Colin McQuillan

The expressibility of functions on the boolean domain, with applications to counting CSPs

An important tool in the study of the complexity of Constraint Satisfaction Problems (CSPs) is the notion of a relational clone, which is the set of all relations expressible using primitive positive formulas over a particular set of base relations. Posts lattice gives a complete classification of all Boolean relational clones, and this has been used to classify the computational difficulty of CSPs. Motivated by a desire to understand the computational complexity of (weighted) counting CSPs, we develop an analogous notion of functional clones and study the landscape of these clones. 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 the conservative case (where all nonnegative unary functions are available), we show 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 to approximate as any problem in #P. Furthermore, we show that any nontrivial functional clone (in a sense that will be made precise) contains the binary function “implies”. As a consequence, in the conservative case, all nontrivial counting CSPs are as hard to approximate as #BIS, the problem of counting independent sets in a bipartite graph. Given the complexity-theoretic results, it is natural to ask whether the “implies” clone is equivalent to the clone of lsm functions. We use the Möbius transform and the Fourier transform to show that these clones coincide precisely up to arity 3. It is an intriguing open question whether the lsm clone is finitely generated. Finally, we investigate functional clones in which only restricted classes of unary functions are available.

Approximating the partition function of planar two-state spin systems

We consider the problem of approximating the partition function of the hard-core model on planar graphs of degree at most 4. We show that when the activity λ is sufficiently large, there is no fully polynomial randomised approximation scheme for evaluating the partition function unless NP = RP . The result extends to a nearby region of the parameter space in a more general two-state spin system with three parameters. We also give a polynomial-time randomised approximation scheme for the logarithm of the partition function. We study approximation of the hard-core partition function.Our graphs are planar with degree at most 4.We show that the problem is intractable for sufficiently high activity.However, the logarithm of the partition function can be approximated.

The complexity of approximating conservative counting CSPs.

We study the complexity of approximation for a weighted counting constraint satisfaction problem #CSP(F). In the conservative case, where F contains all unary functions, a classification is known for the Boolean domain. We give a classification for problems with general finite domain. We define weak log-modularity and weak log-supermodularity, and show that #CSP(F) is in FP if F is weakly log-modular. Otherwise, it is at least as hard to approximate as #BIS, counting independent sets in bipartite graphs, which is believed to be intractable. We further sub-divide the #BIS-hard case. If F is weakly log-supermodular, we show that #CSP(F) is as easy as Boolean log-supermodular weighted #CSP. Otherwise, it is NP-hard to approximate. Finally, we give a trichotomy for the arity-2 case. Then, #CSP(F) is in FP, is #BIS-equivalent, or is equivalent to #SAT, the problem of approximately counting satisfying assignments of a CNF Boolean formula.