boolean Function Complexity. Cambridge University Press, 109--128. Google Scholar
Digital Library
- David A. Mix Barrington, Richard Beigel, and Steven Rudich. 1994. Representing Boolean functions as polynomials modulo composite numbers. Comput. Complex. 4, 4 (1994), 367--382. Google Scholar
Digital Library
- Richard Beigel. 1993. The polynomial method in circuit complexity. In Proceedings of the 8th CCC. 82--95.Google Scholar
Cross Ref
- Abhishek Bhowmick and Shachar Lovett. 2015. Nonclassical polynomials as a barrier to polynomial lower bounds. In Proceedings of the 30th CCC (LIPIcs), Vol. 33. 72--87. Google Scholar
Digital Library
- Hans L. Bodlaender. 2015. Kernelization, exponential lower bounds. In Encyclopedia of Algorithms. Springer.Google Scholar
- Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. 2014. Kernelization lower bounds by cross-composition. SIAM J. Discrete Math. 28, 1 (2014), 277--305.Google Scholar
Digital Library
- Hans L. Bodlaender, Stéphan Thomassé, and Anders Yeo. 2011. Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci. 412, 35 (2011), 4570--4578. Google Scholar
Digital Library
- Andrei A. Bulatov and Dániel Marx. 2014. Constraint satisfaction parameterized by solution size. SIAM J. Comput. 43, 2 (2014), 573--616.Google Scholar
Cross Ref
- Hubie Chen, Bart M. P. Jansen, and Astrid Pieterse. 2019. Best-case and worst-case sparsifiability of Boolean CSPs. In Proceedings of the 13th IPEC Vol. 115. 15:1--15:13.Google Scholar
- Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. 2015. Parameterized Algorithms. Springer. Google Scholar
Digital Library
- Holger Dell, Eun Jung Kim, Michael Lampis, Valia Mitsou, and Tobias Mömke. 2015. Complexity and approximability of parameterized MAX-CSPs. In Proceedings of the 10th IPEC (LIPIcs), Vol. 43. 294--306.Google Scholar
- Holger Dell and Dániel Marx. 2012. Kernelization of packing problems. In Proceedings of the 23rd SODA. 68--81. Google Scholar
Digital Library
- Holger Dell and Dieter van Melkebeek. 2014. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM 61, 4 (2014), 23:1--23:27. Google Scholar
Digital Library
- Michael Dom, Daniel Lokshtanov, and Saket Saurabh. 2009. Incompressibility through colors and IDs. In Proceedings of the 36th ICALP. 378--389. Google Scholar
Digital Library
- Michael Dom, Daniel Lokshtanov, and Saket Saurabh. 2014. Kernelization lower bounds through colors and IDs. ACM Trans. Algor. 11, 2 (2014), 13. Google Scholar
Digital Library
- Rodney G. Downey and Michael R. Fellows. 2013. Fundamentals of Parameterized Complexity. Springer. 1--769 pages. Google Scholar
Digital Library
- J. Flum and M. Grohe. 2006. Parameterized Complexity Theory. Springer-Verlag. Google Scholar
Digital Library
- Lance Fortnow and Rahul Santhanam. 2011. Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77, 1 (2011), 91--106. Google Scholar
Digital Library
- Ben Green and Terence Tao. 2008. The primes contain arbitrarily long arithmetic progressions. Ann. Math. 167, 2 (2008), 481--547.Google Scholar
Cross Ref
- Gregory Gutin. 2015. Kernelization: Constraint satisfaction problems parameterized above average. In Encyclopedia of Algorithms, Ming-Yang Kao (Ed.). Springer.Google Scholar
- Leslie Hogben. 2014. Handbook of Linear Algebra, Second Edition. Chapman and Hall/CRC.Google Scholar
- John A. Howell. 1986. Spans in the module (Z<sub>m</sub>)<sup>s</sup> . Linear Multilin. Alg. 19, 1 (1986), 67--77.Google Scholar
Cross Ref
- Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. 2001. Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63, 4 (2001), 512--530. Google Scholar
Digital Library
- Bart M. P. Jansen. 2015. On sparsification for computing treewidth. Algorithmica 71, 3 (2015), 605--635. Google Scholar
Digital Library
- Bart M. P. Jansen. 2016. Constrained bipartite vertex cover: The easy kernel is essentially tight. In Proceedings of the 33rd STACS (LIPIcs), Vol. 47. 45:1--45:13.Google Scholar
- Bart M. P. Jansen and Astrid Pieterse. 2017. Sparsification upper and lower bounds for graph problems and not-all-equal SAT. Algorithmica 79, 1 (2017), 3--28. Google Scholar
Digital Library
- Bart M. P. Jansen and Astrid Pieterse. 2018. Optimal data reduction for graph coloring using low-degree polynomials. In Proceedings of the 12th IPEC (2017), Vol. 89. 22:1--22:12.Google Scholar
- R. M. Karp. 1972. Reducibility among combinatorial problems. In Complexity of Computer Computations. Plenum Press, 85--103.Google Scholar
- Stefan Kratsch. 2014. Recent developments in kernelization: A survey. Bull. EATCS 113 (2014), 58--97.Google Scholar
- Stefan Kratsch, Dániel Marx, and Magnus Wahlström. 2016. Parameterized complexity and kernelizability of max ones and exact ones problems. ACM Trans. Comput. Theory 8, 1 (2016), 1. Google Scholar
Digital Library
- Stefan Kratsch and Magnus Wahlström. 2010. Preprocessing of min ones problems: A dichotomy. In Proceedings of the 37th ICALP (Lecture Notes in Computer Science), Vol. 6198. 653--665. Google Scholar
Digital Library
- Victor Lagerkvist and Magnus Wahlström. 2017. Kernelization of constraint satisfaction problems: A study through universal algebra. In Proceedings of the 23rd CP. 157--171.Google Scholar
Cross Ref
- Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. 2012. Kernelization—Preprocessing with a guarantee. In The Multivariate Algorithmic Revolution and Beyond—Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday (Lecture Notes in Computer Science), Vol. 7370. 129--161. Google Scholar
Digital Library
- Lásló Lovász. 1976. Chromatic number of hypergraphs and linear algebra. In Studia Scientiarum Mathematicarum Hungarica 11. 113--114. Retrieved from: http://real-j.mtak.hu/5461/.Google Scholar
- Rahul Santhanam and Srikanth Srinivasan. 2012. On the limits of sparsification. In Proceedings of the 39th ICALP. 774--785. Google Scholar
Digital Library
- Thomas J. Schaefer. 1978. The complexity of satisfiability problems. In Proceedings of the 10th ACM Symposium on Theory of Computing. 216--226. Google Scholar
Digital Library
- Arne Storjohann and Thom Mulders. 1998. Fast Algorithms for Linear Algebra Modulo N. Springer Berlin, 139--150. Google Scholar
Digital Library
- Gábor Tardos and David A. Mix Barrington. 1998. A lower bound on the mod 6 degree of the OR function. Comput. Complex. 7, 2 (1998), 99--108. Google Scholar
Digital Library
Index Terms
Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials
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