On self-transformable combinatorial problems

Abstract

A combinatorial problem is called self-transformable if each instance of the problem can be reduced in polynomial time to a set of smaller instances of the same problem. Most natural NP-complete problems are self-transformable as well as other important problems such as graph isomorphism and feasibility of linear inequalities. We prove that the search problem and the decision problem associated with each self-transformable problem are equally hard. This means that self-transformability bridges over the distinction between pure existential proofs and existential proofs constructing the object. This carries over to random algorithms. As a consequence every algorithm which efficiently detects non-isomorphic pairs of graphs, possibly by using random tests and which fails at most on a sparse set of non-isomorphic pairs of graphs, yields an efficient method for constructing isomorphisms between pairs of isomorphic graphs. In an independent section we show that the efficiency of man machine interaction in computing a Boolean function f is strongly limited by the network complexity of f. Any complex Boolean function f can be computed efficiently only if the human problem solver supplies as much information as is necessary to encode the minimal network for f.