Speaker:
Mark Jerrum
Title:
Log-supermodular functions, functional clones and counting CSPs
Abstract:
Joint work with Andrei A. Bulatov, Martin Dyer and Leslie Ann Goldberg
Motivated by a desire to understand the computational complexity
of counting constraint satisfaction problems (counting CSPs), particularly the complexity
of approximation, we study functional clones of functions on the Boolean domain,
which are analogous to the familiar relational clones constituting Post's lattice.
One of these clones is the collection of log-supermodular (lsm) functions,
which turns out to play a significant role in classifying counting CSPs.
In our study, we assume that non-negative unary functions (weights) are available.
Given this, we prove that there are no functional clones lying strictly between
the clone of lsm functions and the total clone (containing all functions).
Thus, any counting CSP that contains a single nontrivial non-lsm function
is computationally as hard as any problem in #P. Furthermore, any non-trivial
functional clone (in a precise sense) contains the binary function "implies".
As a consequence, all non-trivial counting CSPs (with non-negative unary weights
assumed to be available) are computationally at least as difficult as #BIS,
the problem of counting independent sets in a bipartite graph.
There is empirical evidence that #BIS is hard to solve, even approximately.
Finally, we investigate functional clones in which only restricted unary
functions (either favouring 0 or 1) are available.