Abstract
In [Le 82, Le 85, Le 86a, Le 86b] a hierarchical graph model is discussed that allows to exploit the hierarchical description of the graphs for the efficient solution of graph problems. The model is motivated by applications in CAD, and is based on a special form of a graph grammar. The above references contain polynomial time solutions for the hierarchical versions of many classical graph problems. However, there are also graph problems that cannot benefit from the succinctness of hierarchical description of the graphs.
In this paper we investigate whether the complexity of the hierarchical version of a graph problem can be predicted from the complexity of its non-hierarchical version. We find that the correlation between the complexities of the two versions of a graph problem is very loose, i.e., such a prediction is not possible in general. This is contrary to corresponding results about other models of succinct graph description [PY 86, Wa 86].
One specific result of the paper is that the threshold network flow problem, which asks whether the maximum flow in a (hierarchical) directed graph is larger than a given number, is log-space complete for PSPACE in its hierarchical version and log-space complete for P in its non-hierarchical version. The latter result affirms a conjecture by [GS 82].
Other typical results settle the complexities of the hierarchical version of the clique-, independent set-, and Hamiltonian circuit problem.
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References
Chandra, A.K./Kozen, D./Stockmeyer, L.J.: Alternation. Research report RC 7489, IBM Thomas J. Watson Research Center 1979. See also: JACM 28 (1981), 114–133
Dolev, D./Trickey, H.: On linear area embedding of planar graphs. Tech. Rep. STAN-CS-81-876, Comp. Sci. Dept., Stanford University, Stanford CA 94305 (1981)
Galperin, H.: Succinct Representations of Graphs. Ph.D. Thesis, Dept. of Elec. Eng. and Comp. Sci., Princeton University, Princeton (N.J.) (1982)
Garey, M.R./Johnson, D.S.: Computers & Intractability. Freeman, San Francisco (1979)
Goldschlager, L.: The monotone and planar circuit value problems are log-space complete for P. SIGACT News 9,2 (1977), 25–29
Goldschlager, L.M./Shaw, R.A./Staples, J.: The maximum flow problem is log-space complete for P. Theor. Comp. Sci. 21 (1982), 105–111
Galperin, H./Wigderson, A.: Succinct representation of graphs. Information & Control 56 (1983), 183–198
Immerman, N.: Length of predicate calculus formulas as a new complexity measure. 20th IEEE-FOCS (1979), 337–347. See also: JCSS 22 (1981), 384–406
Karp, R.M.: Reducibilities among combinatorial problems. Complexity of Computer Computations (R.E. Miller, J.W. Thatcher, eds.), Plenum Press, N.Y. (1972), 85–103
Lengauer, T.: The complexity of compacting hierarchically specified layouts of integrated circuits. 23rd IEEE-FOCS (1982), 358–368
Lengauer, T.: Efficient solution of connectivity problems on hierarchically defined graphs. "Theoretische Informatik" No. 24, FB 17, Universität-GH Paderborn, Paderborn, West-Germany (1985). Short version in: Proc. of WG '85 (H. Noltemeier, ed.), Trauner Verlag (1985), 201–216
Lengauer, T.: Efficient algorithms for finding minimum spanning forests of hierarchically defined graphs. Proc. of STACS 86, Springer Lecture Notes in Computer Science No. 216 (1986), 153–170
Lengauer, T.: Hierarchical planarity testing algorithms. Proc. of ICALP 86, Springer Lecture Notes in Computer Science No. 226 (1986), 215–225
Lengauer, T.: Exploiting hierarchy in VLSI design. VLSI Algorithms and Architectures (F. Makedon et al., eds.), Springer Lecture Notes in Computer Science No. 227 (1986), 180–193
Meyer, A./Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential space. 13th IEEE-SWAT (1972), 125–129
Papadimitriou, C.H./Yannakakis, M.: A note on succinct representation of graphs. Manuscript (1986), to appear in Information & Control.
Savitch, W.J.: Maze recognizing automata and nondeterministic tape complexity. JCSS 7 (1973), 389–403
Stockmeyer, L.J./Meyer, A.R.: Word problems requiring exponential time. 5th ACM-STOC (1973), 1–9
Sudborough, I.H.: On tape-bounded complexity classes and multihead finite automata. 14th IEEE-SWAT (1973), 138–144
Shyum, S./Valiant, L.G.: A complexity theory based on Boolean algebra. Proc. 22nd IEEE-FOCS (1981), 244–253
Wagner, K.: The complexity of problems concerning graphs with regularities. Proc. of MFCS 84, Springer Lecture Notes in Computer Science No. 176 (1984), 544–552
Wagner, K.: The complexity of combinatorial problems with succinct input representation. Acta Informatica 23 (1986), 325–356
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Lengauer, T., Wagner, K.W. (1987). The correlation between the complexities of the non-hierarchical and hierarchical versions of graph problems. In: Brandenburg, F.J., Vidal-Naquet, G., Wirsing, M. (eds) STACS 87. STACS 1987. Lecture Notes in Computer Science, vol 247. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0039598
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DOI: https://doi.org/10.1007/BFb0039598
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