Random generation of combinatorial structures from a uniform
Theoretical Computer Science
Polynomial-time approximation algorithms for the Ising model
SIAM Journal on Computing
Equivalent Representations of Set Functions
Mathematics of Operations Research
Complexity classifications of boolean constraint satisfaction problems
Complexity classifications of boolean constraint satisfaction problems
Discrete Applied Mathematics
The complexity of partition functions
Theoretical Computer Science - Automata, languages and programming: Algorithms and complexity (ICALP-A 2004)
The Complexity of Ferromagnetic Ising with Local Fields
Combinatorics, Probability and Computing
Structure identification of Boolean relations and plain bases for co-clones
Journal of Computer and System Sciences
Note: The expressive power of binary submodular functions
Discrete Applied Mathematics
The Complexity of Weighted Boolean CSP
SIAM Journal on Computing
Information Processing Letters
An approximation trichotomy for Boolean #CSP
Journal of Computer and System Sciences
Approximate counting for complex-weighted Boolean constraint satisfaction problems
WAOA'10 Proceedings of the 8th international conference on Approximation and online algorithms
Complexity of conservative constraint satisfaction problems
ACM Transactions on Computational Logic (TOCL)
Guest column: complexity dichotomies of counting problems
ACM SIGACT News
The complexity of conservative valued CSPs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
The complexity of weighted and unweighted #CSP
Journal of Computer and System Sciences
Dichotomy for Holant problems of Boolean domain
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
The complexity of approximately counting stable matchings
Theoretical Computer Science
Approximating the partition function of the ferromagnetic Potts model
Journal of the ACM (JACM)
Approximating the Tutte polynomial of a binary matroid and other related combinatorial polynomials
Journal of Computer and System Sciences
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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 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.