arXiv subjects

T. Walsh

Publications and source records attributed to T. Walsh.

4 recordsLinked to original sources

arXiv2011

Dual Modelling of Permutation and Injection Problems

When writing a constraint program, we have to choose which variables should be the decision variables, and how to represent the constraints on these variables. In many cases, there is considerable choice for the decision variables. Consider, for example, permutation problems in which we have as many values as variables, and each variable takes an unique value. In such problems, we can choose between a primal and a dual viewpoint. In the dual viewpoint, each dual variable represents one of the primal values, whilst each dual value represents one of the primal variables. Alternatively, by means of channelling constraints to link the primal and dual variables, we can have a combined model with both sets of variables. In this paper, we perform an extensive theoretical and empirical study of such primal, dual and combined models for two classes of problems: permutation problems and injection problems. Our results show that it often be advantageous to use multiple viewpoints, and to have constraints which channel between them to maintain consistency. They also illustrate a general methodology for comparing different constraint models.

cs.AI↗

arXiv2010

Local search for stable marriage problems

The stable marriage (SM) problem has a wide variety of practical applications, ranging from matching resident doctors to hospitals, to matching students to schools, or more generally to any two-sided market. In the classical formulation, n men and n women express their preferences (via a strict total order) over the members of the other sex. Solving a SM problem means finding a stable marriage where stability is an envy-free notion: no man and woman who are not married to each other would both prefer each other to their partners or to being single. We consider both the classical stable marriage problem and one of its useful variations (denoted SMTI) where the men and women express their preferences in the form of an incomplete preference list with ties over a subset of the members of the other sex. Matchings are permitted only with people who appear in these lists, an we try to find a stable matching that marries as many people as possible. Whilst the SM problem is polynomial to solve, the SMTI problem is NP-hard. We propose to tackle both problems via a local search approach, which exploits properties of the problems to reduce the size of the neighborhood and to make local moves efficiently. We evaluate empirically our algorithm for SM problems by measuring its runtime behaviour and its ability to sample the lattice of all possible stable marriages. We evaluate our algorithm for SMTI problems in terms of both its runtime behaviour and its ability to find a maximum cardinality stable marriage.For SM problems, the number of steps of our algorithm grows only as O(nlog(n)), and that it samples very well the set of all stable marriages. It is thus a fair and efficient approach to generate stable marriages.Furthermore, our approach for SMTI problems is able to solve large problems, quickly returning stable matchings of large and often optimal size despite the NP-hardness of this problem.

cs.AI↗

arXiv1996

A Divergence Critic for Inductive Proof

Inductive theorem provers often diverge. This paper describes a simple critic, a computer program which monitors the construction of inductive proofs attempting to identify diverging proof attempts. Divergence is recognized by means of a ``difference matching'' procedure. The critic then proposes lemmas and generalizations which ``ripple'' these differences away so that the proof can go through without divergence. The critic enables the theorem prover Spike to prove many theorems completely automatically from the definitions alone.

cs.AI↗

arXiv1993

An Empirical Analysis of Search in GSAT

We describe an extensive study of search in GSAT, an approximation procedure for propositional satisfiability. GSAT performs greedy hill-climbing on the number of satisfied clauses in a truth assignment. Our experiments provide a more complete picture of GSAT's search than previous accounts. We describe in detail the two phases of search: rapid hill-climbing followed by a long plateau search. We demonstrate that when applied to randomly generated 3SAT problems, there is a very simple scaling with problem size for both the mean number of satisfied clauses and the mean branching rate. Our results allow us to make detailed numerical conjectures about the length of the hill-climbing phase, the average gradient of this phase, and to conjecture that both the average score and average branching rate decay exponentially during plateau search. We end by showing how these results can be used to direct future theoretical analysis. This work provides a case study of how computer experiments can be used to improve understanding of the theoretical properties of algorithms.

cs.AI↗