NP-intermediate

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Short description: Complexity class of problems

In computational complexity, problems that are in the complexity class NP but are neither in the class P nor NP-complete are called NP-intermediate, and the class of such problems is called NPI. Ladner's theorem, shown in 1975 by Richard E. Ladner,[1] is a result asserting that, if P ≠ NP, then NPI is not empty; that is, NP contains problems that are neither in P nor NP-complete. Since it is also true that if NPI problems exist, then P ≠ NP, it follows that P = NP if and only if NPI is empty.

Under the assumption that P ≠ NP, Ladner explicitly constructs a problem in NPI, although this problem is artificial and otherwise uninteresting. It is an open question whether any "natural" problem has the same property: Schaefer's dichotomy theorem provides conditions under which classes of constrained Boolean satisfiability problems cannot be in NPI.[2][3] Some problems that are considered good candidates for being NP-intermediate are the graph isomorphism problem, and decision versions of factoring and the discrete logarithm.

List of problems that might be NP-intermediate

Algebra and number theory

  • A decision version of factoring integers: for input [math]\displaystyle{ n }[/math] and [math]\displaystyle{ k }[/math], does [math]\displaystyle{ n }[/math] have a factor in the interval [math]\displaystyle{ [2,k] }[/math]?
  • Decision versions of the discrete logarithm problem and others related to cryptographic assumptions
  • Linear divisibility: given integers [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math], does [math]\displaystyle{ y }[/math] have a divisor congruent to 1 modulo [math]\displaystyle{ x }[/math]?[4][5]

Boolean logic

  • IMSAT, the Boolean satisfiability problem for "intersecting monotone CNF": conjunctive normal form, with each clause containing only positive or only negative terms, and each positive clause having a variable in common with each negative clause[6]
  • Minimum circuit size problem: given the truth table of a Boolean function and positive integer [math]\displaystyle{ s }[/math], does there exist a circuit of size at most [math]\displaystyle{ s }[/math] for this function?[7]
  • Monotone dualization: given CNF and DNF formulas for monotone Boolean functions, do they represent the same function?[8]
  • Monotone self-duality: given a CNF formula for a Boolean function, is the function invariant under a transformation that negates all of its variables and then negates the output value?[8]

Computational geometry and computational topology

Game theory

  • Determining the winner in parity games, in which graph vertices are labeled by which player chooses the next step, and the winner is determined by the parity of the highest-priority vertex reached[14]
  • Determining the winner for stochastic graph games, in which graph vertices are labeled by which player chooses the next step, or whether it is chosen randomly, and the winner is determined by reaching a designated sink vertex.[15]

Graph algorithms

Miscellaneous

References

  1. ↑ Ladner, Richard (1975). "On the Structure of Polynomial Time Reducibility". Journal of the ACM 22 (1): 155–171. doi:10.1145/321864.321877. 
  2. ↑ Grädel, Erich; Kolaitis, Phokion G.; Libkin, Leonid; Marx, Maarten; Spencer, Joel; Vardi, Moshe Y.; Venema, Yde; Weinstein, Scott (2007). Finite model theory and its applications. Texts in Theoretical Computer Science. An EATCS Series. Berlin: Springer-Verlag. p. 348. ISBN 978-3-540-00428-8. 
  3. ↑ Schaefer, Thomas J. (1978). "The complexity of satisfiability problems". Proc. 10th Ann. ACM Symp. on Theory of Computing. pp. 216–226. http://www.ccs.neu.edu/home/lieber/courses/csg260/f06/materials/papers/max-sat/p216-schaefer.pdf. 
  4. ↑ Adleman, Leonard; Manders, Kenneth (1977). "Proceedings of the 9th ACM Symp. on Theory of Computing (STOC '77)". doi:10.1145/800105.803405. 
  5. ↑ Papadimitriou, Christos H. (1994). Computational Complexity. Addison-Wesley. p. 236. ISBN 9780201530827. 
  6. ↑ Eiter, Thomas; Gottlob, Georg (2002). "Logics in Artificial Intelligence, European Conference, JELIA 2002, Cosenza, Italy, September, 23-26, Proceedings". in Flesca, Sergio; Greco, Sergio; Leone, Nicola et al.. 2424. Springer. pp. 549–564. doi:10.1007/3-540-45757-7_53. 
  7. ↑ Kabanets, Valentine; Cai, Jin-Yi (2000). "Proc. 32nd Symposium on Theory of Computing". Portland, Oregon, USA. pp. 73–79. doi:10.1145/335305.335314. ECCC TR99-045. 
  8. ↑ 8.0 8.1 Eiter, Thomas; Makino, Kazuhisa; Gottlob, Georg (2008). "Computational aspects of monotone dualization: a brief survey". Discrete Applied Mathematics 156 (11): 2035–2049. doi:10.1016/j.dam.2007.04.017. 
  9. ↑ Sleator, Daniel D.; Tarjan, Robert E.; Thurston, William P. (1988). "Rotation distance, triangulations, and hyperbolic geometry". Journal of the American Mathematical Society 1 (3): 647–681. doi:10.2307/1990951. 
  10. ↑ Skiena, Steven; Smith, Warren D.; Lemke, Paul (1990). "Proceedings of the Sixth Annual Symposium on Computational Geometry, Berkeley, CA, USA, June 6-8, 1990". in Seidel, Raimund. ACM. pp. 332–339. doi:10.1145/98524.98598. 
  11. ↑ Jansen, Klaus; Solis-Oba, Roberto (2011). "A polynomial time OPT + 1 algorithm for the cutting stock problem with a constant number of object lengths". Mathematics of Operations Research 36 (4): 743–753. doi:10.1287/moor.1110.0515. 
  12. ↑ Lackenby, Marc (2021). "The efficient certification of knottedness and Thurston norm". Advances in Mathematics 387: Paper No. 107796. doi:10.1016/j.aim.2021.107796. 
  13. ↑ "24 Geodesics: Lyusternik–Schnirelmann". Geometric folding algorithms: Linkages, origami, polyhedra. Cambridge: Cambridge University Press. 2007. pp. 372–375. doi:10.1017/CBO9780511735172. ISBN 978-0-521-71522-5. .
  14. ↑ Jurdziński, Marcin (1998). "Deciding the winner in parity games is in UP [math]\displaystyle{ \cap }[/math] co-UP". Information Processing Letters 68 (3): 119–124. doi:10.1016/S0020-0190(98)00150-1. 
  15. ↑ Condon, Anne (1992). "The complexity of stochastic games". Information and Computation 96 (2): 203–224. doi:10.1016/0890-5401(92)90048-K. 
  16. ↑ Grohe, Martin; Neuen, Daniel (June 2021). "Recent advances on the graph isomorphism problem". Surveys in Combinatorics 2021. Cambridge University Press. pp. 187–234. doi:10.1017/9781009036214.006. 
  17. ↑ Karpinski, Marek (2002). "Mathematical Foundations of Computer Science 2002, 27th International Symposium, MFCS 2002, Warsaw, Poland, August 26-30, 2002, Proceedings". in Diks, Krzysztof; Rytter, Wojciech. 2420. Springer. pp. 59–67. doi:10.1007/3-540-45687-2_4. 
  18. ↑ Gallian, Joseph A. (December 17, 2021). "A dynamic survey of graph labeling". Electronic Journal of Combinatorics 5: Dynamic Survey 6. https://www.combinatorics.org/Surveys/index.html. 
  19. ↑ Nishimura, N.; Ragde, P.; Thilikos, D.M. (2002). "On graph powers for leaf-labeled trees". Journal of Algorithms 42: 69–108. doi:10.1006/jagm.2001.1195. .
  20. ↑ "Clique-width is NP-complete". SIAM Journal on Discrete Mathematics 23 (2): 909–939. 2009. doi:10.1137/070687256. .
  21. ↑ Gassner, Elisabeth; Jünger, Michael; Percan, Merijam; Schaefer, Marcus; Schulz, Michael (2006). "Graph-Theoretic Concepts in Computer Science: 32nd International Workshop, WG 2006, Bergen, Norway, June 22-24, 2006, Revised Papers". 4271. Berlin: Springer. pp. 325–335. doi:10.1007/11917496_29. http://e-archive.informatik.uni-koeln.de/507/2/zaik2006-507.pdf. .
  22. ↑ "On limited nondeterminism and the complexity of the V-C dimension". Journal of Computer and System Sciences 53 (2, part 1): 161–170. 1996. doi:10.1006/jcss.1996.0058. 

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