28528633
Total Vertex Enumeration in Rooted Planar Maps
18489826
Counting unrooted loopless planar maps
We present a formula for the number of n-edge unrooted loopless planar maps considered up to orientation-preserving isomorphism. The only sum contained in this formula is over the divisors of n.
15854568
Enumeration of eulerian and unicursal planar maps
Sum-free enumerative formulae are derived for several classes of rooted planar maps with no vertices of odd valency (eulerian maps) and with two vertices of odd valency (unicursal maps). As corollaries we obtain simple formulae for the numbers of unrooted eulerian and unicursal planar maps. Also, we obtain a sum-free formula for the number of rooted bi-eulerian (eulerian and bipartite) maps and some related results.
6785300
Difference Matching
Difference matching is a generalization of first-order matching where terms are made identical both by variable instantiation and by structure hiding. After matching, the hidden structure may be removed by a type of controlled rewriting, called rippling, that leaves the rest of the term unaltered. Rippling has proved highly successful in inductive theorem proving. Difference matching allows us to use rippling in other contexts, e.g., equational, inequational, and propositional reasoning. We present a difference matching algorithm, its properties, several applications, and suggest extensions.
7300181
Automatic Concept Formation in Pure Mathematics
The HR program forms concepts and makes conjectures in domains of pure mathematics and uses theorem prover OTTER and model generator MACE to prove or disprove the conjectures. HR measures properties of concepts and assesses the theorems and proofs involving them to estimate the interestingness of each concept and employ a best first search. This approach has led HR to the discovery of interesting new mathematics and enables it to build theories from just the axioms of finite algebras.
15244663
Inverse Consistencies for Non-Binary Constraints
We present a detailed study of two inverse consistencies for non-binary constraints: relational path inverse consistency (rel PIC) and pairwise inverse consistency (PWIC). These are stronger than generalized arc consistency (GAC), even though they also only prune domain values. We propose algorithms to achieve rel PIC and PWIC, that have a time complexity better than the previous generic algorithm for inverse consistencies. One of our algorithms for PWIC has a complexity comparable to that for GAC despite doing more pruning. Our experiments demonstrate that inverse consistencies can be more efficient than GAC on a range of non-binary problems.
8504868
Aggregating preferences cannot be fair
SOMMARIO/ABSTRACT Preferences typically define a partial ordering over outcomes. A number of formalisms like soft constraints and CP-nets exists to specify such partial or-derings. In situations involving multiple agents, we need to combine the preferences of several individuals. In this paper, we consider each agent as voting on whether they prefer one outcome to another. We prove that, under certain conditions on the kind of partial orders that are allowed to express the preferences of the agents and of the result, if there are at least two agents and three outcomes to order, no preference aggregation system can be fair. That is, no preference aggregation system can be free (give any possible result), monotonic (improving a vote for an outcome only ever helps), independent to irrelevant assumptions (the result between two outcomes only depends on how the agents vote on these two outcomes), and non-dictatorial (there is not one agent who is never contradicted). This result generalizes Ar-row's impossibility theorem for combining total orders [1].
14075117
An Empirical Study of Dynamic Variable Ordering Heuristics for the Constraint Satisfaction Problem
The constraint satisfaction community has developed a number of heuristics for variable ordering during backtracking search. For example, in conjunction with algorithms which check forwards, the Fail-First (FF) and Brelaz (Bz) heuristics are cheap to evaluate and are generally considered to be very effective. Recent work to understand phase transitions in NP-complete problem classes enables us to compare such heuristics over a large range of different kinds of problems. Furthermore, we are now able to start to understand the reasons for the success, and therefore also the failure, of heuristics, and to introduce new heuristics which achieve the successes and avoid the failures. In this paper, we present a comparison of the Bz and FF heuristics in forward checking algorithms applied to randomly-generated binary CSP's. We also introduce new and very general heuristics and present an extensive study of these. These new heuristics are usually as good as or better than Bz and FF, and we identify problem classes where our new heuristics can be orders of magnitude better. The result is a deeper understanding of what helps heuristics to succeed or fail on hard random problems in the context of forward checking, and the identification of promising new heuristics worthy of further investigation. 
 
This research was supported by HCM personal fellowship to the last author, by a University of Strathclyde starter grant to the first author, and by an EPSRC ROPA award GR/K/65706 for the first three authors. Authors listed alphabetically. We thank the other members of the APES group, and our reviewers, for their comments.