Randomised Approximation Schemes for Tutte-Gröthendieck Invariants
What is the number of connected subgraphs of G?
How many subgraphs of G are forests?
How many acyclic orientations has G?
the chromatic and flow polynomials of a graph;
the partition function of a Q-state Potts model;
the Jones polynomial of an alternating link;
the weight enumerator of a linear code over GF(q).
It has been shown that apart from a few special points and 2 special hyperbolae, the exact evaluation of any such invariant is #P-hard even for the very restricted class of planar bipartite graphs. However the question of which points have a fully polynomial randomised approximation scheme is wide open. I shall discuss this problem and give a survey of what is currently known.
KeywordsPartition Function Planar Graph Linear Code Connected Subgraph Dense Graph
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