Complexity of Stability

Authors Fabian Frei , Edith Hemaspaandra , Jörg Rothe



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Author Details

Fabian Frei
  • Department of Computer Science, ETH Zürich, Switzerland
Edith Hemaspaandra
  • Department of Computer Science, Rochester Institute of Technology, NY, USA
Jörg Rothe
  • Institut für Informatik, Heinrich-Heine-Universität Düsseldorf, Germany

Acknowledgements

We thank the anonymous referees for their careful reading of this paper and suggestions for improvement.

Cite AsGet BibTex

Fabian Frei, Edith Hemaspaandra, and Jörg Rothe. Complexity of Stability. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ISAAC.2020.19

Abstract

Graph parameters such as the clique number, the chromatic number, and the independence number are central in many areas, ranging from computer networks to linguistics to computational neuroscience to social networks. In particular, the chromatic number of a graph (i.e., the smallest number of colors needed to color all vertices such that no two adjacent vertices are of the same color) can be applied in solving practical tasks as diverse as pattern matching, scheduling jobs to machines, allocating registers in compiler optimization, and even solving Sudoku puzzles. Typically, however, the underlying graphs are subject to (often minor) changes. To make these applications of graph parameters robust, it is important to know which graphs are stable for them in the sense that adding or deleting single edges or vertices does not change them. We initiate the study of stability of graphs for such parameters in terms of their computational complexity. We show that, for various central graph parameters, the problem of determining whether or not a given graph is stable is complete for Θ₂ᵖ, a well-known complexity class in the second level of the polynomial hierarchy, which is also known as "parallel access to NP."

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Complexity classes
  • Mathematics of computing → Extremal graph theory
Keywords
  • Stability
  • Robustness
  • Complexity
  • Local Modifications
  • Colorability
  • Vertex Cover
  • Clique
  • Independent Set
  • Satisfiability
  • Unfrozenness
  • Criticality
  • DP
  • coDP
  • Parallel Access to NP

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References

  1. Douglas Bauer, Frank Harary, Juhani Nieminen, and Charles L. Suffel. Domination alteration sets in graphs. Discrete Mathematics, 47:153-161, 1983. Google Scholar
  2. Adam Beacham and Joseph C. Culberson. On the complexity of unfrozen problems. Discrete Applied Mathematics, 153(1-3):3-24, 2005. Google Scholar
  3. Béla Bollobás. Modern Graph Theory. Springer-Verlag, 1998. Google Scholar
  4. Béla Bollobás. Extremal Graph Theory. London Mathematical Society Monographs. Oxford University Press, London, 1978. Google Scholar
  5. Elisabet Burjons, Fabian Frei, Edith Hemaspaandra, Dennis Komm, and David Wehner. Finding optimal solutions with neighborly help. In Proceedings of the 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019), volume 138 of Leibniz International Proceedings in Informatics (LIPIcs), pages 78:1-78:14. Schloss Dagstuhl, 2019. Google Scholar
  6. Jin-Yi Cai and Gabriele E. Meyer. Graph minimal uncolorability is D^P-complete. SIAM Journal on Computing, 16(2):259-277, 1987. Google Scholar
  7. Gregory J. Chaitin. Register allocation and spilling via graph coloring. ACM SIGPLAN Notices - Proceedings of the 1982 SIGPLAN Symposium on Compiler Construction, 17(6):98-101, 1982. Google Scholar
  8. R. Chang and J. Kadin. The Boolean hierarchy and the polynomial hierarchy: A closer connection. SIAM Journal on Computing, 25(2):340-354, 1996. Google Scholar
  9. Richard Chang and Jim Kadin. On computing boolean connectives of characteristic functions. Mathematical Systems Theory, 28(3):173-198, 1995. Google Scholar
  10. Kamalika Chaudhuri, Fan Chung, and Mohammad Sh. Jamall. A network coloring game. In Proceedings of the 4th International Workshop On Internet And Network Economics, pages 522-530. Springer-Verlag Lecture Notes in Computer Science #5385, December 2008. Google Scholar
  11. Guantao Chen and Guangming Jing. Structural properties of edge-chromatic critical multigraphs. Journal of Combinatorial Theory, Series B, 139:128-162, 2019. Google Scholar
  12. Wyatt J. Desormeaux, Teresa W. Haynes, and Michael A. Henning. Total domination critical and stable graphs upon edge removal. Discrete Applied Mathematics, 158(15):1587-1592, 2010. Google Scholar
  13. Gabriel A. Dirac. Some theorems on abstract graphs. Proceedings of the London Mathematical Society, s3-2(1):69-81, 1952. Google Scholar
  14. Paul Erdős and Tibor Gallai. On the minimal number of vertices representing the edges of a graph. Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 6(1-2):181-203, 1961. Google Scholar
  15. Martin G. Everett and Steve Borgatti. Role colouring a graph. Mathematical Social Sciences, 21(2):183-188, 1991. Google Scholar
  16. Fabian Frei, Edith Hemaspaandra, and Jörg Rothe. Complexity of stability. Technical Report arXiv:1910.00305 [cs.CC], arXiv.org, September 2020. URL: http://arxiv.org/abs/1910.00305.
  17. Georg Gunther, Bert Hartnell, and Douglas F. Rall. Graphs whose vertex independence number is unaffected by single edge addition or deletion. Discrete Applied Mathematics, 46:167-172, 1993. Google Scholar
  18. Frank Harary. Graph Theory. Addison-Wesley, 1969. Google Scholar
  19. Frank Harary and Carsten Thomassen. Anticritcal graphs. Mathematical Proceedings of the Cambridge Philosophical Society, 79:11-18, 1976. Google Scholar
  20. Teresa W. Haynes, Robert C. Brigham, and Ronald D. Dutton. Extremal graphs domination insensitive to the removal of k edges. Discrete Applied Mathematics, 44(1-3):295-304, 1993. Google Scholar
  21. Lane A. Hemachandra. The strong exponential hierarchy collapses. Journal of Computer and System Sciences, 39(3):299-322, 1989. Google Scholar
  22. Edith Hemaspaandra, Lane A. Hemaspaandra, and Jörg Rothe. Exact analysis of Dodgson elections: Lewis Carroll’s 1876 voting system is complete for parallel access to NP. Journal of the ACM, 44(6):806-825, 1997. Google Scholar
  23. Edith Hemaspaandra, Lane A. Hemaspaandra, and Jörg Rothe. The complexity of online manipulation of sequential elections. Journal of Computer and System Sciences, 80(4):697-710, 2014. Google Scholar
  24. Edith Hemaspaandra, Holger Spakowski, and Jörg Vogel. The complexity of Kemeny elections. Theoretical Computer Science, 349(3):382-391, 2005. Google Scholar
  25. Michael A. Henning and Marcin Krzywkowski. Total domination stability in graphs. Discrete Applied Mathematics, 236:246-255, 2018. Google Scholar
  26. Matthew O. Jackson. Social and Economic Networks. Princeton University Press, 2008. Google Scholar
  27. Michael Kearns, Siddharth Suri, and Nick Montfort. An experimental study of the coloring problem on human subject networks. Science, 313(5788):824-827, 2006. Google Scholar
  28. Susan Khor. Application of graph coloring to biological networks. IET Systems Biology, 4(3):185-192, 2010. Google Scholar
  29. Frank Th. Leighton. A graph coloring algorithm for large scheduling problems. Journal of Research of the National Bureau of Standards, 84(6):489-506, 1979. Google Scholar
  30. Rémi Monassen and Riccardo Zecchina. Statistical mechanics of the random k-satisfiability model. The American Physical Society, 56(2):1357-1370, 1997. Google Scholar
  31. Rémi Monassen, Riccardo Zecchina, Scott Kirkpatrick, Bart Selman, and Lidror Troyansky. Determining computational complexity from characteristic `phase transitions'. Nature, 400:133-137, 1998. Google Scholar
  32. Christos H. Papadimitriou and David Wolfe. The complexity of facets resolved. Journal of Computer and System Sciences, 37(1):2-13, 1988. Google Scholar
  33. Christos H. Papadimitriou and Mihalis Yannakakis. The complexity of facets (and some facets of complexity). Journal of Computer and System Sciences, 28(2):244-259, 1984. Google Scholar
  34. Jörg Rothe. Exact complexity of Exact-Four-Colorability. Information Processing Letters, 87(1):7-12, 2003. Google Scholar
  35. Jörg Rothe, Holger Spakowski, and Jörg Vogel. Exact complexity of the winner problem for Young elections. Theory of Computing Systems, 36(4):375-386, 2003. Google Scholar
  36. H. Spakowski and J. Vogel. Θ₂^p-completeness: A classical approach for new results. In Proceedings of the 20th Conference on Foundations of Software Technology and Theoretical Computer Science, volume 1974 of Lecture Notes in Computer Science, pages 348-360. Springer-Verlag, December 2000. Google Scholar
  37. Jimeng Sun, Charalampos E. Tsourakakis, Evan Hoke, Christos Faloutsos, and Tina Eliassi-Rad. Two heads better than one: Pattern discovery in time-evolving multi-aspect data. Data Mining and Knowledge Discovery, 17(1):111-128, 2008. Google Scholar
  38. Klaus W. Wagner. More complicated questions about maxima and minima, and some closures of NP. Theoretical Computer Science, 51(1-2):53-80, 1987. Google Scholar
  39. Klaus W. Wagner. Bounded query classes. SIAM Journal on Computing, 19(5):833-846, 1990. Google Scholar
  40. Walter Wessel. Criticity with respect to properties and operations in graph theory. In László Lovász András Hajnal and Vera T. Sós, editors, Finite and Infinite Sets. (6th Hungarian Combinatorial Colloquium, Eger, 1981), volume 2 of Colloquia Mathematica Societatis Janos Bolyai, pages 829-837. North-Holland, 1984. Google Scholar
  41. Gerhard J. Woeginger. Core stability in hedonic coalition formation. In Proceedings of the 39th International Conference on Current Trends in Theory and Practice of Computer Science, volume 7741 of Lecture Notes in Computer Science, pages 33-50. Springer-Verlag, 2013. Google Scholar
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