# Time and space bounded complexity classes and bandwidth constrained problems

## Abstract

^{k}(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)×log

^{2}n. These results are used to show, for example, that:

- (1)
\(\mathbb{N}\)SPACE( f(n)) (\(\mathbb{D}\)SPACE( f(n)×max(f(n),log n) ), for all functions f,

- (2)
The class SC, called Steve's class in honor of Stephen Cook who showed, for example, that all DCFL's are in SC

^{2}= \(\mathbb{D}\)TISP(poly,log^{2}n), 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 = U_{k≥1}\(\mathbb{D}\)TISP(poly,log^{k}n) ) - (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## Preview

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