Modelling Langford's Problem : a viewpoint for search

Funding: EPSRC (EP/P015638/1).The performance of enumerating all solutions to an instance of Langford's Problem is sensitive to the model and the search strategy. In this paper we compare the performance of a large variety of models, all derived from two base viewpoints. We empirically show that a channelled model with a static branching order on one of the viewpoints offers the best performance out of all the options we consider. Surprisingly, one of the base models proves very effective for propagation, while the other provides an effective means of stating a static search order.Postprin

Cable Tree Wiring -- Benchmarking Solvers on a Real-World Scheduling Problem with a Variety of Precedence Constraints

Cable trees are used in industrial products to transmit energy and information between different product parts. To this date, they are mostly assembled by humans and only few automated manufacturing solutions exist using complex robotic machines. For these machines, the wiring plan has to be translated into a wiring sequence of cable plugging operations to be followed by the machine. In this paper, we study and formalize the problem of deriving the optimal wiring sequence for a given layout of a cable tree. We summarize our investigations to model this cable tree wiring Problem (CTW) as a traveling salesman problem with atomic, soft atomic, and disjunctive precedence constraints as well as tour-dependent edge costs such that it can be solved by state-of-the-art constraint programming (CP), Optimization Modulo Theories (OMT), and mixed-integer programming (MIP) solvers. It is further shown, how the CTW problem can be viewed as a soft version of the coupled tasks scheduling problem. We discuss various modeling variants for the problem, prove its NP-hardness, and empirically compare CP, OMT, and MIP solvers on a benchmark set of 278 instances. The complete benchmark set with all models and instance data is available on github and is accepted for inclusion in the MiniZinc challenge 2020

Model induction: a new source of model redundancy for constraint satisfaction problems.

Law Yat Chiu.Thesis (M.Phil.)--Chinese University of Hong Kong, 2002.Includes bibliographical references (leaves 85-89).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 2 --- Related Work --- p.4Chapter 2.1 --- Equivalence of CSPs --- p.4Chapter 2.2 --- Dual Viewpoint --- p.4Chapter 2.3 --- CSP Reformulation --- p.5Chapter 2.4 --- Multiple Modeling --- p.5Chapter 2.5 --- Redundant Modeling --- p.6Chapter 2.6 --- Minimal Combined Model --- p.6Chapter 2.7 --- Permutation CSPs and Channeling Constraints --- p.6Chapter 3 --- Background --- p.8Chapter 3.1 --- From Viewpoints to CSP Models --- p.8Chapter 3.2 --- Constraint Satisfaction Techniques --- p.10Chapter 3.2.1 --- Backtracking Search --- p.11Chapter 3.2.2 --- Consistency Techniques and Constraint Propagation --- p.12Chapter 3.2.3 --- Incorporating Consistency Techniques into Backtracking Search --- p.18Chapter 4 --- Model Induction --- p.21Chapter 4.1 --- Channeling Constraints --- p.21Chapter 4.2 --- Induced Models --- p.22Chapter 4.3 --- Properties --- p.30Chapter 5 --- Exploiting Redundancy from Model Induction --- p.35Chapter 5.1 --- Combining Redundant Models --- p.35Chapter 5.1.1 --- Model Intersection --- p.36Chapter 5.1.2 --- Model Channeling --- p.38Chapter 5.2 --- Three New Forms of Model Redundancy --- p.39Chapter 5.3 --- Experiments --- p.42Chapter 5.3.1 --- Langford's Problem --- p.44Chapter 5.3.2 --- Random Permutation CSPs --- p.53Chapter 5.3.3 --- Golomb Rulers --- p.72Chapter 5.3.4 --- Circular Golomb Rulers --- p.74Chapter 5.3.5 --- All-Interval Series Problem --- p.78Chapter 6 --- Concluding Remarks --- p.82Chapter 6.1 --- Contributions --- p.82Chapter 6.2 --- Future Work --- p.8

Speeding up weighted constraint satisfaction using redundant modeling.

Woo Hiu Chun.Thesis (M.Phil.)--Chinese University of Hong Kong, 2006.Includes bibliographical references (leaves 91-99).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Constraint Satisfaction Problems --- p.1Chapter 1.2 --- Weighted Constraint Satisfaction Problems --- p.3Chapter 1.3 --- Redundant Modeling --- p.4Chapter 1.4 --- Motivations and Goals --- p.5Chapter 1.5 --- Outline of the Thesis --- p.6Chapter 2 --- Background --- p.8Chapter 2.1 --- Constraint Satisfaction Problems --- p.8Chapter 2.1.1 --- Backtracking Tree Search --- p.9Chapter 2.1.2 --- Local Consistencies --- p.12Chapter 2.1.3 --- Local Consistencies in Backtracking Search --- p.17Chapter 2.1.4 --- Permutation CSPs --- p.19Chapter 2.2 --- Weighted Constraint Satisfaction Problems --- p.20Chapter 2.2.1 --- Branch and Bound Search --- p.23Chapter 2.2.2 --- Local Consistencies --- p.26Chapter 2.2.3 --- Local Consistencies in Branch and Bound Search --- p.32Chapter 2.3 --- Redundant Modeling --- p.34Chapter 3 --- Generating Redundant WCSP Models --- p.37Chapter 3.1 --- Model Induction for CSPs --- p.38Chapter 3.1.1 --- Stated Constraints --- p.39Chapter 3.1.2 --- No-Double-Assignment Constraints --- p.39Chapter 3.1.3 --- At-Least-One-Assignment Constraints --- p.40Chapter 3.2 --- Generalized Model Induction for WCSPs --- p.43Chapter 4 --- Combining Mutually Redundant WCSPs --- p.47Chapter 4.1 --- Naive Approach --- p.47Chapter 4.2 --- Node Consistency Revisited --- p.51Chapter 4.2.1 --- Refining Node Consistency Definition --- p.52Chapter 4.2.2 --- Enforcing m-NC* c Algorithm --- p.55Chapter 4.3 --- Arc Consistency Revisited --- p.58Chapter 4.3.1 --- Refining Arc Consistency Definition --- p.60Chapter 4.3.2 --- Enforcing m-AC*c Algorithm --- p.62Chapter 5 --- Experiments --- p.67Chapter 5.1 --- Langford's Problem --- p.68Chapter 5.2 --- Latin Square Problem --- p.72Chapter 5.3 --- Discussion --- p.75Chapter 6 --- Related Work --- p.77Chapter 6.1 --- Soft Constraint Satisfaction Problems --- p.77Chapter 6.2 --- Other Local Consistencies in WCSPs --- p.79Chapter 6.2.1 --- Full Arc Consistency --- p.79Chapter 6.2.2 --- Pull Directional Arc Consistency --- p.81Chapter 6.2.3 --- Existential Directional Arc Consistency --- p.82Chapter 6.3 --- Redundant Modeling and Channeling Constraints --- p.83Chapter 7 --- Concluding Remarks --- p.85Chapter 7.1 --- Contributions --- p.85Chapter 7.2 --- Future Work --- p.87List of Symbols --- p.88Bibliograph

Realizations of common channeling constraints in constraint satisfaction: theory and algorithms.

Lam Yee Gordon.Thesis (M.Phil.)--Chinese University of Hong Kong, 2006.Includes bibliographical references (leaves 109-117).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Constraint Satisfaction Problems --- p.1Chapter 1.2 --- Motivations and Goals --- p.2Chapter 1.3 --- Outline of the Thesis --- p.4Chapter 2 --- Background --- p.5Chapter 2.1 --- CSP --- p.5Chapter 2.2 --- Classes of Variable --- p.6Chapter 2.3 --- Solution of a CSP --- p.7Chapter 2.4 --- Constraint Solving Techniques --- p.8Chapter 2.4.1 --- Local Consistencies --- p.8Chapter 2.4.2 --- Constraint Tightness --- p.10Chapter 2.4.3 --- Tree Search --- p.10Chapter 2.5 --- Graph --- p.14Chapter 3 --- Common Channeling Constraints --- p.16Chapter 3.1 --- Models --- p.16Chapter 3.2 --- Channeling Constraints --- p.17Chapter 3.2.1 --- Int-Int Channeling Constraint (II) --- p.18Chapter 3.2.2 --- Set-Int Channeling Constraint (SI) --- p.21Chapter 3.2.3 --- Set-Set Channeling Constraint (SS) --- p.24Chapter 3.2.4 --- Int-Bool Channeling Constraint (IB) --- p.25Chapter 3.2.5 --- Set-Bool Channeling Constraint (SB) --- p.27Chapter 3.2.6 --- Discussions --- p.29Chapter 4 --- Realization in Existing Solvers --- p.31Chapter 4.1 --- Implementation by if-and-only-if constraint --- p.32Chapter 4.1.1 --- "Realization of iff in CHIP, ECLiPSe, and SICStus Prolog" --- p.32Chapter 4.1.2 --- Realization of iff in Oz and ILOG Solver --- p.32Chapter 4.2 --- Implementations by Element Constraint --- p.38Chapter 4.2.1 --- "Realization of ele in CHIP, ECLiPSe, and SICStus Prolog" --- p.40Chapter 4.2.2 --- Realization of ele in Oz and ILOG Solver --- p.40Chapter 4.3 --- Global Constraint Implementations --- p.41Chapter 4.3.1 --- "Realization of glo in CHIP, SICStus Prolog, and ILOG Solver" --- p.42Chapter 5 --- Consistency Levels --- p.43Chapter 5.1 --- Int-Int Channeling (II) --- p.44Chapter 5.2 --- Set-Int Channeling (SI) --- p.49Chapter 5.3 --- Set-Set Channeling Constraints (SS) --- p.53Chapter 5.4 --- Int-Bool Channeling (IB) --- p.55Chapter 5.5 --- Set-Bool Channeling (SB) --- p.57Chapter 5.6 --- Discussion --- p.59Chapter 6 --- Algorithms and Implementation --- p.61Chapter 6.1 --- Source of Inefficiency --- p.62Chapter 6.2 --- Generalized Element Constraint Propagators --- p.63Chapter 6.3 --- Global Channeling Constraint --- p.66Chapter 6.3.1 --- Generalization of Existing Global Channeling Constraints --- p.66Chapter 6.3.2 --- Maintaining GAC on Int-Int Channeling Constraint --- p.68Chapter 7 --- Experiments --- p.72Chapter 7.1 --- Int-Int Channeling Constraint --- p.73Chapter 7.1.1 --- Efficient AC implementations --- p.74Chapter 7.1.2 --- GAC Implementations --- p.75Chapter 7.2 --- Set-Int Channeling Constraint --- p.83Chapter 7.3 --- Set-Set Channeling Constraint --- p.89Chapter 7.4 --- Int-Bool Channeling Constraint --- p.89Chapter 7.5 --- Set-Bool Channeling Constraint --- p.91Chapter 7.6 --- Discussion --- p.93Chapter 8 --- Related Work --- p.101Chapter 8.1 --- Empirical Studies --- p.101Chapter 8.2 --- Theoretical Studies --- p.102Chapter 8.3 --- Applications --- p.103Chapter 8.4 --- Other Kinds of Channeling Constraints --- p.104Chapter 9 --- Concluding Remarks --- p.106Chapter 9.1 --- Contributions --- p.106Chapter 9.2 --- Future Work --- p.108Bibliography --- p.10

Permutation Problems and Channelling Constraints

When writing a constraint program, we have to decide what to make the decision variable, and how to represent the constraints on these variables. In many cases, there is considerable choice for the decision variables. For example, with permutation problems, we can choose between a primal and a dual representation. In the dual representation, dual variables stand for the primal values, whilst dual values stand for the primal variables. By means of channelling constraints, a combined model can have both primal and dual variables. In this paper, we perform an extensive theoretical and empirical study of these different models. Our results will aid constraint programmers to choose a model for a permutation problem. They also illustrate a general methodology for comparing different constraint models

Permutation Problems and Channelling Constraints

We perform an extensive study of several different models of permutation problems proposed by Smith in [Smith, 2000]. We first define a measure of constraint tightness parameterized by the level of local consistency being enforced. We then compare the constraint tightness in these different models with respect to a large number of local consistency properties including arc-consistency, (restricted) path-consistency, path inverse consistency, singleton arc-consistency and bounds consistency. We also compare the constraint tightness in SAT encodings of these permutation problems. These results will aid users of constraints to choose a model for a permutation problem, and a local consistency property to enforce upon it. They also illustrate a methodology, as well as a measure of constraint tightness, that can be used to compare different constraint models.