  CSMA (Carrier Sense Multiple Access) algorithms based on Gibbs sampling can
achieve throughput optimality if certain parameters called the fugacities are
appropriately chosen. However, the problem of computing these fugacities is
NP-hard. In this work, we derive estimates of the fugacities by using a
framework called the regional free energy approximations. In particular, we
derive explicit expressions for approximate fugacities corresponding to any
feasible service rate vector. We further prove that our approximate fugacities
are exact for the class of chordal graphs. A distinguishing feature of our work
is that the regional approximations that we propose are tailored to conflict
graphs with small cycles, which is a typical characteristic of wireless
networks. Numerical results indicate that the fugacities obtained by the
proposed method are quite accurate and significantly outperform the existing
Bethe approximation based techniques.
