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

Authors Bart M. P. Jansen, Astrid Pieterse

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  • sparsification
  • graph coloring
  • Hamiltonian cycle
  • satisfiability

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Abstract

We present several sparsification lower and upper bounds for classic problems in graph theory and logic. 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, of an arbitrary problem, with bitsize O(n^{2-epsilon}) for epsilon > 0, unless NP is a subset of coNP/poly and the polynomial-time hierarchy collapses. These results imply that existing linear-vertex kernels for k-Nonblocker and k-Max Leaf Spanning Tree (the parametric duals of (Connected) Dominating Set) cannot be improved to have O(k^{2-epsilon}) edges, unless NP is a subset of NP/poly. We also present a positive result and exhibit a non-trivial sparsification algorithm for d-Not-All-Equal-SAT. We give an algorithm that reduces an n-variable input with clauses of size at most d to an equivalent input with O(n^{d-1}) clauses, for any fixed d. Our algorithm is based on a linear-algebraic proof of Lovász that bounds the number of hyperedges in critically 3-chromatic d-uniform n-vertex hypergraphs by binom{n}{d-1}. We show that our kernel is tight under the assumption that NP is not a subset of NP/poly.

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Bart M. P. Jansen and Astrid Pieterse. Sparsification Upper and Lower Bounds for Graphs Problems and Not-All-Equal SAT. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 163-174, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.IPEC.2015.163

Author Details

Bart M. P. Jansen
Astrid Pieterse

References

  1. 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. Google Scholar
  2. Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernel bounds for path and cycle problems. Theor. Comput. Sci., 511:117-136, 2013. Google Scholar
  3. 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. Google Scholar
  4. Hans L. Bodlaender, Stéphan Thomassé, and Anders Yeo. Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci., 412(35):4570-4578, 2011. Google Scholar
  5. Marek Cygan, Fabrizio Grandoni, and Danny Hermelin. Tight kernel bounds for problems on graphs with small degeneracy. In Proc. 21st ESA, pages 361-372, 2013. Google Scholar
  6. Frank K. H. A. Dehne, Michael R. Fellows, Henning Fernau, Elena Prieto, and Frances A. Rosamond. NONBLOCKER: parameterized algorithmics for minimum dominating set. In Proc. 32nd SOFSEM, pages 237-245, 2006. Google Scholar
  7. Holger Dell and Dániel Marx. Kernelization of packing problems. In Proc. 23rd SODA, pages 68-81, 2012. Google Scholar
  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. Google Scholar
  9. Michael Dom, Daniel Lokshtanov, and Saket Saurabh. Kernelization lower bounds through colors and IDs. ACM Trans. Algorithms, 11(2):13:1-13:20, 2014. Google Scholar
  10. David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Amnon Nissenzweig. Sparsification - a technique for speeding up dynamic graph algorithms. J. ACM, 44(5):669-696, 1997. Google Scholar
  11. Vladimir Estivill-Castro, Michael Fellows, Michael Langston, and Frances Rosamond. FPT is P-time extremal structure I. In Proc. 1st ACiD, pages 1-41, 2005. Google Scholar
  12. Michael R. Fellows and Bart M. P. Jansen. FPT is characterized by useful obstruction sets: Connecting algorithms, kernels, and quasi-orders. TOCT, 6(4):16, 2014. Google Scholar
  13. Lance Fortnow and Rahul Santhanam. Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci., 77(1):91-106, 2011. Google Scholar
  14. Michael R. Garey and David S. Johnson. Computers and Intractability. W.H. Freeman, 1979. Google Scholar
  15. Danny Hermelin and Xi Wu. Weak compositions and their applications to polynomial lower bounds for kernelization. In Proc. 23rd SODA, pages 104-113, 2012. Google Scholar
  16. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. Google Scholar
  17. Bart M. P. Jansen. On sparsification for computing treewidth. Algorithmica, 71(3):605-635, 2015. Google Scholar
  18. Bart M. P. Jansen and Stefan Kratsch. Data reduction for graph coloring problems. Information and Computation, 231:70-88, 2013. Google Scholar
  19. Bart M. P. Jansen and Astrid Pieterse. Sparsification upper and lower bounds for graphs problems and not-all-equal SAT. CoRR, abs/1509.07437, 2015. Google Scholar
  20. Richard M. Karp. Reducibility Among Combinatorial Problems. In Complexity of Computer Computations, pages 85-103. Plenum Press, 1972. Google Scholar
  21. Lásló Lovász. Chromatic number of hypergraphs and linear algebra. In Studia Scientiarum Mathematicarum Hungarica 11, pages 113-114, 1976. Google Scholar
  22. George L. Nemhauser and Leslie E. Trotter Jr. Vertex packings: structural properties and algorithms. Math. Program., 8:232-248, 1975. Google Scholar
  23. Bjarne Toft. On the maximal number of edges of critical k-chromatic graphs. Studia Scientiarum Mathematicarum Hungarica, 5:461-470, 1970. Google Scholar
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