Certifying Solvers for Clique and Maximum Common (Connected) Subgraph Problems

An algorithm is said to be certifying if it outputs, together with a solution to the problem it solves, a proof that this solution is correct. We explain how state of the art maximum clique, maximum weighted clique, maximal clique enumeration and maximum common (connected) induced subgraph algorithms can be turned into certifying solvers by using pseudo-Boolean models and cutting planes proofs, and demonstrate that this approach can also handle reductions between problems. The generality of our results suggests that this method is ready for widespread adoption in solvers for combinatorial graph problems

CONJURE: automatic generation of constraint models from problem specifications

Funding: Engineering and Physical Sciences Research Council (EP/V027182/1, EP/P015638/1), Royal Society (URF/R/180015).When solving a combinatorial problem, the formulation or model of the problem is critical tothe efficiency of the solver. Automating the modelling process has long been of interest because of the expertise and time required to produce an effective model of a given problem. We describe a method to automatically produce constraint models from a problem specification written in the abstract constraint specification language Essence. Our approach is to incrementally refine the specification into a concrete model by applying a chosen refinement rule at each step. Any nontrivial specification may be refined in multiple ways, creating a space of models to choose from. The handling of symmetries is a particularly important aspect of automated modelling. Many combinatorial optimisation problems contain symmetry, which can lead to redundant search. If a partial assignment is shown to be invalid, we are wasting time if we ever consider a symmetric equivalent of it. A particularly important class of symmetries are those introduced by the constraint modelling process: modelling symmetries. We show how modelling symmetries may be broken automatically as they enter a model during refinement, obviating the need for an expensive symmetry detection step following model formulation. Our approach is implemented in a system called Conjure. We compare the models producedby Conjure to constraint models from the literature that are known to be effective. Our empirical results confirm that Conjure can reproduce successfully the kernels of the constraint models of 42 benchmark problems found in the literature.Publisher PDFPeer reviewe

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

Models and Algorithms for School Timetabling

In constraint programming, combinatorial problems are specified declaratively in terms of constraints. Constraints are relations over problem variables that define the space of solutions by specifying restrictions on the values that variables may take simultaneously. To solve problems stated in terms of constraints, the constraint programmer typically combines chronological backtracking with constraint propagation that identifies infeasible value combinations and prunes the search space. In recent years, constraint programming has emerged as a key technology for combinatorial optimization in industrial applications. In this success, global constraints have been playing a vital role. Global constraints are carefully designed abstractions that, in a concise and natural way, allow to model problems that arise in different fields of application. For example, the alldiff constraint allows to state that variables must take pairwise distinct values; it has numerous applications in timetabling and scheduling. In school timetabling, we are required to schedule a given set of meetings between students and teachers s.t. the resulting timetables are feasible and acceptable to all people involved. Since schools differ in their educational policies, the school-timetabling problem occurs in several variations. Nevertheless, a set of entities and constraints among them exist that are common to these variations. This common core still gives rise to NP-complete combinatorial problems. In the first place, this thesis proposes to model the common core of school-timetabling problems by means of global constraints. The presentation continues with a series of operational enhancements to the resulting problem solver which are grounded on the "track parallelization problem" (TPP). A TPP is specified by a set of task sets which are called "tracks". The problem of solving a TPP consists in scheduling the tasks s.t. the tracks are processed in parallel. We show how to infer TPPs in school timetabling and we investigate two ways of TPP propagation: On the one hand, we utilize TPPs to down-size our models. On the other hand, we propagate TPPs to prune the search space. To this end, we introduce the TPP constraint along with a suitable constraint solver for modeling and solving TPPs in a finite-domain constraint programming framework. To investigate our problem solvers' behavior, we performed a large-scale empirical study. When designing the experiment, the top priority was to obtain results that are both reliable from a statistical point of view and practically relevant. To this end, the sample sizes have been chosen accordingly - for each school, our problem set contains 1000 problems - and the problems have been generated from detailed models of ten representative schools. Our timetabling engine essentially embeds network-flow techniques and value sweep pruning into chronological backtracking