The Correlation between the Complexities of the Nonhierarchical and Hierarchical Versions of Graph Problems

@article{Lengauer1992TheCB,
  title={The Correlation between the Complexities of the Nonhierarchical and Hierarchical Versions of Graph Problems},
  author={Thomas Lengauer and K. Wagner},
  journal={J. Comput. Syst. Sci.},
  year={1992},
  volume={44},
  pages={63-93}
}
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… Expand
A Parametrized Analysis of Algorithms on Hierarchical Graphs
TLDR
The complexity in the hierarchical setting is higher, but all “jumps” in complexity up to an exponential one are exhibited, including no jumps in some problems. Expand
The Complexity of Searching Succinctly Represented Graphs
TLDR
A model for the analysis of algorithms on graphs given by vertex expansion procedures is introduced, based on previously studied concepts of “succinct representation” techniques, and allows us to prove PSPACE-completeness or EXPTIME-completion of specific, natural problems on implicit graphs, such as those solved by A*, AO*; and other best-first search strategies. Expand
The Complexity of Searching Implicit Graphs
TLDR
A model for the analysis of algorithms on graphs given by vertex expansion procedures is introduced, based on previously studied concepts of “succinct representation” techniques, and allows us to prove PSPACE-completeness or EXPTIME-Completeness of specific, natural problems on implicit graphs, such as those solved by A ∗, AO ∗ , and other best-first search strategies. Expand
Fixpoint Logics on Hierarchical Structures
TLDR
In this paper, the model-checking problem for the modal μ-calculus and (monadic) least fixpoint logic on hierarchically defined graphs is investigated and parity games on hierarchic defined graphs are studied. Expand
Fixpoint Logics over Hierarchical Structures
TLDR
In this paper, the model-checking problem for the modal μ-calculus and (monadic) least fixpoint logic on hierarchically defined input graphs is investigated, and a restriction on hierarchical graph definitions that leads to more efficient model- checking algorithms is presented. Expand
Model-checking hierarchical structures
  • Markus Lohrey
  • Computer Science, Mathematics
  • J. Comput. Syst. Sci.
  • 2012
TLDR
Two restrictions on the structure of hierarchical graph definitions that lead to more efficient model-checking algorithms are presented, based on classical results of Gaifman and Courcelle. Expand
The Complexity of Connectivity Problems on Context-Free Graph Languages
  • Egon Wanke
  • Computer Science, Mathematics
  • J. Comput. Syst. Sci.
  • 1994
TLDR
This work analyzes the complexity of connectivity problems on sets of graphs defined by context-free graph rewriting systems under various restrictions and shows that L is P-complete for simple context- free graph rewriting Systems, but NP-complete (co-NP-complete, respectively), for boundary node label controlled graph grammars and more powerful systems. Expand
The Complexity of Approximating PSPACE-Complete Problems for Hierarchical Specifications (Extended Abstract)
We extend the concept of polynomial time approximation algorithms to apply to problems for hierarchically specified graphs, many of which are PSPACE-complete. Assuming P # PSPACE, the existence orExpand
Hierarchically Speciied Unit Disk Graphs
We characterize the complexity of a number of basic optimization problems for unit disk graphs speciied hierarchically as in BOW83, LW87a, Le88, LW92]. Both PSPACE-hardness results and polynomialExpand
The Expressive Power and Complexity of Dynamic Process Graphs
TLDR
The lower bound is obtained by showing that this kind of dynamic process graph s can represent certain Boolean formulas in a highly succint way and implies a quadratic bound on the maximal deadline in contrast to the general case, where the execution time may be exponential. Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 19 REFERENCES
Hierarchical planarity testing algorithms
TLDR
A hierarchical graph model that permits taking advantage of the hierarchy is presented and algorithms are given that test planarity of a hierarchically described graph in linear time in the length of the hierarchical description. Expand
Efficient Solution of Connectivity Problems on Hierarchically Defined Graphs
TLDR
This paper presents the bottom-up method that solves graph problems without expanding the hierarchical description, which allows solutions that are efficient in terms of the hierarchical graph description instead of the size of the expanded graph. Expand
Succinct Representations of Graphs
TLDR
The main result is characterizing a large class of graph properties for which the respective “succinct problem” is NP-hard, and shows that the succinct versions of polynomially equivalent problems may not be polynomial equivalent. Expand
Efficient Algorithms for Finding Minimum Spanning Forests of Hierarchically Defined Graphs
TLDR
A hierarchical graph model is defined that allows the exploitation of the hierarchy for the more efficient solution of graph problems on very large graphs and it is shown how to efficiently find minimum spanning forests in this graph model. Expand
The Binary Network Flow Problem is Logspace Complete for P
Abstract It is shown that the problem of whether the maximum flow in a given network exceeds a given natural number is logspace many-one complete for P if the edge capacities are presented in binaryExpand
On linear area embedding of planar graphs
Planar embedding with minimal area of graphs on an integer grid is one of the major issues in VLSI. Valiant [1981] gave an algorithm to construct a planar embedding for trees in linear area; he alsoExpand
The monotone and planar circuit value problems are log space complete for P
TLDR
It is shown that Ladner's simulation of Turing mac]hines by boolean circuits seems to require an "adequate" set of gates, such as AND and NOT, but the same simulation is possible with monotone circuits using AND and OR gates only. Expand
Number of Quantifiers is Better Than Number of Tape Cells
  • N. Immerman
  • Computer Science, Mathematics
  • J. Comput. Syst. Sci.
  • 1981
TLDR
A new complexity measure is introduced, QN[f(n), which measures the size of sentences from predicate calculus needed to express a given property, and Fraisse-Ehrenfeucht games are used to prove sharp lower bounds in the measure. Expand
Length of predicate calculus formulas as a new complexity measure
  • N. Immerman
  • Mathematics, Computer Science
  • 20th Annual Symposium on Foundations of Computer Science (sfcs 1979)
  • 1979
We introduce a new complexity measure, QR[f(n)], which clocks the size of formulas from predicate calculus needed to express a given property. Techniques from logic are used to prove sharp lowerExpand
The Maximum Flow Problem is Log Space Complete for P
TLDR
It is shown that the problem is log space complete for deterministic polynomial time, so the maximum flow problem probably has no algorithm which needs only O(logk n) storage space for any constant k. Expand
...
1
2
...