  As it follows from G\"odel's incompleteness theorems, any consistent formal
system of axioms and rules of inference should imply a true unprovable
statement. Actually, this fundamental principle can be efficiently applicable
in computational mathematics and complexity theory concerning the computational
complexity of problems from the class NP, particularly and especially the
NP-complete ones. While there is a wide set of algorithms for these problems
that we call heuristic, the correctness or/and complexity of each concrete
algorithm (or the probability of its correct and polynomial-time work) on a
class of instances is often too difficult to determine, although we may also
assume the existence of a variety of algorithms for NP-complete problems that
are both correct and polynomial-time on all the instances from a given class
(where the given problem remains NP-complete), but whose correctness or/and
polynomial-time complexity on the class is impossible to prove as an example
for G\"odel's theorems. However, supposedly such algorithms should possess a
certain complicatedness of processing the input data and treat it in a certain
algebraically "entangled" manner. The same algorithmic analysis in fact
concerns all the other significant problems and subclasses of NP, such as the
graph isomorphism problem and its associated complexity class GI.
  The following short article offers a couple of algebraically entangled
polynomial-time algorithms for the graph isomorphism and clique problems whose
correctness is yet to be determined either empirically or through attempting to
find proofs. The authors are grateful to Prof. Anuj Dawar (University of
Cambridge) for kindly endorsing the present article for publishing in arXiv.
