A Linearly-Growing Conversion from the Set Splitting Problem to the Directed Hamiltonian Cycle Problem

Abstract

We consider a direct conversion of the, classical, set splitting problem to the directed Hamiltonian cycle problem. A constructive procedure for such a conversion is given, and it is shown that the input size of the converted instance is a linear function of the input size of the original instance. A proof that the two instances are equivalent is given, and a procedure for identifying a solution to the original instance from a solution of the converted instance is also provided. We conclude with two examples of set splitting problem instances, one with solutions and one without, and display the corresponding instances of the directed Hamiltonian cycle problem, along with a solution in the first example.

Change ringing and Hamiltonian cycles: The search for Erin and Stedman triples

“…This difficulty is interesting in its own right, since truly difficult instances of HCP are not widely known in literature and there has not been a truly difficult benchmark set produced to date. Recently, an HCP challenge was held, called the FHCP Challenge [6], to take strides in this direction. In the FHCP Challenge, which ran for one year, a set of 1001 graphs known to be Hamiltonian needed to be solved.…”