Toby Walsh

A theory of abstraction

Abstract Informally , abstraction can be described as the process of mapping a representation of a problem onto a new representation. The aim of this paper is to propose the beginnings of a theory of reasoning with abstraction which captures and generalizes most previous work in the area. The theory allows us to study the properties of abstraction mappings and provides the foundations for the mechanization of abstraction inside an abstract proof checker.

CSPLIB: A Benchmark Library for Constraints

Constraint satisfaction algorithms are often benchmarked on hard, random problems. There are, however, many reasons for wanting a larger class of problems in our benchmark suites. For example, we may wish to benchmark algorithms on more realistic problems, to run competitions, or to study the impact on modelling and problem reformulation. Whilst there are many other constructive benefits of a benchmark library, there are also several potential pitfalls. For example, if the library is small, we run the risk of over-fitting our algorithms. Even if the library is large, certain problem features may be rare or absent. A model benchmark library should be easy to find and easy to use. It should contain as diverse and large a set of problems as possible. It should be easy to extend, and as comprehensive and up to date as possible. It should also be independent of any particular constraint solver, and contain neither just hard (nor just easy) problems.

Principles and Practice of Constraint Programming — CP 2001

Hybrid Benders Decomposition Algorithms in Constraint Logic Programming.- Branch-and-Check: A Hybrid Framework Integrating Mixed Integer Programming and Constraint Logic Programming.- Towards Inductive Constraint Solving.- Collaborative Learning for Constraint Solving.- Towards Stochastic Constraint Programming: A Study of Onine Multi-Choice Knapsack with Deadlines.- Global Cut Framework for Removing Symmetries.- Symmetry Breaking.- The Non-existence of (3,1,2)-Conjugate Orthogonal Idempotent Latin Square of Order 10.- Random 3-SAT and BDDs: The Plot Thickens Further.- Capturing Structure with Satisfiability.- Phase Transitions and Backbones of 3-SAT and Maximum 3-SAT.- Solving Non-binary CSPs Using the Hidden Variable Encoding.- A Filtering Algorithm for the Stretch Constraint.- Network Flow Problems in Constraint Programming.- Pruning for the Minimum Constraint Family and for the Number of Distinct Values Constraint Family.- A Constraint Programming Approach to the Stable Marriage Problem.- Components for State Restoration in Tree Search.- Adaptive Constraint Handling with CHR in Java.- Consistency Maintenance for ABT.- Constraint-Based Verification of Client-Server Protocols.- A Temporal Concurrent Constraint Programming Calculus.- Lower Bounds for Non-binary Constraint Optimization Problems.- New Lower Bounds of Constraint Violations for Over-Constrained Problems.- A General Scheme for Multiple Lower Bound Computation in Constraint Optimization.- Solving Disjunctive Constraints for Interactive Graphical Applications.- Sweep as a Generic Pruning Technique Applied to the Non-overlapping Rectangles Constraint.- Non-overlapping Constraints between Convex Polytopes.- Formal Models of Heavy-Tailed Behavior in Combinatorial Search.- The Phase Transition of the Linear Inequalities Problem.- In Search of a Phase Transition in the AC-Matching Problem.- Specific Filtering Algorithms for Over-Constrained Problems.- Specializing Russian Doll Search.- A CLP Approach to the Protein Side-Chain Placement Problem.- Fast, Constraint-Based Threading of HP-Sequences to Hydrophobic Cores.- One Flip per Clock Cycle.- Solving Constraints over Floating-Point Numbers.- Optimal Pruning in Parametric Differential Equations.- Interaction of Constraint Programming and Local Search for Optimisation Problems.- Partition-k-AC: An Efficient Filtering Technique Combining Domain Partition and Arc Consistency.- Neighborhood-Based Variable Ordering Heuristics for the Constraint Satisfaction Problem.- The Expressive Power of Binary Linear Programming.- Constraint Generation via Automated Theory Formation.- The Traveling Tournament Problem Description and Benchmarks.- Deriving Explanations and Implications for Constraint Satisfaction Problems.- Generating Tradeoffs for Interactive Constraint-Based Configuration.- Structural Constraint-Based Modeling and Reasoning with Basic Configuration Cells.- Composition Operators for Constraint Propagation:An Application to Choco.- Solving Boolean Satisfiability Using Local Search Guided by Unit Clause Elimination.- GAC on Conjunctions of Constraints.- Dual Models of Permutation Problems.- Boosting Local Search with Artificial Ants.- Fast Optimal Instruction Scheduling for Single-Issue Processors with Arbitrary Latencies.- Evaluation of Search Heuristics for Embedded System Scheduling Problems.- Interpreting Sloppy Stick Figures with Constraint-Based Subgraph Matching.- Selecting and Scheduling Observations for Agile Satellites: Some Lessons from the Constraint Reasoning Community Point of View.- A Dynamic Distributed Constraint Satisfaction Approach to Resource Allocation.- A Constraint Optimization Framework for Mapping a Digital Signal Processing Application onto a Parallel Architecture.- iOpt: A Software Toolkit for Heuristic Search Methods.- AbsCon: A Prototype to Solve CSPs with Abstraction.- A Constraint Engine for Manufacturing Process Planning.- On the Dynamic Detection of Interchangeability in Finite Constraint Satisfaction Problems.- Automatic Generation of Implied Clauses for SAT.- Verification of Infinite-State Systems by Specialization of CLP Programs.- Partially Ordered Constraint Optimization Problems.- Translations for Comparing Soft Frameworks.- Counting Satisfiable k-CNF Formulas.- High-Level Modelling and Reformulation of Constraint Satisfaction Problems.- Distributed Constraint Satisfaction as a Computational Model of Negotiation via Argumentation.- Aircraft Assignment Using Constraint Programming.- Labelling Heuristics for CSP Application Domains.- Improving SAT Algorithms by Using Search Pruning Techniques.- Optimum Symmetry Breaking in CSPs Using Group Theory.- Distributed Dynamic Backtracking.- Constraint Programming for Distributed Resource Allocation.- Exploiting the CSP Structure by Interchangeability.- Constraint Processing Techniques for Model-Based Reasoning about Dynamic Systems.- Distributed Constraint Satisfaction with Cooperating Asynchronous Solvers.- Building Negative Reduced Cost Paths Using Constraint Programming.- An Incremental and Non-binary CSP Solver: The Hyperpolyhedron Search Algorithm.- Partial Stable Generated Models of Generalized Logic Programs with Constraints.- Heterogeneous Constraint Problems An Outline of the Field of Work.- Comparing SAT Encodings for Model Checking.- Asynchronous Search for Numeric DisCSPs.- Temporal Concurrent Constraint Programming.

Symmetry Breaking using Value Precedence

I find a class of black hole solutions to a (3+1) dimensional theory gravity coupled to abelian gauge fields with negative cosmological constant that has been proposed as the dual theory to a Lifshitz theory describing critical phenomena in (2+1) dimensions. These black holes are all asymptotic to a Lifshitz fixed point geometry and depend on a single parameter that determines both their area (or size) and their charge. Most of the solutions are obtained numerically, but an exact solution is also obtained for a particular value of this parameter. The thermodynamic behaviour of large black holes is almost the same regardless of genus, but differs considerably for small black holes. Screening behaviour is exhibited in the dual theory for any genus, but the critical length at which it sets in is genus-dependent for small black holes.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.

Breaking Row and Column Symmetries in Matrix Models

We identify an important class of symmetries in constraint programming, arising from matrices of decision variables where rows and columns can be swapped. Whilst lexicographically ordering the rows (columns) breaks all the row (column) symmetries, lexicographically ordering both the rows and the columns fails to break all the compositions of the row and column symmetries. Nevertheless, our experimental results show that this is effective at dealing with these compositions of symmetries. We extend these results to cope with symmetries in any number of dimensions, with partial symmetries, and with symmetric values. Finally, we identify special cases where all compositions of the row and column symmetries can be eliminated by the addition of only a linear number of symmetry-breaking constraints.

Random Constraint Satisfaction: Flaws and Structure

A recent theoretical result by Achlioptas et al. shows that many models of random binary constraint satisfaction problems become trivially insoluble as problem size increases. This insolubility is partly due to the presence of ‘flawed variables,’ variables whose values are all ‘flawed’ (or unsupported). In this paper, we analyse how seriously existing work has been affected. We survey the literature to identify experimental studies that use models and parameters that may have been affected by flaws. We then estimate theoretically and measure experimentally the size at which flawed variables can be expected to occur. To eliminate flawed values and variables in the models currently used, we introduce a ‘flawless’ generator which puts a limited amount of structure into the conflict matrix. We prove that such flawless problems are not trivially insoluble for constraint tightnesses up to 1/2. We also prove that the standard models B and C do not suffer from flaws when the constraint tightness is less than the reciprocal of domain size. We consider introducing types of structure into the constraint graph which are rare in random graphs and present experimental results with such structured graphs.

Easy problems are sometimes hard

We present a detailed experimental investigation of the easy-hard-easy phase transition for randomly generated instances of satisfiability problems. Problems in the hard part of the phase transition have been extensively used for benchmarking satisfiability algorithms. This study demonstrates that problem classes and regions of the phase transition previously thought to be easy can sometimes be orders of magnitude more difficult than the worst problems in problem classes and regions of the phase transition considered hard. These difficult problems are either hard unsatisfiable problems or are satisfiable problems which give a hard unsatisfiable subproblem following a wrong split. Whilst these hard unsatisfiable problems may have short proofs, these appear to be difficult to find, and other proofs are long and hard.

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 CSPs. 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.

Global Constraints for Lexicographic Orderings

We propose some global constraints for lexicographic orderings on vectors of variables. These constraints are very useful for breaking a certain kind of symmetry arising in matrices of decision variables. We show that decomposing such constraints carries a penalty either in the amount or the cost of constraint propagation. We therefore present a global consistency algorithm which enforces a lexicographic ordering between two vectors of n variables in O(nb) time, where b is the cost of adjusting the bounds of a variable. The algorithm can be modified very slightly to enforce a strict lexicographic ordering. Our experimental results on a number of domains (balanced incomplete block design, social golfer, and sports tournament scheduling) confirm the efficiency and value of these new global constraints.

The TSP phase transition

The traveling salesman problem is one of the most famous combinatorial problems. We identify a natural parameter for the two-dimensional Euclidean traveling salesman problem. We show that for random problems there is a rapid transition between soluble and insoluble instances of the decision problem at a critical value of this parameter. Hard instances of the traveling salesman problem are associated with this transition. Similar results are seen both with randomly generated problems and benchmark problems using geographical data. Surprisingly, finite-size scaling methods developed in statistical mechanics describe the behaviour around the critical value in random problems. Such phase transition phenomena appear to be ubiquitous. Indeed, we have yet to find an NP-complete problem which lacks a similar phase transition.