Boolean approximate counting CSPs with weak conservativity, and implications for ferromagnetic two-spin

We analyse the complexity of approximate counting constraint satisfactions problems $\mathrm{\#CSP}(\mathcal{F})$, where $\mathcal{F}$ is a set of nonnegative rational-valued functions of Boolean variables. A complete classification is known in the conservative case, where $\mathcal{F}$ is assumed to contain arbitrary unary functions. We strengthen this result by fixing any permissive strictly increasing unary function and any permissive strictly decreasing unary function, and adding only those to $\mathcal{F}$: this is weak conservativity. The resulting classification is employed to characterise the complexity of a wide range of two-spin problems, fully classifying the ferromagnetic case. In a further weakening of conservativity, we also consider what happens if only the pinning functions are assumed to be in $\mathcal{F}$ (instead of the two permissive unaries). We show that any set of functions for which pinning is not sufficient to recover the two kinds of permissive unaries must either have a very simple range, or must satisfy a certain monotonicity condition. We exhibit a non-trivial example of a set of functions satisfying the monotonicity condition.Comment: 37 page

Holant clones and the approximability of conservative holant problems

We construct a theory of holant clones to capture the notion of expressibility in the holant framework. Their role is analogous to the role played by functional clones in the study of weighted counting Constraint Satisfaction Problems. We explore the landscape of conservative holant clones and determine the situations in which a set F of functions is “universal in the conservative case”, which means that all functions are contained in the holant clone generated by F together with all unary functions. When F is not universal in the conservative case, we give concise generating sets for the clone. We demonstrate the usefulness of the holant clone theory by using it to give a complete complexity-theory classification for the problem of approximating the solution to conservative holant problems. We show that approximation is intractable exactly when F is universal in the conservative case

The computational complexity of approximation of partition functions

This thesis studies the computational complexity of approximately evaluating partition functions. For various classes of partition functions, we investigate whether there is an FPRAS: a fully polynomial randomised approximation scheme. In many of these settings we also study “expressibility”, a simple notion of defining a constraint by combining other constraints, and we show that the results cannot be extended by expressibility reductions alone. The main contributions are: -� We show that there is no FPRAS for evaluating the partition function of the hard-core gas model on planar graphs at fugacity 312, unless RP = NP. -� We generalise an argument of Jerrum and Sinclair to give FPRASes for a large class of degree-two Boolean #CSPs. -� We initiate the classification of degree-two Boolean #CSPs where the constraint language consists of a single arity 3 relation. -� We show that the complexity of approximately counting downsets in directed acyclic graphs is not affected by restricting to graphs of maximum degree three. -� We classify the complexity of degree-two #CSPs with Boolean relations and weights on variables. -� We classify the complexity of the problem #CSP(F) for arbitrary finite domains when enough non-negative-valued arity 1 functions are in the constraint language. -� We show that not all log-supermodular functions can be expressed by binary logsupermodular functions in the context of #CSPs

Counting Small Induced Subgraphs Satisfying Monotone Properties

Given a graph property $\Phi$, the problem $\#\mathsf{IndSub}(\Phi)$ asks, on input a graph $G$ and a positive integer $k$, to compute the number of induced subgraphs of size $k$ in $G$ that satisfy $\Phi$. The search for explicit criteria on $\Phi$ ensuring that $\#\mathsf{IndSub}(\Phi)$ is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and D\"orfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property $\Phi$, the problem $\#\mathsf{IndSub}(\Phi)$ cannot be solved in time $f(k)\cdot |V(G)|^{o(k/ {\log^{1/2}(k)})}$ for any function $f$, unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a $\#\mathsf{W}[1]$-completeness result

Counting Small Induced Subgraphs Satisfying Monotone Properties

Given a graph property $\Phi$, the problem $\#\mathsf{IndSub}(\Phi)$ asks, on input a graph $G$ and a positive integer $k$, to compute the number of induced subgraphs of size $k$ in $G$ that satisfy $\Phi$. The search for explicit criteria on $\Phi$ ensuring that $\#\mathsf{IndSub}(\Phi)$ is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and D\"orfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property $\Phi$, the problem $\#\mathsf{IndSub}(\Phi)$ cannot be solved in time $f(k)\cdot |V(G)|^{o(k/ {\log^{1/2}(k)})}$ for any function $f$, unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a $\#\mathsf{W}[1]$-completeness result.Comment: 33 pages, 2 figure

Boolean approximate counting CSPs with weak conservativity, and implications for ferromagnetic two-spin

We analyse the complexity of approximate counting constraint satisfactions problems #CSP(F), where F is a set of nonnegative rational-valued functions of Boolean variables. A complete classification is known if F contains arbitrary unary functions. We strengthen this result by fixing any permissive strictly increasing unary function and any permissive strictly decreasing unary function, and requiring only those to be in F. The resulting classification is employed to characterise the complexity of a wide range of two-spin problems, fully classifying the ferromagnetic case. Furthermore, we also consider what happens if only the pinning functions are assumed to be in F. We show that any set of functions for which pinning is not sufficient to recover the two kinds of permissive unaries must either have a very simple range, or must satisfy a certain monotonicity condition. We exhibit a non-trivial example of a set of functions satisfying the monotonicity condition