Thermodynamics of Physical Approximations to Non Deterministic Polynomial Complete Problems

Non deterministic polynomial complete problems are playing an essential role in nowadays communications, security and many other fields. One of the well-known NP-complete problems is the Examination Timetabling problem. Given the information about the courses and the students that have taken some of those courses as well as the time slots for the exams of those courses the problem asks if there is a timetable such that no student has two exams simultaneously. We are modeling this NP-complete problem as a system of particles with one dimensional motion. After adding some interactions between the particles based on the problem we are simulating the motion and arriving to an equilibrium. We then use the equilibrium state to group the particles together where each cluster represents a time slot. To discuss the physics of this model we use the Replica method to find the free energy functional of this system. A hypothesis is formulated that all or at least most numerical solutions for NP-complete problems can be brought to this model with some configuration of interactions between the particles.