Constructing concrete hard instances of the maximum independent set problem

Abstract

We provide a deterministic construction of hard instances for the maximum independent set problem (MIS). The constructed hard instances form an infinite graph sequence with increasing size, which possesses similar characteristics to sparse random graphs and in which MIS cannot be solved efficiently. We analytically and numerically show that the linear programming relaxation with cutting planes, which is one of the standard method to solve the MIS problem, cannot upper bound the size of the maximum independent set tightly.