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Theory of Computing Systems

, Volume 62, Issue 5, pp 1125–1143 | Cite as

Bounded-Depth Succinct Encodings and the Structure they Imply on Graphs

  • Patrick ScharpfeneckerEmail author
Article
  • 85 Downloads

Abstract

We study the complexity of graph problems succinctly encoded by bounded depth circuits and the existence of upward translation theorems for these models. While almost all succinct encodings have an upward translation theorem for some type of reduction, we prove that such theorems for CNF- and DNF-encoded graphs and for the most studied reductions do not exist. In contrast, we show that there are upward translation theorems for AC0 circuits of depth at least 3. This implies that the complexity of the succinct versions of problems complete for NP (under quantifier-free reductions), encoded by such circuits, have an exponential blow-up. We adapt these results to problems on explicitly given graphs with the same structural properties as graphs encoded by bounded depth circuits. We define a graph class hierarchy \(\mathcal {I}^{k}\) which consists of at most k alternating unions and intersections of edge-sets in \(\mathcal {I}^{0}\), a class which only consists of single bicliques. We show that the complexity of every NP-complete problem (under quantifier-free reductions) collapses to the second level: on graphs in \(\mathcal {I}^{2}\) the problem is already NP-complete. Finally, we show that by restricting \(\mathcal {I}^{2}\) to use only sub-logarithmic many intersections we get graphs for which Dominating Set is not NP-complete unless the Exponential Time Hypothesis is false. In contrast, a degree of O(log n) is enough for NP-completeness. Therefore, Dominating Set on \(\mathcal {I}^{2}\) graphs with intersection-degree O(log δ (n)) has either a spontaneous transition from P (for all δ < 1) to NP-complete (for δ = 1), or is NP-intermediate on the restricted graph class (for some δ < 1).

Keywords

Complexity Succinct Circuits Biclique Graphs NP 

References

  1. 1.
    Alon, N.: Covering graphs by the minimum number of equivalence relations. Combinatorica 6(3), 201–206 (1986). doi: 10.1007/BF02579381 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Babai, L., Frankl, P., Simon, J.: Complexity classes in communication complexity theory Proceedings of the 27th Annual Symposium on Foundations of Computer Science, SFCS’86, pp 337–347. IEEE Computer Society, Washington (1986), doi: 10.1109/SFCS.1986.15 Google Scholar
  3. 3.
    Balcázar, J.L., Lozano, A., Torán, J.: The complexity of algorithmic problems on succinct instances. Computer Science, Research and Applications, pp 351–377. Springer, US (1992)Google Scholar
  4. 4.
    Chlebík, M., Chlebíková, J.: Approximation hardness of dominating set problems in bounded degree graphs. Inf. Comput. 206(11), 1264–1275 (2008). doi: 10.1016/j.ic.2008.07.003 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cook, S.A.: A hierarchy for nondeterministic time complexity. J. Comput. Syst. Sci. 7(4), 343–353 (1973). doi: 10.1016/S0022-0000(73)80028-5 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dahlhaus, E.: Reduction to NP-complete problems by interpretations Logic and Machines: Decision Problems and Complexity, Proceedings of the Symposium “Rekursive Kombinatorik” held from May 23-28, 1983 at the Institut für Mathematische Logik und Grundlagenforschung der Universität Münster/Westfalen, pp. 357–365 (1983), doi: 10.1007/3-540-13331-3_51
  7. 7.
    Das, B., Scharpfenecker, P., Torán, J.: CNF And DNF succinct graph encodings Information and Computation. doi: 10.1016/j.ic.2016.06.009(2016)
  8. 8.
    Eaton, N., Rödl, V.: Graphs of small dimensions. Combinatorica 16(1), 59–85 (1996). doi: 10.1007/BF01300127 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ebbinghaus, H., Flum, J.: Finite Model Theory. Perspectives in Mathematical Logic, Springer (2005)Google Scholar
  10. 10.
    Eiter, T., Gottlob, G., Mannila, H.: Adding disjunction to datalog (extended abstract) Proceedings of the 13th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems - PODS’94, pp 267–278. ACM Press, New York (1994), doi: 10.1145/182591.182639 Google Scholar
  11. 11.
    Galperin, H., Wigderson, A.: Succinct representations of graphs. Inf. Control. 56(3), 183–198 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Garey, M., Johnson, D., Stockmeyer, L.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976). doi: 10.1016/0304-3975(76)90059-1 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Impagliazzo, R., Paturi, R.: On the Complexity of k-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001). doi: 10.1006/jcss.2000.1727 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Impagliazzo, R., Paturi, R., Zane, F.: Which Problems Have Strongly Exponential Complexity?. J. Comput. Syst. Sci. 63(4), 512–530 (2001). doi: 10.1006/jcss.2001.1774 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jukna, S.: On graph complexity. Comb. Probab. Comput. 15(6), 855–876 (2006). doi: 10.1017/S0963548306007620 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jukna, S.: On covering graphs by complete bipartite subgraphs. Discret. Math. 309(10), 3399–3403 (2009). doi: 10.1016/j.disc.2008.09.036 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jukna, S.: On set intersection representations of graphs. Journal of Graph Theory 61(1), 55–75 (2009). doi: 10.1002/jgt.20367 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mahaney, S.R.: Sparse complete sets for NP: Solution of a Conjecture of Berman and Hartmanis. J. Comput. Syst. Sci. 25(2), 130–143 (1982). doi: 10.1016/0022-0000(82)90002-2 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rödl, V., Ruciński, A.: Bipartite coverings of graphs. Comb. Probab. Comput. 6(3), 349–352 (1997). doi: 10.1017/S0963548397003064 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Seiferas, J.I., Fischer, M.J., Meyer, A.R.: Separating nondeterministic time complexity classes. J. ACM 25(1), 146–167 (1978). doi: 10.1145/322047.322061 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Stewart, I.A.: On completeness for NP via projection translations. Mathematical Systems Theory 27(2), 125–157 (1994). doi: 10.1007/BF01195200 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Veith, H.: Languages represented by Boolean formulas. Inf. Process. Lett. 63 (5), 251–256 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Veith, H.: How to encode a logical structure by an OBDD Proceedings 13th IEEE Conference on Computational Complexity, pp 122–131. IEEE Computer Society (1998)Google Scholar

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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institute of Theoretical Computer ScienceUniversity of UlmUlmGermany

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