We present a way to create small yet difficult model counting instances. Our generator is highly parameterizable: the number of variables of the instances it produces, as well as their number of clauses and the number of literals in each clause, can all be set to any value. Our instances have been tested on state of the art model counters, against other difficult model counting instances, in the Model Counting Competition. The smallest unsolved instances of the competition, both in terms of number of variables and number of clauses, were ours. We also observe a peak of difficulty when fixing the number of variables and varying the number of clauses, in both random instances and instances built by our generator. Using these results, we predict the parameter values for which the hardest to count instances will occur.

The approach used by our algorithm mimics the way a human would tr y to solve the same problem. Every progress made during the solvi ng process is accompanied by a detailed explanation of our program's re asoning. Since this reasoning is based on the same heuristics that a hu man would employ, the user can easily follow the given explanation.

When creating benchmarks for SAT solvers, we need SAT instances that are easy to build but hard to solve. A recent development in the search for such methods has led to the Balanced SAT algorithm, which can create k-SAT instances with m clauses of high difficulty, for arbitrary k and m. In this paper we introduce the No-Triangle SAT algorithm, a SAT instance generator based on the cluster coefficient graph statistic. We empirically compare the two algorithms by fixing the arity and the number of variables, but varying the number of clauses. The hardest instances that we find are produced by No-Triangle SAT. Furthermore, difficult instances from No-Triangle SAT have a different number of clauses than difficult instances from Balanced SAT, potentially allowing a combination of the two methods to find hard SAT instances for a larger array of parameters.

A variable elimination rule allows the polynomial-time identification of certain variables whose elimination does not affect the satisfiability of an instance. Variable elimination in the constraint satisfaction problem (CSP) can be used in preprocessing or during search to reduce search space size. We show that there are essentially just four variable elimination rules defined by forbidding generic sub-instances, known as irreducible patterns, in arc-consistent CSP instances. One of these rules is the Broken Triangle Property, whereas the other three are novel.

A var(a) v} to v. If cpt(a, b) T then the two assignments (points) a, b are compatible and {a, b} is a compatibility In a In a CSP instance the aim is to determine the existence pattern, the compatibility of a pair of points a, b such that of an assignment of values to variables such that a set var(a) var(b) and (a, b) / E is undefined. A fundamental research question is the identification of tractable subproblems A binary CSP instance is a pattern 〈V, A, var, E, cpt〉 of CSP.

Although the CSP (constraint satisfaction problem) is NP-complete, even in the case when all constraints are binary, certain classes of instances are tractable. We study classes of instances defined by excluding subproblems. This approach has recently led to the discovery of novel tractable classes. The complete characterisation of all tractable classes defined by forbidding patterns (where a pattern is simply a compact representation of a set of subproblems) is a challenging problem. We demonstrate a dichotomy in the case of forbidden patterns consisting of either one or two constraints. This has allowed us to discover new tractable classes including, for example, a novel generalisation of 2SAT.