The expressibility of functions on the Boolean domain, with applications to Counting CSPs

Abstract

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. Post's 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 non-trivial functional clone (in a sense that will be made precise) contains the binary function "implies". As a consequence, in the conservative case, all non-trivial counting CSPs are as hard 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 Mobius 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.

corrected typo in title :-)

References in corpus (7)

• Approximating the partition function of the ferromagnetic Potts model
• The Complexity of Weighted Boolean #CSP
• The complexity of weighted and unweighted #CSP
• The Expressive Power of Binary Submodular Functions
• The Complexity of Approximately Counting Stable Matchings
• Approximating the Tutte polynomial of a binary matroid and other related combinatorial polynomials
• LSM is not generated by binary functions

Cited by in corpus (8)

• #BIS-Hardness for 2-Spin Systems on Bipartite Bounded Degree Graphs in the Tree Nonuniqueness Region
• FPTAS for #BIS with Degree Bounds on One Side
• The complexity of approximating conservative counting CSPs
• The Complexity of Approximately Counting Tree Homomorphisms
• Functional Clones and Expressibility of Partition Functions
• Boolean approximate counting CSPs with weak conservativity, and implications for ferromagnetic two-spin
• Holant clones and the approximability of conservative holant problems
• Hierarchies of Minion Tests for PCSPs through Tensors