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.
782417
The Backbone of the Travelling Salesperson
We study the backbone of the travelling salesperson optimization problem. We prove that it is intractable to approximate the backbone with any performance guarantee, assuming that P≠NP and there is a limit on the number of edges falsely returned. Nevertheless, in practice, it appears that much of the backbone is present in close to optimal solutions. We can therefore often find much of the backbone using approximation methods based on good heuristics. We demonstrate that such backbone information can be used to guide the search for an optimal solution. However, the variance in runtimes when using a backbone guided heuristic is large. This suggests that we may need to combine such heuristics with randomization and restarts. In addition, though backbone guided heuristics are useful for finding optimal solutions, they are less help in proving optimality.
3264013
Symmetry Breaking Using Value Precedence
We present a comprehensive study of the use of value precedence constraints to break value symmetry. We first give a simple encoding of value precedence into ternary constraints that is both efficient and effective at breaking symmetry. We then extend value precedence to deal with a number of generalizations like wreath value and partial interchangeability. We also show that value precedence is closely related to lexicographical ordering. Finally, we consider the interaction between value precedence and symmetry breaking constraints for variable symmetries.
9178924
Incompleteness and Incomparability in Preference Aggregation
We consider how to combine the preferences of multiple agents despite the presence of incompleteness and incomparability in their preference orderings. An agent's preference ordering may be incomplete because, for example, there is an ongoing preference elicitation process. It may also contain incomparability as this is useful, for example, in multi-criteria scenarios. We focus on the problem of computing the possible and necessary winners, that is, those outcomes which can be or always are the most preferred for the agents. Possible and necessary winners are useful in many scenarios including preference elicitation. First we show that computing the sets of possible and necessary winners is in general a difficult problem as is providing a good approximation of such sets. Then we identify general properties of the preference aggregation function which are sufficient for such sets to be computed in polynomial time. Finally, we show how possible and necessary winners can be used to focus preference elicitation.
2340000
A General Purpose Reasoner for Abstraction
The goal of the work described in this paper is the development of a system, called ABSFOL, which allows the user to state declaratively abstractions and to use them according to the desired control strategy. ABSFOL has been successfully tested on many examples. So far we have failed to find an interesting abstraction whose implementation requires a major programming effort.
1373490
The SAT Phase Transition
We describe a detailed experimental investigation of the phase transition for several different classes of randomly generated satisfiability problems. We observe a remarkable consistency of fea- tures in the phase transition despite the presence in some of the problem classes of clauses of mixed lengths. For instance, each of the problem classes considered has a sharp transition from satisfiable to unsatisfiable problems at a critical value. In addition, there is a com- mon easy-hard-easy pattern in the median difficulty of the problems, with the hardest problems being associated with the phase transition. However, the difficulty of problems of mixed clause lengths is much more variable than that of fixed clause length. Indeed, whilst the me- dian difficulty of random problems of mixed clause lengths can be orders of magnitude easier than that of equivalently sized problems of fixed clause length, the hardest problems of mixed clause lengths can be orders of magnitude harder than the hardest equivalently sized problems of fixed clause length. Such very hard random problems may be of considerable practical and theoretical use in analysing algorithm performance.