Experimental Evaluation of Branching Schemes for the CSP

The search strategy of a CP solver is determined by the variable and value ordering heuristics it employs and by the branching scheme it follows. Although the effects of variable and value ordering heuristics on search effort have been widely studied, the effects of different branching schemes have received less attention. In this paper we study this effect through an experimental evaluation that includes standard branching schemes such as 2-way, d-way, and dichotomic domain splitting, as well as variations of set branching where branching is performed on sets of values. We also propose and evaluate a generic approach to set branching where the partition of a domain into sets is created using the scores assigned to values by a value ordering heuristic, and a clustering algorithm from machine learning. Experimental results demonstrate that although exponential differences between branching schemes, as predicted in theory between 2-way and d-way branching, are not very common, still the choice of branching scheme can make quite a difference on certain classes of problems. Set branching methods are very competitive with 2-way branching and outperform it on some problem classes. A statistical analysis of the results reveals that our generic clustering-based set branching method is the best among the methods compared.Comment: To appear in the 3rd workshop on techniques for implementing constraint programming systems (TRICS workshop at the 16th CP Conference), St. Andrews, Scotland 201

Reasoning from Last Conflict(s) in Constraint Programming

International audienceConstraint programming is a popular paradigm to deal with combinatorial problems in arti cial intelligence. Backtracking algorithms, applied to constraint networks, are commonly used but su er from thrashing, i.e. the fact of repeatedly exploring similar subtrees during search. An extensive literature has been devoted to prevent thrashing, often classi ed into look-ahead (constraint propagation and search heuristics) and look-back (intelligent backtracking and learning) approaches. In this paper, we present an original look-ahead approach that allows to guide backtrack search toward sources of conicts and, as a side e ect, to obtain a behavior similar to a backjumping technique. The principle is the following: after each conict, the last assigned variable is selected in priority, so long as the constraint network cannot be made consistent. This allows us to find, following the current partial instantiation from the leaf to the root of the search tree, the culprit decision that prevents the last variable from being assigned. This way of reasoning can easily be grafted to many variations of backtracking algorithms and represents an original mechanism to reduce thrashing. Moreover, we show that this approach can be generalized so as to collect a (small) set of incompatible variables that are together responsible for the last conict. Experiments over a wide range of benchmarks demonstrate the e ectiveness of this approach in both constraint satisfaction and automated arti cial intelligence planning

Local consistency and sat-solvers

Abstract Local consistency techniques such as k-consistency are a key component of specialised solvers for constraint satisfaction problems. In this paper we show that the power of using k-consistency techniques on a constraint satisfaction problem is precisely captured by using a particular inference rule, which we call negative-hyper-resolution, on the standard direct encoding of the problem into Boolean clauses. We also show that current clauselearning SAT-solvers will discover in expected polynomial time any inconsistency that can be deduced from a given set of clauses using negative-hyper-resolvents of a fixed size. We combine these two results to show that, without being explicitly designed to do so, current clause-learning SAT-solvers efficiently simulate k-consistency techniques, for all fixed values of k. We then give some experimental results to show that this feature allows clause-learning SAT-solvers to efficiently solve certain families of constraint problems which are challenging for conventional constraint-programming solvers

Constraint satisfaction problems in clausal form

This is the report-version of a mini-series of two articles on the foundations of satisfiability of conjunctive normal forms with non-boolean variables, to appear in Fundamenta Informaticae, 2011. These two parts are here bundled in one report, each part yielding a chapter. Generalised conjunctive normal forms are considered, allowing literals of the form "variable not-equal value". The first part sets the foundations for the theory of autarkies, with emphasise on matching autarkies. Main results concern various polynomial time results in dependency on the deficiency. The second part considers translations to boolean clause-sets and irredundancy as well as minimal unsatisfiability. Main results concern classification of minimally unsatisfiable clause-sets and the relations to the hermitian rank of graphs. Both parts contain also discussions of many open problems.Comment: 91 pages, to appear in Fundamenta Informaticae, 2011, as Constraint satisfaction problems in clausal form I: Autarkies and deficiency, Constraint satisfaction problems in clausal form II: Minimal unsatisfiability and conflict structur

2-way vs d-way branching for CSP

Abstract. Most CSP algorithms are based on refinements and extensions of backtracking, and employ one of two simple “branching schemes”: 2-way branching or d-way branching, for domain size d. The schemes are not equivalent, but little is known about their relative power. Here we compare them in terms of how efficiently they can refute an unsatisfiable instance with optimal branching choices, by studying two variants of the resolution proof system, denoted C-RES and NG-RES, which model the reasoning of CSP algorithms. The tree-like restrictions, tree-C-RES and tree-NG-RES, exactly capture the power of backtracking with 2-way branching and d-way branching, respectively. We give a family instances which require exponential sized search trees for backtracking with d-way branching, but have size O(d 2 n) search trees for backtracking with 2way branching. We also give a natural branching strategy with which backtracking with 2-way branching finds refutations of these instances in time O(d 2 n 2). The unrestricted variants of C-RES and NG-RES can simulate the reasoning of algorithms which incorporate learning and k-consistency enforcement. We show exponential separations between C-RES and NG-RES, as well as between the tree-like and unrestricted versions of each system. All separations given are nearly optimal.

Solving of constraint satisfaction problems by reduction to SAT

Mnogi realni problemi se danas mogu predstaviti u obliku problema zadovoljenja ogranicenja (CSP) i zatim rešiti nekom od mnogobrojnih tehnika za rešavanje ovog problema. Jedna od tehnika podrazumeva svođenje na problem SAT, tj. problem iskazne zadovoljivosti. Promenljive i ogranicenja problema CSP se prevode (kodiraju) u SAT instancu, ona se potom rešava pomocu modernih SAT rešavaca i rešenje se, ako postoji, prevodi u rešenje problema CSP. Glavni cilj ove teze je unapređenje rešavanja problema CSP svođenjem na SAT. Razvijena su dva nova hibridna kodiranja (prevođenja u SAT formulu) koja kombinuju dobre strane postojecih kodiranja. Dat je dokaz korektnosti jednog od kodiranja koji do sada nije postojao u literaturi. Razvijen je sistem meSAT koji omogucava svođenje problema CSP na SAT pomocu cetiri osnovna i dva hibridna kodiranja, kao i rešavanje problema CSP svođenjem na dva problema srodna problemu SAT, SMT i PB. Razvijen je portfolio za automatski odabir kodiranja/rešavaca za ulaznu instancu koju je potrebno rešiti i pokazano je da je razvijeni portfolio uporediv sa najefikasnijim savremenim pristupima. Prikazan je novi pristup zasnovan na kratkim vremenskim ogranicenjima sa ciljem da se znacajno smanji vreme pripreme portfolija. Pokazano je da se ovim pristupom dobijaju rezultati konkurentni onima koji se dobijaju korišcenjem standardnog vremena za pripremu. Izvršeno je poređenje nekoliko tehnika mašinskog ucenja, sa ciljem da se ustanovi koja od njih je pogodnija za pristup sa kratkim treniranjem. Prikazan je jedan realan problem, problem raspoređivanja kontrolora leta, kao i tri njegova modela. Veliki broj metoda rešavanja i raznovrsnih rešavaca je upotrebljeno za rešavanje ovog problema. Razvijeno je više optimizacionih tehnika koje imaju za cilj pronalaženje optimalnih rešenja problema. Pokazuje se da je najefikasnija hibridna tehnika koja kombinuje svođenje na SAT i lokalnu pretragu. Razmotren je i problem sudoku, kao i postojece tehnike rešavanja sudoku zagonetki vecih dimenzija od 9 x 9. Pokazuje se da je u rešavanju ovih zagonetki najefikasnije vec postojece svođenje na SAT. Unapređen je postojeci algoritam za generisanje velikih sudoku zagonetki. Pokazano je da jednostavna pravila preprocesiranja dodatno unapređuju brzinu generisanja sudokua.Many real-world problems can be modeled as constraint satisfaction problems (CSPs) and then solved by one of many available techniques for solving these problems. One of the techniques is reduction to SAT, i.e. Boolean Satisfiability Problem. Variables and constraints of CSP are translated (encoded) to SAT instance, that is then solved by state-of-the-art SAT solvers and solution, if exists, is translated to the solution of the original CSP. The main aim of this thesis is to improve CSP solving techniques that are using reduction to SAT. Two new hybrid encodings of CSPs to SAT are presented and they combine good sides of the existing encodings. We give the proof of correctness of one encoding that did not exist in literature. We developed system meSAT that enables reduction of CSPs to SAT by using 4 basic and 2 hybrid encodings. The system also enables solving of CSPs by reduction to two problems related to SAT, SMT and PB. We developed a portfolio for automated selection of encoding/solver to be used on some new instance that needs to be solved. The developed portfolio is comparable with the state-of-the-art portfolios. We developed a hybrid approach based on short solving timeouts with the aim of significantly reducing the preparation time of a portfolio. By using this approach, we got results comparable to the ones obtained by using preparation time of usual length. We made comparison between several machine learning techniques with the aim to find out which one is the best suited for the short training approach. The problem of assigning air traffic controllers to shifts is described and three models of this problem are presented. We used a large number of different solving methods and a diverse set of solvers for solving this problem. We developed optimization techniques that aim to find optimal solutions of the problem. A hybrid technique combining reduction to SAT and local search is shown to be the most efficient one. We also considered sudoku puzzles and the existing techniques of solving the puzzles of greater size than 9x9. Amongst the used techniques, the existing reduction to SAT is the most efficient in solving these puzzles. We improved the existing algorithm for generating large sudoku puzzles. It is shown that simple preprocessing rules additionally improve speed of generating large sudokus

Identifying sources of global contention in constraint satisfaction search

Much work has been done on learning from failure in search to boost solving of combinatorial problems, such as clause-learning and clause-weighting in boolean satisfiability (SAT), nogood and explanation-based learning, and constraint weighting in constraint satisfaction problems (CSPs). Many of the top solvers in SAT use clause learning to good effect. A similar approach (nogood learning) has not had as large an impact in CSPs. Constraint weighting is a less fine-grained approach where the information learnt gives an approximation as to which variables may be the sources of greatest contention. In this work we present two methods for learning from search using restarts, in order to identify these critical variables prior to solving. Both methods are based on the conflict-directed heuristic (weighted-degree heuristic) introduced by Boussemart et al. and are aimed at producing a better-informed version of the heuristic by gathering information through restarting and probing of the search space prior to solving, while minimizing the overhead of these restarts. We further examine the impact of different sampling strategies and different measurements of contention, and assess different restarting strategies for the heuristic. Finally, two applications for constraint weighting are considered in detail: dynamic constraint satisfaction problems and unary resource scheduling problems