Sabin, Mihaela C.

English

UNH Scholars' Repository

University of New HampshireUniversity of New Hampshire Scholars' RepositoryApplied Engineering and Sciences Scholarship Applied Engineering and Sciences1-1-2006Evaluation of solving methods for conditionalconstraint satisfaction problemMihaela C. SabinUniversity of New Hampshire, Manchester, mihaela.sabin@unh.eduFollow this and additional works at: https://scholars.unh.edu/unhmcis_facpubThis Article is brought to you for free and open access by the Applied Engineering and Sciences at University of New Hampshire Scholars' Repository.It has been accepted for inclusion in Applied Engineering and Sciences Scholarship by an authorized administrator of University of New HampshireScholars' Repository. For more information, please contact nicole.hentz@unh.edu.Recommended CitationMihaela Sabin and Esther Gelle, Evaluation of solving methods for conditional constraint satisfaction problems, Proceedings of theAAAI, 2006.Evaluation of Solving Methods for Conditional Constraint Satisfaction ProblemsMihaela SabinDepartment of Mathematics and Computer Science, Rivier College420 Main Street, Nashua, NH 03060, U.S.A., msabin@rivier.eduEsther GelleABB Switzerland Ltd, Corporate ResearchCH-5405 Baden, Switzerland, esther.gelle@ch.abb.comAbstractConditional constraint satisfaction problems (CondCSPs) ad-equately capture problem change at solving time by condi-tionally identifying those variables and constraints that arerelevant to final solutions. Real-world tasks with dynamicbehavior, such as configuration, design, diagnosis, planning,and hardware test generation, have been modeled more nat-urally with CondCSPs. Such interest has been matched bythe development of more effective algorithms that departfrom classical backtracking and incorporate local consistencychecking. Although performance results have been reportedfor these specialized algorithms, the experimental analysishas been conducted separately, using different test suites, andlittle is known about the algorithms’ relative performance. Inthis abstract we present a CondCSP solver that implementsdirect and reformulation-based algorithms, each of which us-ing forward checking and maintaining local consistency. Inour experimental analysis we have considered randomly gen-erated CondCSPs of diverse topologies in terms of problemdensity and satisfiability of the standard and conditional prob-lem components. Execution time results show that there is notone winner but that reformulation solving in conjunction withforward checking performs better on problems with larger so-lution sets, while direct solving in conjunction with maintain-ing arc consistency is always preferred over direct solving us-ing forward checking.A conditional constraint satisfaction problem (CondCSP)extends the standard CSP with a condition-based componentthat models problem change by allowing for “on-the-fly” se-lection of subsets of variables that participate in problem so-lutions. The formalism, introduced by (Mittal & Falken-hainer 1990) under the name of dynamic CSP, has been re-named to conditional CSP (Sabin & Freuder 1998) to qualifythe control component that models dynamic changes of thesolution space with predefined conditions. The original ap-plication domain in (Mittal & Falkenhainer 1990) is productconfiguration, in which a changing rather than fixed numberof components are part of final solutions. Selecting optionalcomponents to assemble configuration variants is naturallyrepresented in CondCSP with condition-based constraints.More recently, conditional constraint satisfaction has beenadapted to the planning domain (Do & Kambhampati 2000;Copyright c⃝ 2006, American Association for Artificial Intelli-gence (www.aaai.org). All rights reserved.Ambite et al. 2005; Miguel, Jarvis, & Shen 2000) and hard-ware test generation (Geller & Veksler 2005).Each application domain has produced specialized al-gorithms for solving CondCSPs. Following the exampleof standard CSP solving, local consistency methods havebeen incorporated in CondCSP solvers and improved perfor-mance results have been reported. Although this solving ap-proach has been observed across application domains, littleis known about the algorithms’ relative performance. More-over, in striking contrast with the state-of-the-art of standardCSP solving, CondCSP class lacks systematic findings withregard to how algorithm efficiency correlates with problemtopology, such as density, satisfiability, and conditionality.This challenge is compounded by an almost inexistent li-brary of CondCSP benchmark problems.To address these challenges, we have developed a Cond-CSP solver that includes two representative algorithms thathave initially been proposed, implemetned, and evaluatedseparately. One algorithm has direct solving methods thatadapt standard consistency checking, such as forward check-ing and maintaining arc consistency, to the special con-straints that enforce conditionality in a CondCSP (Sabin2003; Sabin, Freuder, & Wallace 2003). The other algorithmreformulates the original problem into intermediate Cond-CSPs with incrementally lesser conditionality as they areultimately transformed into standard CSPs. Standard con-sistency checking is interleaved with problem reformulationto eliminate inconsistent subproblems and solve the result-ing standard CSPs (Gelle 1998; Gelle & Faltings 2003). Toovercome the lack of publicly known benchmark problems,we have used random CondCSPs and designed test suites forboth direct and reformulation solving algorithms.The first complete description of CondCSP backtracksearch (Gelle 1998) solves a partially reformulated Cond-CSP, in which activity constraints of exclusion are rewrittenas compatibility constraints. (Sabin 2003) proposed Cond-CSP analogs to CSP backtrack (CondBt), forward checking(CondFc), and maintaining arc consistency (CondMac)search algorithms. CondMac interleaves backtrack searchwith maintaining arc consistency (Mac) adapted to propa-gate consistency checking on both compatibility and activityconstraints in the original CondCSP. Experimental evalua-tion on random CondCSPs (Wallace 1996) shows up to twoorders of magnitude of efficiency improvement over plainbacktrack search.A different solving approach is to successively process ac-tivation conditions of inclusion into an equivalent reformu-lation (Gelle 1998; Gelle & Faltings 2003). The reformula-tion algorithm, Gt, generates a tree whose internal nodesare CondCSPs and the leaves are standard CSPs. Gt re-formulates inclusion activity constraints into compatibilityconstraints by creating intermediate CondCSPs with lesserconditionality until the leaves level is reached. Standardconsistency checking is interleaved with tree generation toeliminate inconsistent subproblems and to solve in the endthe resulting standard CSPs. Experimental evaluation resultsfor Gt using forward checking were reported for solving abridge design problem.These algorithms contribute to CondCSP solving, buttheir development and evaluation took place separately andused different implementation frameworks (C++ vs. Lisp)and different experimental test suites. For the purpose ofexamining these algorithms’ relative performance and howproblem topology impacts performance we have integratedboth implementations within the same solver.We have analyzed experimentally the relative perfor-mance of the two types of algorithms by using randomlygenerated CondCSPs. The random CondCSP generator(Wallace 1996) extends the random standard CSP generatorwith activity parameters. In addition to the standard param-eters of density and satisfiability of the problem compati-bility constraints, dc and sc, we have da, density of activity,the probability of generating a non-initial variable as a targetvariable, and sa, satisfiability of activity, the probability ofgenerating a value in a domain as a condition variable. Thetest suites we designed generated different problem topolo-gies with a problem size of number of variables n = 10,and number of values per domain dsize = 10. Density andsatisfiability of compatibility and activity were varied.We designed two test suites for our experiments. Wecompared the execution time, in number of seconds, forCondFc, CondMac, GtFc and GtMac. In the first testsuite density parameters were kept constant, dc = 0.3 andda = 0.5, while satisfiability parameters varied: sc ∈[0.2 . . . 0.4], and sa ∈ [0.5 . . .0.8] (increments of 0.1). Inthe second test suite compatibility density was increased todc = 0.5, and the variation of the compatibility satisfiabilitywas shifted up into the interval sc ∈ [0.5 . . . 0.7]. The activ-ity parameters were maintained the same as in the first testsuite.The problems in the first test suite have lower density ofcompatibility and variable satisfiability of compatibility ina lower range than the problems in the second suite. Weobserve that in all cases CondMac outperforms CondFcwhile GtFc outperforms GtMac in most cases. There aretwo contributing factors in support of the latter observation.First, Gt transforms all exclusion constraints into compat-ibility constraints prior to the tree generation. Thus, theadvantage of MacActivity pruning in detecting activationconflicts does not manifest in Gt. Secondly, it is knownthat Mac’s performance deteriorates with increasing prob-lem density. In the case of Gt, the compatibility density ofthe resulting standard CSPs is larger than the original Cond-CSP’s compatibility density, since all activity constraintshave been reformulated into compatibility constraints. Over-all, the pruning power of MacCompatibility is overcomeby the implementation overhead which, in the end, does notalways pay off.Another observation is that GtFc has the best perfor-mance, thus outperforming CondMac, in the second testsuite for problems with higher density and satisfiability ofcompatibility. Finally, we observe that, in general, de-creasing satisfiability of activity costs both CondFc andCondMac more, while just the opposite is seen for Gt al-gorithms. Our test suites show that the number of solutionsincreases with decreasing satisfiability of activity. Gt’s ad-vantage of pruning whole subproblems during reformulationcopes better with larger solution spaces (in the tens of thou-sands) than the direct solving methods.ReferencesAmbite, J.; Knoblock, C.; Muslea, M.; and Minton, S.2005. Conditional constraint networks for interleaved plan-ning and information gathering. Intelligent Systems, IEEE[see also IEEE Intelligent Systems and Their Applications]20(2):25–33.Do, M. B., and Kambhampati, S. 2000. Solving planning-graph by compiling it into CSP. In Artificial IntelligencePlanning Systems, 82–91.Gelle, E., and Faltings, B. 2003. Solving mixed and con-ditional constraint satisfaction problems. In Constraints,8(2):107–141, 2003.Gelle, E. 1998. On the generation of locally consistent so-lution spaces. Ph.D. Thesis, Ecole Polytechnique Fe´de´ralede Lausanne, Switzerland.Geller, F., and Veksler, M. 2005. Assumption-based prun-ing in conditional CSP. In Principles and Practice of Con-straint Programming.Miguel, I.; Jarvis, P.; and Shen, Q. 2000. Flexible graph-plan. In Proceedings of the Fourteenth European Confer-ence on Artificial Intelligence, 506–510.Mittal, S., and Falkenhainer, B. 1990. Dynamic constraintsatisfaction problems. In Proceedings of the 8th NationalConference on Artificial Intelligence, 25–32. The MITPress.Sabin, M., and Freuder, E. 1998. Detecting and resolv-ing inconsistency and redundancy in conditional constraintsatisfaction problems. In CP’98 Workshop on ConstraintProblem Reformulation.Sabin, M.; Freuder, E. C.; and Wallace, R. J. 2003. Greaterefficiency for conditional constraint satisfaction. In Princi-ples and Practice of Constraint Programming, 649–663.Sabin, M. 2003. Towards Improving Solving of ConditionalConstraint Satisfaction Problems. Ph.D. Dissertation, Uni-versity of New Hampshire, Durham, NH, U.S.A.Wallace, R. 1996. Random CSP Generator. Con-straint Computation Center, University of New Hampshire,Durham, NH, U.S.A.

Evaluation of solving methods for conditional constraint satisfaction problem

https://scholars.unh.edu/cgi/viewcontent.cgi?article=1023&amp;context=unhmcis_facpub

Evaluation of solving methods for conditional constraint satisfaction problem

Abstract

Similar works

Full text

Available Versions

UNH Scholars' Repository