Backtracking

It is shown that a process similar to Earley's algorithm can be generated by a simple top-down backtracking parser, when augmented by automatic memoization. The memoized parser has the same complexity as Earley's algorithm, but parses constituents in a different order. Techniques for deriving memo functions are described, with a complete implementation in Common Lisp, and an outline of a macro-based approach for other languages.

Local search is often able to solve larger problems than systematic backtracking. To apply it to a constraint satisfaction problem, the problem is often treated as an optimization problem in which the search space is the total assignments, and the number of constraint violations is to be minimized to zero. Though often successful, this approach is sometimes unsuitable for structured problems with few solutions. An alternative is to explore the consistent partial assignments, minimizing the number of unassigned variables to zero. A local search algorithm of this type violates no constraints and can exploit cost and propagation techniques. This paper describes such an algorithm for balanced incomplete block design generation. On a large set of instances it out-performs several backtrackers and a neural network with simulated annealing.

Many constraint satisfaction and optimisation problems can be expressed using linear constraints on pseudo-Boolean (0/1) variables. Problems expressed in this form are usually solved by integer programming techniques, but good results have also been obtained using generalisations of SAT algorithms based on both backtracking and local search. A recent class of algorithm uses randomised backtracking to combine constraint propagation with

Systematic backtracking is used in many constraint solvers and combinatorial optimisation al- gorithms. It is complete and can be combined with powerful search pruning techniques such as branch- and-bound, constraint propagation and dynamic variable ordering. However, it often scales poorly to large problems. Local search is incomplete, and has the additional drawback that it cannot exploit pruning tech- niques, making it uncompetitive on some problems. Nevertheless its scalability makes it superior for many large applications. This paper describes a hybrid approach called Incomplete Dynamic Backtracking, a very flexible form of backtracking that sacrifices completeness to achieve the scalability of local search. It is combined with forward checking and dynamic variable ordering, and evaluated on three combinatorial problems: on the n-queens problem it out-performs the best local search algorithms; it finds large optimal Golomb rulers much more quickly than a constraint-based backtr...

Many combinatorial problems can be modeled as 0/1 integer linear programs. Problems expressed in this form are usually solved by Operations Research algorithms, but good results have also been obtained using generalised SAT algorithms based on backtracking or local search, after transformation to pseudo-Boolean form. A third class of SAT algorithm uses non-systematic backtracking to combine constraint propagation with local search-like scalability, at the cost of completeness. This paper describes such an algorithm for pseudo-Boolean models. Experimental results on a variety of problems are encouraging, in some cases yielding improved solutions or performance compared to previous algorithms.

A variety of hybrids of Constraint Programming, Artificial Intelligence and Operations Research techniques have given impressive results. Three recent approaches are (i) the use of relaxations in constraint systems, (ii) non-systematic backtracking to boost the scalability of constraint solvers, and (iii) non-systematic backtracking to boost the scalability of branch-and-bound search. This paper describes a hybrid of all three approaches that combines non-systematic backtracking, SAT-based inference and linear relaxation. It is intended for large MIPs that are hard for reasons of both optimisation and feasibility. Such problems are of practical as well as theoretical interest and we expect the hybrid to find many applications. It is currently under development and results will be reported at the workshop.

Logic programming provides a declarative framework for modeling and solving many combinatorial problems. Until recently, it was not competitive with state of the art planning techniques partly due to search capabilities limited to backtracking. Recent development brought more advanced search techniques to logic programming such as tabling that simplifies implementation and exploitation of more sophisticated search algorithms. Together with rich modeling capabilities this progress brings tabled logic programing on a par with current best planners. The paper brings an initial experimental study comparing various approaches to search for sequential plans in the Picat planning module.

The Distributed Constraint Optimization Problem (DCOP) is a fundamental framework of multi-agent systems. With DCOPs a multi-agent system is represented as a set of variables and a set of constraints/cost functions. Distributed task scheduling and distributed resource allocation can be formalized as DCOPs. In this paper, we propose an efficient method that applies directed soft arc consistency to a DCOP. In particular, we focus on DCOP solvers that employ pseudo-trees. A pseudo-tree is a graph structure for a constraint network that represents a partial ordering of variables. Some pseudo-tree-based search algorithms perform optimistic searches using explicit/implicit backtracking in parallel. However, for cost functions taking a wide range of cost values, such exact algorithms require many search iterations. Therefore additional improvements are necessary to reduce the number of search iterations. A previous study used a dynamic programming-based preprocessing technique that estimates the lower bound values of costs. However, there are opportunities for further improvements of efficiency. In addition, modifications of the search algorithm are necessary to use the estimated lower bounds. The proposed method applies soft arc consistency (soft AC) enforcement to DCOP. In the proposed method, directed soft AC is performed based on a pseudo-tree in a bottom up manner. Using the directed soft AC, the global lower bound value of cost functions is passed up to the root node of the pseudo-tree. It also totally reduces values of binary cost functions. As a result, the original problem is converted to an equivalent problem. The equivalent problem is efficiently solved using common search algorithms. Therefore, no major modifications are necessary in search algorithms. The performance of the proposed method is evaluated by experimentation. The results show that it is more efficient than previous methods.

In this paper, we extend the traditional formalization of distributed constraint satisfaction problems (DisCSP) to a quantified DisCSP. A quantified DisCSP includes several universally quantified variables, while all of the variables in a traditional DisCSP are existentially quantified. A universally quantified variable represents a choice of nature or an adversary. A quantified DisCSP formalizes a situation where a team of agents is trying to make a robust plan against nature or an adversary. In this paper, we present the formalization of such a quantified DisCSP and develop an algorithm for solving it by generalizing the asynchronous backtracking algorithm used for solving a DisCSP. In this algorithm, agents communicate a value assignment called a good in addition to the nogood used in asynchronous backtracking. Interestingly, the procedures executed by an adversarial/cooperative agent for good/nogood are completely symmetrical. Furthermore, we develop a method that improves this basic algorithm. Experimental evaluation results illustrate that we observe an easy-hard-easy transition by changing the tightness of the constraints, while very loose problem instances are relatively hard. The modification of the basic algorithm is also effective and reduces the number of cycles by approximately 25% for the hardest problem instances.

Mobile ad-hoc network is a prioritized mobility based wireless network It can be easily deployed in every aspect, without underlying foundation and base support. It is a collection of mobile nodes that help and comply with their neighbor nodes to convey packets from source to destination. There is a need to discover a more power efficient technique for gainful network access and to avoid network deficiency. So the existing protocol, an energy adequate AODV protocol is blended for perfect results and the four parameters that are concentrated are Bandwidth, Power, Hop count and Time. A dominant part of the process is changing the route table values using (PCBR) Position Changes Based Routing protocol. Our proposed methods and protocol will give enhanced results than the earlier technique.

Chapter 2 Algorithms from the Literature : : : : : : : : : : : : : 17 2.1 Overview of the Chapter : : : : : : : : : : : : : : : : : : : : : : : : 17 2.2 De nitions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 18 2.3 Backtracking : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20 2.4 Backmarking : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 25 ...

Sudoku is one of the most famous puzzles in the world, it has variants such as Hyper
Sudoku. These puzzles can be solved by different AI algorithms. This project consists of a
solver that can solve standard Sudokus and Hyper Sudokus using different computer
algorithms and compares the performance of these methods. The solver has a command
line interface, and allows the user to type his own Sudoku or solve a Sudoku in the database
using one of the algorithms.

In the setting of session behaviours, we study an extension of the concept of compliance when a disciplined form of backtracking and of output skipping is present. After adding checkpoints to the syntax of session behaviours, we formalise the operational semantics via a LTS, and define natural notions of checkpoint compliance and sub-behaviour, which we prove to be both decidable. Then we extend the operational semantics with skips and we show the decidability of the obtained compliance.

Behavioral contracts are abstract descriptions of the communications that clients and servers perform. Behavioral contracts come naturally equipped with a notion of compliance: when a client and a server follow compliant contracts, their interaction is guaranteed to progress or successfully complete. We study two extensions of contracts, dealing respectively with backtracking and with speculative execution. We show that the two extensions give rise to the same notion of compliance. As a consequence, they also give rise to the same subcontract relation, which determines when one server can be replaced by another preserving compliance. Moreover, compliance and subcontract relation are both decidable in polynomial time.

The inclusion of universal quantification and a form of implication in goals in logic programming is considered. These additions provide a logical basis for scoping but they also raise new implementation problems. When universal and existential quantifiers are permitted to appear in mixed order in goals, the devices of logic variables and unification that are employed in solving existential goals must be modified to ensure that constraints arising out of the order of quantification are respected. Suitable modifications that are based on attaching numerical tags to constants and variables and on using these tags in unification are described. The resulting devices are amenable to an efficient implementation and can, in fact, be assimilated easily into the usual machinery of the Warren Abstract Machine (WAM). The provision of implications in goals results in the possibility of program clauses being added to the program for the purpose of solving specific subgoals. A naive scheme based on asserting and retracting program clauses does not suffice for implementing such additions for two reasons. First, it is necessary to also support the resurrection of an earlier existing program in the face of backtracking. Second, the possibility for implication goals to be surrounded by quantifiers requires a consideration of the parameterization of program clauses by bindings for their free variables. Devices for supporting these additional requirements are described as also is the integration of these devices into the WAM. Further extensions to the machine are outlined for handling higher-order additions to the language. The ideas presented here are relevant to the implementation of the higher-order logic programming language λProlog.

Puji syukur penulis panjatkan ke hadirat Tuhan Yang Maha Esa karena atas rahmat-Nya, karya tulis ilmiah berjudul "Algoritma Pencarian: Penerapan dan Optimalisasi dalam Ilmu Komputer" ini dapat diselesaikan. Karya ini disusun untuk memberikan wawasan mengenai pentingnya algoritma pencarian dalam dunia teknologi, khususnya dalam meningkatkan efisiensi proses data. Penulis mengucapkan terima kasih kepada semua pihak yang telah memberikan dukungan selama penyusunan karya ini. Kritik dan saran sangat diharapkan demi kesempurnaan karya ini di masa mendatang. Semoga karya ini bermanfaat bagi para pembaca dan menambah wawasan di bidang ilmu komputer.

One of the programming problems in the 2002 Pacific Northwest regional ACM ICPC contest provides a new way to teach backtracking and also provides a very powerful example of a forward-looking bounding function. This article presents the problem, the bounding function, and timing information of implementations with and without the bounding function. It also provides the URL for access to the programs themselves.

We describe a reduction from temporal property verification to a program analysis problem. Our reduction is an encoding which, with the use of procedures and nondeterminism, enables existing interprocedural program analysis tools to naturally perform the reasoning necessary for proving temporal properties (eg. backtracking, eventuality checking, tree counterexamples for branching-time properties, abstraction refinement, etc.). Our reduction is state-based in nature but also forms the basis of an efficient algorithm for verifying trace-based properties, when combined with an iterative symbolic determinization technique, due to Cook and Koskinen. In this extended version, we formalize our encoding as a guarded transition system G, parameterized by a finite set of ranking functions and the temporal logic property. We establish soundness between a safety property of G and the validity of a branching time temporal logic property ∀CTL. ∀CTL is a sufficient logic for proving properties wri...

Multi-task learning is a powerful method for solving several tasks jointly by learning robust representation. Optimization of the multi-task learning model is a more complex task than a single-task due to task conflict. Based on theoretical results, convergence to the optimal point is guaranteed when step size is chosen through line search. But, usually, line search for the step size is not the best choice due to the large computational time overhead. We propose a novel idea for line search algorithms in multi-task learning. The idea is to use latent representation space instead of parameter space for finding step size. We examined this idea with backtracking line search. We compare this fast backtracking algorithm with classical backtracking and gradient methods with a constant learning rate on MNIST, CIFAR-10, Cityscapes tasks. The systematic empirical study showed that the proposed method leads to more accurate and fast solution, than the traditional backtracking approach and keep competitive computational time and performance compared to the constant learning rate method.

This study presents a novel Equiangular Direction Method (EDM) to solve a multi-objective optimization problem. We consider optimization problems, where multiple differentiable losses have to be minimized. The presented method computes descent direction in every iteration to guarantee equal relative decrease of objective functions. This descent direction is based on the normalized gradients of the individual losses. Therefore, it is appropriate to solve multi-objective optimization problems with multi-scale losses. We test the proposed method on the imbalanced classification problem and multi-task learning problem, where standard datasets are used. EDM is compared with other methods to solve these problems.

Multi-task learning is a powerful method for solving several tasks jointly by learning robust representation. Optimization of the multi-task learning model is a more complex task than a single-task due to task conflict. Based on theoretical results, convergence to the optimal point is guaranteed when step size is chosen through line search. But, usually, line search for the step size is not the best choice due to the large computational time overhead. We propose a novel idea for line search algorithms in multi-task learning. The idea is to use latent representation space instead of parameter space for finding step size. We examined this idea with backtracking line search. We compare this fast backtracking algorithm with classical backtracking and gradient methods with a constant learning rate on MNIST, CIFAR-10, Cityscapes tasks. The systematic empirical study showed that the proposed method leads to more accurate and fast solution, than the traditional backtracking approach and kee...

This study presents a novel Equiangular Direction Method (EDM) to solve a multi-objective optimization problem. We consider optimization problems, where multiple differentiable losses have to be minimized. The presented method computes descent direction in every iteration to guarantee equal relative decrease of objective functions. This descent direction is based on the normalized gradients of the individual losses. Therefore, it is appropriate to solve multi-objective optimization problems with multi-scale losses. We test the proposed method on the imbalanced classification problem and multi-task learning problem, where standard datasets are used. EDM is compared with other methods to solve these problems.

One of the key factors in the efficiency of backtracking algorithms is the rule they use to decide on which variable to branch next (namely, the variable ordering heuristics). In this paper, we give a formulation of dynamic variable ordering heuristics that takes into account the properties of the neighborhood of the variable.

Cet article présente plusieurs contributions au "Conflict Driven Clauses Learning" (CDCL), qui est une des composantes clés des solveurs SAT modernes. Tout d'abord, nous montrons que, à partir du graphe d'implication, les clauses assertives obtenues en utilisant le principe du premier point d'implication unique ("First Unique Implication Point" (FUIP)) sont optimales en terme de saut arrière. Puis nous proposons une extension du graphe d'implication contenant de nouveaux arcs appelés arcs arrières. Ces arcs sont obtenus en tenant compte des clauses satisfaites qui sont habituellement ignorées par l'analyse du conflit. Cette extension capture plus fidèlement l'ensemble du processus de propagation et ouvre de nouvelles perspectives pour les approches fondées sur CDCL. Entre autres avantages, notre extension du graphe d'implication conduit à un nouveau schéma d'analyse des conflits qui exploite les arcs ajoutés et permet des retours arrières plus haut dans l'arbre de recherche. Les résultats expérimentaux montrent que l'intégration de notre système d'analyse des conflits généralisés au sein de solveurs dernier-cri améliore sensiblement leurs performances.

Dans cet article, nous proposons une analyse statique des différentes contraintes d'un réseau afin d'identifier certaines propriétés (ou caractéristiques) générales. L'utilisation de ces propriétés rend possible une inférence de supports qui permet de réduire le nombre de tests de consistance. En effet, l'exploitation de certaines propriétés identifiées lors d'une phase de pré-traitement peut apporter une amélioration substantielle des algorithmes de recherche qui maintiennent une forme de consistance locale telle que la consistance d'arc. Les résultats d'expérimentations menées sur de nombreuses classes d'instances démontrent l'intérêt de cette approche.

One of the key factors in the efficiency of backtracking algorithms is the rule they use to decide on which variable to branch next (namely, the variable ordering heuristics). In this paper, we give a formulation of dynamic variable ordering heuristics that takes into account the properties of the neighborhood of the variable.

In this paper we give a short introduction to logic programming approach to knowledge representation and reasoning. The intention is to help the reader to develop a 'feel' for the field's history and some of its recent developments. The discussion is mainly limited to logic programs under the answer set semantics. For understanding of approaches to logic programming built on well-founded semantics, general theories of argumentation, abductive reasoning, etc., the reader is referred to other publications.

PρLog is an experimental extension of logic programming with strategic conditional transformation rules, combining Prolog with ρLog calculus. The rules perform nondeterministic transformations on hedges. Queries may have several results that can be explored on backtracking. Strategies provide a control on rule applications in a declarative way. With strategy combinators, the user can construct more complex strategies from simpler ones. Matching with four different kinds of variables provides a flexible mechanism of selecting (sub)terms during execution. We give an overview on programming with strategies in PρLog and demonstrate how rewriting strategies can be expressed.

In the scope of distributed constraint reasoning, the main algorithms presented so far have a feature in common: the addition of links between previously unrelated agents, before or during search. Our work presents a new search procedure for finding a solution in a distributed constraint satisfaction problem. This algorithm makes use of some of the good properties of centralized dynamic backtracking. It is sound, complete and allows a high level of asynchronism by sidestepping the unnecessary addition of links.

In the scope of distributed constraint reasoning, the main algorithms presented so far have a feature in common: the addition of links between previously unrelated agents, before or during search. Our work presents a new search procedure for finding a solution in a distributed constraint satisfaction problem. This algorithm makes use of some of the good properties of centralized dynamic backtracking. It is sound, complete and allows a high level of asynchronism by sidestepping the unnecessary addition of links.

In the last years, the AI community has shown an increasing interest in distributed problem solving. In the scope of distributed constraint reasoning, several asynchronous backtracking procedures have been proposed for nding solutions in a constraint network distributed among several computers. They dier in the way they store failing combinations of values (nogoods), and in the way they check the

Following the pioneer work of Yokoo and colleagues on the ABT (asynchronous backtracking) algorithm, several ABT-based procedures have been proposed for solving distributed constraint networks. They differ in the way they store nogoods, but they all use additional communication links between unconnected agents to detect obsolete information. In this paper, we propose a new asynchronous backtracking algorithm which does not need to add links between initially unconnected agents. To make the description simpler and to facilitate the comparisons between algorithms, we present a unifying framework from which the new algorithm we propose, as well as existing ones, are derived. We provide an experimental evaluation of these algorithms.

With the aim of supporting the process of adapting railway infrastructure to present and future traffic needs, we have developed a method to build train timetables efficiently. In this work, we describe the problem in terms of constraints derived from railway infrastructure, user requirements and traffic constraints, and we propose a method to solve it efficiently. This method carries out the search by assigning values to variables in a given order and verifying the satisfaction of constraints where these are involved. When a constraint is not satisfied, a guided backtracking is done. The technique reduces the search space allowing us to solve real and complex problems efficiently.

Student evaluation is an essential part of education and is done through the system of examinations. Examinations generally use question papers as an important component to determine the quality of the students. Examination question paper generation is a multi-constraint concurrent optimization problem. Question papers generated with random and backtracking algorithms are inefficient in handling multiple constraints such as total time for completion of the paper, total number of questions, module weightages, question types, knowledge points, difficulty level of questions etc,. In this paper we have proposed an innovative evolutionary approach that handles multi-constraints while generating question papers from a very large question bank. The proposed Elitist Multi-objective Differential Evolution Approach (EMODEA) has its advantage of simple structure, ease of use, better computational speed and good robustness. It is identified to be more suitable for combinatorial problems as comp...

In today's competitive market designing of digital systems (hardware as well as software) faces tremendous challenges. In fact, notwithstanding an ever decreasing project budget, time to market and product lifetime, designers are faced with an ever increasing system complexity and customer expected quality. The above situation calls for better and better formal verification techniques at all steps of the design flow. This special issue is devoted to publishing revised versions of contributions first presented at the 12th Advanced Research Working Conference on Correct Hardware Design and Verification Methods (CHARME) held 21-24 October 2003 in L'Aquila, Italy. Authors of well regarded papers from CHARME'03 were invited to submit to this special issue. All papers included here have been suitably extended and have undergone an independent round of reviewing.

In this paper we introduce a methodology for utilizing concurrency in place of backtracking in the implementation of nondeterministic algorithms. This is achieved in an applicative setting through the use of the Friedman-Wise multiprogramming primitive frons, and a paradigm which views the action of nondeterministic algorithms as one of data structure construction. The element by element nondeterminism arising from a linearized search is replaced by a control structure which is oriented towards constructing sets of partial computations. This point of view is facilitated by the use of suspensions, which allow control disciplines to be embodied in the form of conceptual data structures that in reality manifest themselves only for purposes of control. We apply this methodology to the class of problems usually solved through the use of simple backtracking (e.g. "'eight queens"), and to a problem presented by Lindstrom to illustrate the use of coroutine controlled backtracking, to produce backtrack-free solutions. Our solution to the latter illustrates the coroutine capability of suspended structures, but also demonstrates a need for further investigations into resolving problems of process communication in applicative languages. Nondeterministic algorithms Concurrency structures Applicative languages Multiprocessing Multisets Suspended

We introduce a new programming language construct, Interactors, supporting the agent-oriented view that programming is a dialog between simple, self-contained, autonomous building blocks. We define Interactors as an abstraction of answer generation and refinement in Logic Engines resulting in expressive language extension and metaprogramming patterns, including emulation of Prolog's dynamic database. A mapping between backtracking based answer generation in the callee and "forward" recursion in the caller enables interaction between different branches of the callee's search process and provides simplified design patterns for algorithms involving combinatorial generation and infinite answer streams. Interactors extend language constructs like Ruby, Python and C#'s multiple coroutining block returns through yield statements and they can emulate the action of monadic constructs and catamorphisms in functional languages.

Prolog's ability to return multiple answers on backtracking provides an elegant mechanism to derive reversible encodings of combinatorial objects as Natural Numbers i.e. ranking and unranking functions. Starting from a generalization of Ackerman's encoding of Hereditarily Finite Sets with Urelements and a novel tupling/untupling operation, we derive encodings for Finite Functions and use them as building blocks for an executable theory of Hereditarily Finite Functions. The more difficult problem of ranking and unranking Hereditarily Finite Permutations is then tackled using Lehmer codes and factoradics.

With sound unification, Definite Clause Grammars and compact expression of combinatorial generation algorithms, logic programming is shown to conveniently host a declarative playground where interesting properties and behaviors emerge from the interaction of heterogenous but deeply connected computational objects. Compact combinatorial generation algorithms are given for several families of lambda terms, including open, closed, simply typed and linear terms as well as type inference and normal order reduction algorithms. We describe a Prolog-based combined lambda term generator and type-inferrer for closed well-typed terms of a given size, in de Bruijn notation. We introduce a compressed de Bruijn representation of lambda terms and define its bijections to standard representations. Our compressed terms facilitate derivation of size-proportionate ranking and unranking algorithms of lambda terms and their inferred simple types. By taking advantage of Prolog's unique bidirectional execution model and sound unification algorithm, our generator can build "customized" closed terms of a given type. This relational view of terms and their types enables the discovery of interesting patterns about frequently used type expressions occurring in well-typed functional programs. Our study uncovers the most "popular" types that govern function applications among a about a million small-sized lambda terms and hints toward practical uses to combinatorial software testing. The S and K combinator expressions form a well-known Turing-complete subset of the lambda calculus. We specify evaluation, type inference and combinatorial generation algorithms for SKcombinator trees. In the process, we unravel properties shedding new light on interesting aspects of their structure and distribution. We study the proportion of well-typed terms among size-limited SK-expressions as well as the type-directed generation of terms of sizes smaller then the size of their simple types. We also introduce the well-typed frontier of an untypable term and we use it to design a simplification algorithm for untypable terms taking advantage of the fact that well-typed terms are normalizable. A uniform representation, as binary trees with empty leaves, is given to expressions built with Rosser's X-combinator, natural numbers, lambda terms and simple types. Using this shared representation, ranking/unranking algorithm of lambda terms to tree-based natural numbers are described. Our algorithms, expressed as an incrementally developed literate Prolog program, implement a declarative playground for exploration of representations, encodings and computations with uniformly represented lambda terms, types, combinators and tree-based arithmetic.

The workflow satisfiability problem (WSP) asks whether there exists an assignment of authorised users to the steps in a workflow specification, subject to certain constraints on the assignment. (Such an assignment is called valid.) The problem is NP-hard even when restricted to the large class of user-independent constraints. Since the number of steps k is relatively small in practice, it is natural to consider a parametrisation of the WSP by k. We propose a new fixed-parameter algorithm to solve the WSP with user-independent constraints. The assignments in our method are partitioned into equivalence classes such that the number of classes is exponential in k only. We show that one can decide, in polynomial time, whether there is a valid assignment in an equivalence class. By exploiting this property, our algorithm reduces the search space to the space of equivalence classes, which it browses within a backtracking framework, hence emerging as an efficient yet relatively simple-to-implement or generalise solution method. We empirically evaluate our algorithm against the state-of-the-art methods and show that it clearly wins the competition on the whole range of our test problems and significantly extends the domain of practically solvable instances of the WSP.

Sudoku is a puzzle game played on a grid that consists of 9 x 9 cells each belonging to three groups: one of nine rows, one of nine columns and one of nine sub grids (sometimes called regions). The game of Sudoku is basically based on Latin squares. The Sudoku was incepted in the year 1979 and was first published in the Dell Magazines as "Number Place" in the year 1984. The term Sudoku means a single number. The game begins with numbers already printed in some cells. The player must fill in the empty cells with the numbers 1 to 9 such that each column, row and region contains that number exactly once. There are several Sudoku applications that have already been developed by many programmers around the globe. In this paper, we give an overview of the work that we have performed on the development of the game of Sudoku that generates a 9 x 9 puzzle grid with various difficulty levels. The application also enables users to input their own puzzle and to be solved by the computer. The developed application also includes advanced features such as save, load and fast input validation. The solving algorithm of the developed Sudoku application has also been compared to some existing Sudoku applications for analysis.

Problem Solving systems customarily use backtrackingto deal with obstaclesthat they encounterin the courseoftryingto salvea problem. This paper outlines an approach in which the possible obstacles are investigated priorto the search for a solution.This providesa solution strategy that avoidsbacktracking.

Three connections between Dynamic Programming (DP) and Constraint Programming (CP) have previously been explored in the literature: DP-based global constraints, DPlike memoisation during tree search to avoid recomputing results, and subsumption of both by bucket elimination. In this paper we propose a new connection: any discrete DP algorithm can be directly modelled and solved as a constraint satisfaction problem (CSP) without backtracking. This has applications including the design of monolithic CP models for bilevel optimisation. We show that constraint filtering can occur between leader and follower variables in such models, and demonstrate the method on network interdiction.

In this paper we analyze the effect of selecting the root in a tree decomposition when using decomposition-based backtracking algorithms. We focus on optimization tasks for Graphical Models using the BTD algorithm. We show that the choice of the root typically has a dramatic effect in the solving performance. Then we investigate different simple measures to predict near optimal roots. Our study shows that correlations are often low, so the automatic selection of a near optimal root will require more sophisticated techniques.

This paper describes the main algorithms for solving constraint satisfaction problems, and includes the corresponding encodings (the main different with respect [3] is that, in their paper, they provide a more informal description of the algorithms). We consider three main algorithmic approaches: search, inference and hybrid methods. Search methods can be divided into systematic and non-systematic. We present backtracking as an example of systematic search, and local search as an example of nonsystematic search. Inference methods can be divided into complete and incomplete. We describe adaptive consistency as an example of complete inference, and several local consistency algorithms as incomplete inference. We also present some examples of hybrid methods which combine search and inference.

Collins introduced the cylindrical algebraic decomposition method for eliminating quantifiers in real arithmetic formulas. In our work we use this method for satisfiability checking in satisfiability modulo theories solver technologies, and tune it by trying to avoid some computation steps that are needed for quantifier elimination but not for satisfiability checking. We further propose novel data structures and adapt the method to work incrementally and to support backtracking. We verify the effectiveness experimentally by comparing different versions of the cylindrical algebraic decomposition used within a state-of-the-art satisfiability modulo theories solver.