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On the Efficient Approximability of “HARD” Problems: A Survey

  • Harry B. HuntIII
  • Madhav V. Marathe
  • Richard E. Stearns
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 42)

Abstract

By a HARD problem, we mean a problem that is PSPACE-, DEXPTIME-, NEXPTIME-hard, etc. Many basic algorithmically-solvable problems, for quantified formulas, for sequential circuits, for combinatorial games, and for problems when instances are specified hierarchically or periodically are known to be HARD. Analogous to what has occurred for NP-complete problems, it often makes sense to talk about the complexities of the approxi-mation problems associated with these HARD problems. Here, we survey our results on the complexities of such approximation problems, emphasizing our results for hierarchically-and periodically-specified problems. These results include the first collection of PTASs, for natural PSPACE-complete, DEXPTIME-complete, and NEXPTIME-complete problems in the literature. In contrast, these results also include a number of new results showing that related approximation problems are HARD.

Our results support the following conclusions:
  1. 1.

    “Local” approximation-preserving reductions between problems can be extended to efficient approximation-preserving reductions between these problems, when instances are hierarchically- or periodically-specified. Such reductions can be used both to obtain efficient approximation algorithms and to prove approximation problems are HARD.

     
  2. 2.

    Hierarchically- and periodically-specified problems are often HARD. But, they also are often efficiently approximable.

     
  3. 3.

    The efficient decomposability of problems and problem instances and the compatibility of such decompositions with the structure of hierarchical- or periodic-specifications play central roles in the development of efficient approximation algorithms, for the problems when hierarchically- or periodically-specified.

     

Keywords

Succinct Specifications Computational Complexity Efficient and Non-Efficient Approximability. 

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References

  1. [1]
    S. Agarwal and A. Condon, “On approximation algorithms for hierarchical MAX-SAT,” Journal of Algorithms, 26 (1): 141–165, January 1998.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy, “Proof verification and hardness of approximation problems,” Journal of the ACM (JACM), 45 (3): 501–555, May 1998.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    B.S. Baker, “Approximation Algorithms for NP-complete Problems on Planar Graphs,” J. ACM, Vol. 41, No. 1, pp. 153–180, 1994.MATHCrossRefGoogle Scholar
  4. [4]
    J.L. Bentley, T. Ottman and P. Widmayer, “The complexity of manipulating hierarchically defined sets of rectangles”, Advances in Computing Research, Vol. 1, F.P. Preparata, ed., JAI Press Inc., pp. 127–158, 1983.Google Scholar
  5. [5]
    E. Cohen and N. Megiddo, “Recognizing Properties of Periodic graphs”, Applied Geometry and Discrete Mathematics, The Victor Klee Festschrift, Vol. 4, P. Gritzmann and B. Strumfels, eds., ACM, New York, pp. 135–146, 1991.Google Scholar
  6. [6]
    E. Cohen and N. Megiddo, “Strongly polynomial-time and NC algorithms for detecting cycles in dynamic graphs”, Journal of the ACM (JACM), 40, pp.791–830, Sept, 1993.MATHCrossRefGoogle Scholar
  7. [7]
    A. Condon, J. Feigenbaum, C. Lund and P. Shor, “Probabilistically Checkable Debate Systems and Approximation Algorithms for PSPACE-Hard Functions,” in Chicago Journal of Theoretical Computer Science, Vol. 1995, No. 4. http://www.cs.uchicago.edu/publications/cjtcs/articles/1995/4/contents.html.
  8. [8]
    A Condon, “Approximate Solutions to Problems in PSPACE,” SIGACT News: Introduction to Complexity Theory Column 9, Guest Column, July, 1995.Google Scholar
  9. [9]
    A. Condon, J. Feigenbaum, C. Lund and P. Shor. Condon, J. Feigenbaum, C. Lund and P. Shor, “Random debaters and the hardness of approximating stochastic functions,” SIAM Journal on Computing, 26 (2), pp. 369–400, April 1997.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    T. Feder and M.Y. Vardi, “Monotone Monadic SNP and Constraint Satisfaction,” Proc. 25th Annual ACM Symposium on the Theory of Computing, pp. 612–622, 1993.Google Scholar
  11. [11]
    L.R. Ford and D.R. Fulkerson, “Constructing Maximal Dynamic Flows from Static Flows,” Operations Research, No. 6, pp. 419–433, 1958.MathSciNetCrossRefGoogle Scholar
  12. [12]
    D. Gale, “Transient Flows in Networks,” Michigan Mathematical Journal, No. 6, pp. 59–63, 1959.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman, New York, 1979.MATHGoogle Scholar
  14. [14]
    D.S. Hochbaum and W. Maass, “Approximation Schemes for Covering and Packing Problems in Image Processing and VLSI,” Journal of the ACM (JACM), 32 (1), pp. 130–136, 1985.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    F. Höfting, T. Lengauer and E. Wanke, “Processing of Hierarchically Defined Graphs and Graph Families,” in Data Structures and Efficient Algorithms (Final Report on the DFG Special Joint Initiative), Springer-Verlag, LNCS 594, pp. 44–69, 1992.CrossRefGoogle Scholar
  16. [16]
    F. Hofting and E. Wanke, “Minimum Cost Paths in Periodic Graphs,” SIAM Journal on Computing, Vol. 24, No. 5, pp. 1051–1067, Oct. 1995.MathSciNetCrossRefGoogle Scholar
  17. [17]
    H. B. Hunt III, M. V. Marathe, and R.E. Stearns, “Generalized CNF satisfiability problems and non-efficient approximability”, Proc. 9th Annual IEEE Conf. on Structure in Complexity Theory, Amsterdam, Netherlands, pp. 356–366, June 1994.Google Scholar
  18. [18]
    H.B. Hunt III, M.V. Marathe, V. Radhakrishnan, S.S. Ravi, D.J. Rosenkrantz, and R.E. Stearns, “A Unified approach to approximation schemes for NP- and PSPACE-hard problems for geometric graphs,” Journal of Algorithms, 26, pp. 238–274, 1998.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    H. B. Hunt III, M. V. Marathe, and R.E. Stearns, “Local reductions, generalized satisfiability problems, complexity, and Efficient Approximability” in preparation, March 1999.Google Scholar
  20. [20]
    K. Iwano and K. Steiglitz, “Testing for Cycles in Infinite Graphs with Periodic Structure,” Proc. 19th Annual ACM Symposium on Theory of Computing, (STOC), pp. 46–53, 1987.Google Scholar
  21. [21]
    K. Iwano and K. Steiglitz, “Planarity Testing of Doubly Connected Periodic Infinite Graphs,” Networks, No. 18, pp. 205–222, 1988.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    P. Jeavons, D. Cohen, and M. Gyssens, “Closure Properties of Constraints,” Journal of the ACM (JACM), 44, pp. 527–549, 1997.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    R.M. Karp, R.E. Miller and S. Winograd, “The Organization of Computations for Uniform Recurrence Equations,” Journal of the ACM (JACM), 14 (3), pp. 563–590, 1967.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    M. Kodialam and J.B. Orlin, “Recognizing Strong Connectivity in Periodic graphs and its relation to integer programming,” Proc. 2nd ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 131–135, 1991.Google Scholar
  25. [25]
    K. R. Kosaraju and G.F. Sullivan, “Detecting Cycles in Dynamic Graphs in Polynomial Time,” Proc. 27th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 398–406, 1988.Google Scholar
  26. [26]
    T. Lengauer, “Efficient algorithms for finding minimum spanning forests of hierarchically defined graphs,” Journal of Algorithms, 8, pp. 260–284, 1987.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    T. Lengauer, “Hierarchical planarity testing algorithms,” Journal of the ACM (JACM), 36, pp. 474–509, 1989.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    T. Lengauer and K.W. Wagner, “The correlation between the complexities of the non-hierarchical and hierarchical versions of graph problems,” Journal of Computer and System Sciences (JCSS), 44, pp. 63–92, 1992.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    M.V. Marathe, V. Radhakrishnan, H.B. Hunt III and S.S. Ravi, “Hierarchically Specified Unit Disk Graphs,” Theoretical Computer Science, 174 (1–2), pp. 23–65, March 1997.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    M.V. Marathe, H.B. Hunt III, V. Radhakrishnan, and R.E. Stearns, “Approximation algorithms for PSPACE-Hard hierarchically and periodically specified problems,” SIAM Journal on Computing, 27 (5), pp. 1237–1261, Oct. 1998.MathSciNetMATHCrossRefGoogle Scholar
  31. [31]
    M.V. Marathe, H.B. Hunt III, R.E. Stearns, and V. Radhakrishnan, “Complexity of hierarchically and 1-dimensional periodically specified problems,” AMSDIMACS Volume Series on Discrete Mathematics and Theoretical Computer Science: Workshop on Satisfiability Problem: Theory and Application, 35, pp. 225–259, Nov. 1996.MathSciNetGoogle Scholar
  32. [32]
    M.V. Marathe, H.B. Hunt III, D.J. Rosenkrantz, and R.E. Stearns, “Theory of periodically specified problems I: complexity and approximability,” Proc. 13th Annual IEEE Conference on Computational Complexity, Buffalo, NY, June 1998.Google Scholar
  33. [33]
    M.V.Marathe, H.B.Hunt III, D.J.Rosenkrantz, and R.E.Stearns, “Theory of periodically specified problems II: Applications,” in preparation, March 1999.Google Scholar
  34. [34]
    G.L. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization, John Wiley & Sons, 1988.Google Scholar
  35. [35]
    J.B. Orlin, “The Complexity of Dynamic/Periodic Languages and Optimization Problems,” Sloan W.P. No. 1679–86 July 1985, Working paper, Alfred P. Sloan School of Management, MIT, Cambridge, MA 02139. A Preliminary version of the paper appears in Proc. 13th ACM Annual Symposium on Theory of Computing (STOC), pp. 218–227, 1978.Google Scholar
  36. [36]
    J.B. Orlin, “Some problems on dynamic/periodic graphs,” Progress in Combinatorial Optimization, W.R. Pulleybank, ed., Academic Press, Orlando, FL, 1984.Google Scholar
  37. [37]
    J.B. Orlin, “Minimum convex cost dynamic network flows,” Mathematics of Operations Research, 9, pp. 190–207, 1984.MathSciNetMATHCrossRefGoogle Scholar
  38. [38]
    A. Panconesi and D. Ranjan, “Quantifiers and approximation,” Theoretical Computer Science, 107(1): 145–163, 4, January 1993.MathSciNetMATHCrossRefGoogle Scholar
  39. [39]
    C.H. Papadimitriou, “Games against nature,” Journal of Computer and System Sciences (JCSS), 31, pp. 288–301, 1985.MathSciNetMATHCrossRefGoogle Scholar
  40. [40]
    C. Papadimitriou, Computational Complexity, Addison-Wesley, Reading, Massachusetts, 1994.MATHGoogle Scholar
  41. [41]
    C.H. Papadimitriou and M. Yannakakis, “Optimization, approximation, and complexity classes,” Journal of Computer and System Sciences (JCSS), 43, pp. 425–440, 1991.MathSciNetMATHCrossRefGoogle Scholar
  42. [42]
    D.J. Rosenkrantz and H.B. Hunt III, “The complexity of processing hierarchically specifications,” SIAM Journal on Computing, 22, pp. 627–649, 1993.MathSciNetMATHCrossRefGoogle Scholar
  43. [43]
    T.J. Schaefer, “The complexity of satisfiability problems,” Proc. 10th Annual ACM Symposium on Theory of Computing, (STOC), pp. 216–226, 1978.Google Scholar
  44. [44]
    T.J. Schaefer, “Complexity of some two-person perfect-information games,” Journal of Computer and System Sciences (JCSS), 16, pp. 185–225, 1978.MathSciNetMATHCrossRefGoogle Scholar
  45. [45]
    R.E. Stearns, “Turing Award Lecture: Its Time to Reconsider Time,” Communications of the ACM (CACM), 37 (11), pp. 95–99, Nov. 1994.CrossRefGoogle Scholar
  46. [46]
    R.E. Stearns and H.B. Hunt III, “Power Indices and Easier Hard Problems”, Mathematical Systems Theory, 23, pp. 209–225, 1990.MathSciNetMATHCrossRefGoogle Scholar
  47. [47]
    L.J. Stockmeyer and A.R. Meyer, “Word problems requiring exponential time,” Proc. 5th Annual ACM Symposium on Theory of Computing, (STOC), Texas, 1973, pages 1–9.Google Scholar
  48. [48]
    E. Wanke, “Paths and cycles in finite periodic graphs,” Proc. 20th Symposium. on Math. Foundations of Computer Science(MFCS), Springer-Verlag LNCS 711, pp. 751–760, 1993.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Harry B. HuntIII
    • 1
  • Madhav V. Marathe
    • 2
  • Richard E. Stearns
    • 1
  1. 1.Department of Computer ScienceUniversity at Albany — SUNYAlbanyUSA
  2. 2.Los Alamos National LaboratoryLos AlamosUSA

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