Nondeterminism and the Clique Problem

Abstract

The Clique problem is known to be NP-Complete [7,3] 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(n3) 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.