Improved Inapproximability Results for Counting Independent Sets in the Hard-Core Model

Computer Science – Computational Complexity

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

Scientific paper

We study the computational complexity of approximately counting the number of independent sets of a graph with maximum degree Delta. More generally, for an input graph G=(V,E) and an activity lambda>0, we are interested in the quantity Z_G(lambda) defined as the sum over independent sets I weighted as w(I) = lambda^|I|. In statistical physics, Z_G(lambda) is the partition function for the hard-core model, which is an idealized model of a gas where the particles have non-negibile size. Recently, an interesting phase transition was shown to occur for the complexity of approximating the partition function. Weitz showed an FPAS for the partition function for any graph of maximum degree Delta when Delta is constant and lambda< lambda_c(Tree_Delta):=(Delta-1)^(Delta-1)/(Delta-2)^Delta. The quantity lambda_c(Tree_Delta) is the critical point for the so-called uniqueness threshold on the infinite, regular tree of degree Delta. On the other side, Sly proved that there does not exist efficient (randomized) approximation algorithms for lambda_c(Tree_Delta) < lambda < lambda_c(Tree_Delta)+epsilon(Delta), unless NP=RP, for some function epsilon(Delta)>0. We remove the upper bound in the assumptions of Sly's result for Delta not equal to 4 and 5, that is, we show that there does not exist efficient randomized approximation algorithms for all lambda>lambda_c(Tree_Delta) for Delta=3 and Delta>= 6. Sly's inapproximability result uses a clever reduction, combined with a second-moment analysis of Mossel, Weitz and Wormald which prove torpid mixing of the Glauber dynamics for sampling from the associated Gibbs distribution on almost every regular graph of degree Delta for the same range of lambda as in Sly's result. We extend Sly's result by improving upon the technical work of Mossel et al., via a more detailed analysis of independent sets in random regular graphs.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Improved Inapproximability Results for Counting Independent Sets in the Hard-Core Model does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Improved Inapproximability Results for Counting Independent Sets in the Hard-Core Model, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Improved Inapproximability Results for Counting Independent Sets in the Hard-Core Model will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-192492

All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.