Phase Transition in Reduction between 3-SAT and Graph Colorability for Channel Assignment in Cellular Network

Since, channel assignment problem has been shown to be an NP-hard problem. Also, it is shown that the channel assignment problem is very similar to the graph k-color ability problem. But Graph k-Color ability (for k ≥ 3) Problem (GCP) is still a well known NP-complete problem. There are many approaches have been proposed to solve NP-complete problem, but none of the approaches could give the deterministic solution. One of the recent approach to solve NP-complete problem in deterministic way is Boolean satisfiability (SAT). Reduction between graph k-color ability problem to/from satisfiability expression can be a important concept to solve channel assignment problem in cellular network. For analyzing the behaviors of NP-Complete problems, in recent years, there has been much interest in study of phase transitions. Analytical and experimental research has shown that the âphase transitionâ phenomenon is often associated with the hardness of complexity. Each of the problems has a standard known phase transition. Previously, in [1] [2], we have reduced graph k-color ability problem to/from 3-satisfiability expression in polynomial way. In this paper, we analyzed and calculated the phase transition of systematically generated 3-colorable graph and 3-CNF-SAT expression by our reduction method of 3-SAT to/from 3-colorable graph. We observed that calculated phase transitions are lower than the know phase transition as well as phase transition obtained by Alaxander [3]. This lower phase transition shows that our reduction method is better than previously proposed methods to transform two NP-complete problems into each other more efficiently.