A Graph-Transformational Approach for Proving the Correctness of Reductions between NP-Problems

Abstract

The complexity class NP of decision problems that can be solved nondeterministically in polynomial time is of great theoretical and practical importance where the notion of polynomial-time reductions between NP-problems is a key concept for the study of NP. As many typical NP-problems are naturally described as graph problems, they and their reductions are obvious candidates to be investigated by graph-transformational means. In this paper, we propose such a graph-transformational approach for proving the correctness of reductions between NP-problems.

Modeling NP-Problems with Families of Extended Graph-based Reaction Systems

“…Old and often used results for various types of graph transformation are the local Church-Rosser theorems concerning the interchangeability of rule applications. In Kreowski et al [9], it is shown that these results are very useful to prove the correctness of reductions between NP-problems in a systematic and procedural way. It may be worthwhile to investigate whether corresponding results and applications work for steps of interactive processes.…”

“…The complexity class NP of decision problems solvable nondeterministically in polynomial time can be formally defined as the class of string languages that can be recognized by nondeterministic Turing machines in polynomial time, i.e., in a polynomially bounded number of computation steps. As strings may be seen as special graphs and the processing of Turing machines as a special kind of graph transformation, an alternative definition of NP is given by rule-based graph transformation systems the lengths of derivations of which are polynomially bounded (cf., e.g., [9]). The question is whether graph-based reaction systems may play a similar role.…”