Cylindrical algebraic decomposition (CAD) is a key tool for problems in real
algebraic geometry and beyond. When using CAD there is often a choice over the
variable ordering to use, with some problems infeasible in one ordering but
simple in another. Here we discuss a recent experiment comparing three
heuristics for making this choice on thousands of examples

Davenport, James H.

England, Matthew

Huang, Zongyan

Paulson, Lawrence C.

Wilson, David

English

arXiv

Cylindrical algebraic decomposition (CAD) is a key tool for problems in real algebraic geometry and beyond. When using CAD there is often a choice over the variable ordering to use, with some problems infeasible in one ordering but simple in another. Here we discuss a recent experiment comparing three heuristics for making this choice on thousands of examples

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        Citation for published version:Huang, Z, England, M, Wilson, D, Davenport, JH & Paulson, LC 2015, A comparison of three heuristics tochoose the variable ordering for CAD. in E Zima, M Caboara, J-G Dumas, L Gonzalez-Vega, M Wester & L Zhi(eds), ACM Communications in Computer Algebra. 3-4 edn, vol. 48, Association for Computing Machinery, NewYork, U. S. A., pp. 121-123. https://doi.org/10.1145/2733693.2733706DOI:10.1145/2733693.2733706Publication date:2015Document VersionPeer reviewed versionLink to publication© ACM, YYYY. This is the author's version of the work. It is posted here by permission of ACM for your personaluse. Not for redistribution. The definitive version was published in ACM Communications in Computer Algebra,48, 3/4, (2014)} http://doi.acm.org/10.1145/2733693.2733706University of BathGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 13. May. 2019A comparison of three heuristics to choose the variableordering for cylindrical algebraic decompositionZongyan Huang1, Matthew England2, David Wilson2,James H. Davenport2 and Lawrence C. Paulson11 University of Cambridge Computer Laboratory, Cambridge CB3 0FD, U.K.{zh242,lp15}@cam.ac.uk2 Department of Computer Science, University of Bath, Bath, BA2 7AY, U.K.{M.England,D.J.Wilson,J.H.Davenport}@bath.ac.ukAbstractCylindrical algebraic decomposition (CAD) is a key tool for problems in real algebraicgeometry and beyond. When using CAD there is often a choice over the variable ordering touse, with some problems infeasible in one ordering but simple in another. Here we discuss arecent experiment comparing three heuristics for making this choice on thousands of examples.1 BackgroundA cylindrical algebraic decomposition (CAD) dissects Rn into cells, each described by polynomialrelations and arranged cylindrically so that the projection of any two cells into lower coordinates isequal or disjoint. CAD is a key tool, both for its original motivation of quantifier elimination (QE)over real-closed fields and many other applications discovered since. For a more detailed introductionsee Huang et al. [4] and the references within. When using CAD there may be a choice for thevariable ordering and it is well known that this can have a great effect on the tractability of aproblem. We recently tested the following three heuristics for picking this ordering.Brown: Suggested by Brown [2], this heuristic chooses a variable ordering according to threecriteria on the input system. We start with the first and break ties with successive ones. Theadvice is to eliminate a variable first if:1. it has lower overall degree in the input;2. it has lower (maximum) total degree of those terms in the input in which it occurs;3. there is a smaller number of terms in the input which contain the variable.sotd: Suggested by Dolzmann et al. [3], this heuristic constructs the full set of projection polyno-mials for each ordering and selects the ordering whose set has the lowest sum of total degreefor each of the monomials in each of the polynomials.ndrr: Suggested by Bradford et al. [1], this heuristic also constructs the full projection set, andselects the one with the lowest number of distinct real roots of the univariate polynomials.1A comparison of three heuristics to choose the variable ordering for CADBrown’s heuristic is the cheapest, checking only simple properties of the input. Ndrr is the mostexpensive (requiring real root isolation) but is the only one to consider the real geometry.All three heuristics may identify more than one variable ordering as a suitable choice. In this casewe took the heuristic’s choice to be the first of these after they had been ordered lexicographically(when written as a tuple in the reverse order to which they are projected).2 Experiment and resultsWe ran a machine learning experiment involving these heuristics [4]. Two experiments were under-taken, one for CAD and another for QE by CAD, in both cases implemented by Qepcad-B1. 7001three-variable QE problems were taken from the nlsat database2, all of which were fully existential(satisfiability or SAT problems). Removing all quantifiers gave a corresponding problem set forevaluating CAD alone. In Huang et al. [4] the problems were split into training, evaluation andtest sets but here we report on the performance of the heuristics for all problems. In each casethe heuristic’s selections were compared according to the number of cells produced (as opposedto computation time: so the experiment concerns the CAD theory rather than just the Qepcadimplementation). Note that cell counts are usually smaller for quantified problems as partial CADtechniques can be used to stop the lifting process early when the outcome is already determined.We first observed for how many problems (and thus for what percentage of the problem set)each heuristic made the most competitive selection of the three:sotd ndrr BrownQuantifier free 4221 (60.29%) 3620 (51.71%) 4523 (64.61%)Quantified 4603 (65.75%) 4000 (57.13%) 5166 (73.79%)Hence we see that Brown’s heuristic is most likely to make the best choice for our problems, bothwhen quantified and when quantifier free. We next investigate how much of a cell count saving isoffered by each heuristic. We made the following calculations for each problem:1. The average cell count of the six orderings;2. The difference between the cell count for each heuristic’s pick and the problem average;3. The value of (2) as a percentage of (1).These calculations were made for all problems in which no variable ordering timed out (5262 of thequantifier free problems and 5332 of the quantified problems). The data is visualised in the plotbelow, where the boxes indicate the second and third quartiles. The mean and median values aregiven below (and marked on the plots with circles and lines respectively).Mean average Median valuesotd ndrr Brown sotd ndrr BrownQuantifier free 27.32% -0.20% 25.28% 29.47% 0.00% 32.28%Quantified 19.47% 4.15% 21.03% 14.68% 0.00% 16.67%While Brown’s heuristic makes the best choice the most frequently, for the quantifier free problemsthe average saving of using sotd is actually higher.1Interactive command-line program, freely available from http://www.usna.edu/CS/∼qepcad/B/QEPCAD.html.2Benchmarks for solving nonlinear arithmetic freely available from http://cs.nyu.edu/∼dejan/nonlinear/.2Zongyan Huang, Matthew England, David Wilson, James H. Davenport and Lawrence C. PaulsonNdrr performs the worst on average, but there are classes of problems where it makes a betterchoice than the others (see Huang et al. [4]). For example, consider the remaining problems (thosewhere at least one ordering timed out). The following table describes how often each heuristicavoids a time out. We see that for quantified problems ndrr does the best.sotd ndrr BrownQuantifier free 559 537 594Quantified 512 530 4783 ConclusionsWe note first that the conclusion of which heuristic is the best varies depending on the criteriaused to judge, whether we consider quantified problems or not (and indeed various other featuresof the problem which is why machine learning was useful [4]). We highlight the strong performanceof Brown’s heuristic, surprising both because it requires the least computation and since it wasnot formally published. (To the best of our knowledge it is mentioned only in notes to a tutorialat ISSAC 2004 [2]). The problems in our example set are all from genuine applications, but arequite different to those for which CAD is normally used. Hence further experimentation would bebeneficial to see if the results can be verified more generally. Other future work will include thetesting of greedy and combination heuristics, perhaps leading to the development of a new heuristic.References[1] R. Bradford, J.H. Davenport, M. England, and D. Wilson. Optimising problem formulationsfor cylindrical algebraic decomposition. In Intelligent Computer Mathematics (LNCS vol. 7961),pages 19–34. Springer Berlin Heidelberg, 2013.[2] C.W. Brown. Companion to the tutorial: Cylindrical algebraic decomposition, (ISSAC 2004).Available from: www.usna.edu/Users/cs/wcbrown/research/ISSAC04/handout.pdf.[3] A. Dolzmann, A. Seidl, and T. Sturm. Efficient projection orders for CAD. In Proc. ISSAC ’04,pages 111–118. ACM, 2004.[4] Z. Huang, M. England, D. Wilson, J.H. Davenport, L.C. Paulson, and J. Bridge. Applyingmachine learning to the problem of choosing a heuristic to select the variable ordering forcylindrical algebraic decomposition. To appear: Proc. CICM ’14 (LNAI 8543) pages 92-107.Preprint at: http://arxiv.org/abs/1404.6369, 2014.3

A comparison of three heuristics to choose the variable ordering for CAD

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