Nondeterminism and the clique problem

Abstract

The Clique problem is known to be NP-Complete and the question whether P=NP is unresolved. This paper examines the relative power of nondeterminism versus determinism in a restricted setting. Specifically, we consider solving the clique problem using non-deterministic and deterministic Turing machines. We impose a (reasonable) format in which a problem instance is encoded. We also impose constraints on the computation of both deterministic and non-deterministic Turing machines: both have two tapes, the input tape is read-only and one-way, and once a certain stop point in the input tape is reached, no additional writing on the work tape is allowed. We consider two cases for the position of the stop point: immediately after the number of graph nodes and the size of the clique are specified, or controlled by a parameter q that indicates what portion of the graph nodes' edge specifications have been scanned. The parameter q may be arbitrarily close to 1, e.g., $q=0.99999$. We show, for both cases in our setting, that a non-deterministic Turing machine can solve the problem in $O\left(n^3\right)$ time whereas no deterministic Turing machine can solve the problem in polynomial time. However, we exhibit an exponential time deterministic single work tape, two-heads Turing machine that solves the clique problem in our setting.