# Time and space bounded complexity classes and bandwidth constrained problems

A survey
• Burkhard Monien
• Ivan Hal Sudborough
Invited Lectures
Part of the Lecture Notes in Computer Science book series (LNCS, volume 118)

## Abstract

Bandwidth constraints on familiar natural computational problems are considered. It is seen that generally as the bandwidth of a problem decreases its space complexity decreases. More interestingly, for problems complete for a complexity class $$\mathbb{K}$$, often as one decreases the bandwidth one obtains complete problems for space restricted subclasses of $$\mathbb{K}$$. For example, (1) the $$\mathbb{N}$$SPACE(log n) complete graph accessibility problem (GAP), when restricted to graphs of bandwidth fk(n), for some k≥1, forms a complete family of problems for $$\mathbb{N}$$SPACE(log f(n)), (2) the $$\mathbb{P}$$complete and/or graph accessibility problem (AGAP), when restricted to graphs of bandwidth f(n), is complete for the simultaneous time-space complexity class $$\mathbb{D}$$TISP(poly,f(n)), (3) the $$\mathbb{N}$$P complete graph problems 3COLOR, SIMPLE MAX CUT, INDEPENDENT SET, VERTEX COVER, DOMINATING SET, and several others, when restricted to graphs of bandwidth f(n) are complete for the simultaneous time-space complexity class $$\mathbb{N}$$TISP(poly,f(n)), and (4) the $$\mathbb{P}$$-Space complete PEBBLE problem, when restricted to graphs of bandwidth f(n), can be solved in space f(n)×log2n. These results are used to show, for example, that:
1. (1)

$$\mathbb{N}$$SPACE( f(n)) ($$\mathbb{D}$$SPACE( f(n)×max(f(n),log n) ), for all functions f,

2. (2)

The class SC, called Steve's class in honor of Stephen Cook who showed, for example, that all DCFL's are in SC2 = $$\mathbb{D}$$TISP(poly,log2n), is identical to the log space closure of the class of sets accepted by one-way alternating Turing machines within loglog n space. ( Note: SC = Uk≥1$$\mathbb{D}$$TISP(poly,logkn) )

3. (3)

The graph problems 3COLOR, SIMPLE MAX CUT, INDEPENDENT SET, VERTEX COVER, DOMINATING SET, and several other $$\mathbb{N}$$P complete problems, when restricted to graphs of bandwidth f(n), can be solved in polynomial time if and only if $$\mathbb{N}$$TISP(poly,f(n)) ($$\mathbb{P}$$.

## Keywords

Polynomial Time Turing Machine Vertex Cover Negative Instance Complete Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
A. K. Chandra, D. C. Kozen, and L. J. Stockmeyer, Alternation, J. ACM 28,1 (1981), pp. 114–133.Google Scholar
2. 2.
M-J Chung, W. M. Evangelist, and I. H. Sudborough, Some Additional Examples of Bandwidth Constrained NP-Complete Problems, Proc. 15th Conf. on Information Sciences and Systems (1981), The Johns Hopkins University, Baltimore, Md., U.S.A., to appear.Google Scholar
3. 3.
S. A. Cook, The Complexity of Theorem-Proving Procedures, Proc. 3rd Annual ACM Theory of Computing Symp. (1971), Assoc. for Comput. Mach., New York, pp. 151–158.Google Scholar
4. 4.
S. A. Cook, Deterministic CFL's are Accepted Simultaneously in Polynomial Time and Log Squared Space, Proc. 11th Annual ACM Theory of Computing Symp. (1979), Assoc. for Comput. Mach., New York, pp. 338–345.Google Scholar
5. 5.
S. A. Cook, Towards a Complexity Theory of Synchronous Parallel Computation, Tech. Report #141/80 (1980), Dept. of Computer Science, University of Toronto, Toronto, Canada.Google Scholar
6. 6.
S. Even and R. E. Tarjan, A Combinatorial Problem which is Complete in Polynomial Space, J. ACM 23,4 (1976), pp. 710–719.Google Scholar
7. 7.
M. R. Garey, D. S. Johnson, and L. Stockmeyer, Some Simplified NP-Complete Graph Problems, Theoretical Computer Science 1 (1976), pp. 237–267.Google Scholar
8. 8.
M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth, Complexity Results for Bandwidth Minimization, SIAM J. Appl. Math. 34 (1978), pp. 477–495.Google Scholar
9. 9.
M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Co., San Francisco, 1979.Google Scholar
10. 10.
J. R. Gilbert, T. Lengauer, and R. E. Tarjan, The Pebbling Problem is Complete in Polynomial Space, SIAM J. Comput. 9,3 (1980), pp. 513–524.Google Scholar
11. 11.
J. E. Hopcroft and J. D. Ullman, Introduction to Automata Theory, Languages, and Computation, Addison-Wesley Publishing Co., Reading, Mass., U.S.A., 1979.Google Scholar
12. 12.
N. Immerman, Length of Predicate Calculus Formulas as a new Complexity Measure, Proc. 20th Annual Symp. on Foundations of Computer Sci. (1979), IEEE Computer Society, Long Beach, Calif., U.S.A., pp. 337–347.Google Scholar
13. 13.
N. D. Jones, Space Bounded Reducibility Among Combinatorial Problems, J. Comput. System Sci. 11 (1975), pp. 68–85.Google Scholar
14. 14.
R. M. Karp, Reducibility Among Combinatorial Problems, in R. E. Miller and J. W. Thatcher (eds.), Complexity of Computer Computations, Plenum Press, New York, 1972, pp. 85–103.Google Scholar
15. 15.
R. E. Ladner, The Circuit Value Problem is Log Space Complete for P, SIGACT News 7,1 (1975), pp. 18–20.Google Scholar
16. 16.
B. Monien and I. H. Sudborough, On Eliminating Nondeterminism from Turing Machines which use less than Logarithm Worktape Space, in Vol. 72 Lecture Notes in Computer Science, Springer Verlag (1979), pp. 431–445.Google Scholar
17. 17.
B. Monien and I. H. Sudborough, Bandwidth Constrained NP-Complete Problems, Proc. 13th Annual ACM Theory of Computing Symp. (1981), Assoc. for Comput. Mach., New York, to appear.Google Scholar
18. 18.
B. Monien and I. H. Sudborough, Bandwidth Problems in Graphs, Proc. 18th Annual Allerton Conference on Communication, Control, and Computing (1980), Dept. of Computer Science, University of Illinois, Champaign-Urbana, Illinois, U.S.A., to appear.Google Scholar
19. 19.
20. 20.
C. H. Papadimitriou, The NP-Completeness of the Bandwidth Minimization Problem, Computing 16 (1976), pp. 263–270.Google Scholar
21. 21.
N. Pippenger, On Simultaneous Resource Bounds, Proc. 20th Annual Symp. on Foundations of Computer Sci. (1979), IEEE Computer Society, Long Beach, Calif., U.S.A., pp. 307–311.Google Scholar
22. 22.
W. J. Savitch, Relationships between Nondeterministic and Deterministic Tape Complexities, J. Comput. and System Sci. 4 (1970), pp. 177–192.Google Scholar
23. 23.
J. B. Saxe, Dynamic-Programming Algorithms for Recognizing Small Bandwidth Graphs in Polynomial Time, Proc. 17th Annual Allerton Conference on Communication, Control, and Computing (1979), Dept. of Computer Science, University of Illinois, Champaign-Urbana, Illinois, U.S.A.Google Scholar
24. 24.
T. J. Schaefer, Complexity of Some Two-Person Perfect-Information Games, J. Comput. System Sci. 16 (1978), pp. 185–225.Google Scholar
25. 25.
I. H. Sudborough, Efficient Algorithms for Path System Problems and Applications to Alternating and Time-Space Complexity Classes, Proc. 21st Annual Symp. on Foundations of Computer Sci. (1980), IEEE Computer Society, Long Beach, Calif., U.S.A., pp. 62–73.Google Scholar
26. 26.
I. H. Sudborough, Pebbling and Bandwidth, Proc. Fundamentals of Computation Theory Symp. (1981), to appear in Springer Verlag Lecture Notes in Computer Science series.Google Scholar
27. 27.
O. Vornberger, Komplexität von Wegeproblemen in Graphen, Ph.D. dissertation, Fachbereich Mathematik/Informatik, Universität Paderborn, Paderborn, West Germany.Google Scholar