Anytime AND/OR Depth-first Search for Combinatorial Optimization

17 Pages·2012·0.52 MB·English

1
Anytime AND/OR Depth-first Search for
Combinatorial Optimization
LarsOtten∗ andRinaDechter avoid redundant computation using caching schemes,
DepartmentofComputerScience, and most significantly, because they take advantage
UniversityofCalifornia,Irvine,U.S.A. of problem decomposition by exploring an AND/OR
{lotten,dechter}@ics.uci.edu searchspace[19]oranequivalentrepresentation.The
efficiency of these algorithms was established in sev-
eral evaluations, including recent UAI competitions
One popular and efficient scheme for solving combinato-
[7],andtheirpropertieswhenusedforexactcomputa-
rial optimization problems over graphical models exactly
tionarewelldocumented[10,16,17].
is depth-first Branch and Bound. However, when the al-
A principled alternative is presented by best-first
gorithm exploits problem decomposition using AND/OR
searchspaces,itsanytimebehaviorbreaksdown.Thisarticle schemes,butwhileprovablysuperiorintermsofnum-
1)analyzesanddemonstratesthisinherentconflictbetween ber of node expansions, these often fail when a prob-
effective exploitation of problem decomposition (through lem has large induced width due to the generally ex-
AND/ORsearchspaces)andtheanytimebehaviorofdepth- ponentialsizeofthealgorithm’sOPENlist;moreover,
first search (DFS), 2) presents a new search scheme to ad- they can only provide a solution at termination [17].
dress this issue while maintaining desirable DFS memory Depth-firstsearchisthereforeoftenpreferredbecause
properties, and 3) analyzes and demonstrates its effective-
of its flexibility in working with bounded memory –
nessthroughcomprehensiveempiricalevaluation.Ourwork
theOPENlistofnodesgrowslinearly–andbecauseof
isapplicabletoanyproblemthatcanbecastassearchover
itsanytimebehavior.Namely,whenfindingafeasible
anAND/ORsearchspace.
solutioniseasybutanoptimaloneishard,depth-first
Keywords: combinatorial optimization, graphical models,
Branch and Bound generates solutions that get better
Bayesian and constraint networks, anytime performance,
and better over time, until it eventually discovers an
AND/ORsearch,problemdecomposition.
optimal one. Thus it can function also as an approx-
imation scheme for otherwise infeasible problems or
whentimeislimited[24].
1. Introduction
Indeed, in the 2010 UAI Approximate Inference
ChallengeparticipatingBranchandBoundsolversper-
Max-productproblemsovergraphicalmodels,gen-
formed competitively with respect to approximation
erally known as MPE (most probable explanation) or
(placing 1st and 3rd in some categories). But we also
MAP (maximum a posteriori) inference, have many
observedaninabilitytoproduceevenasinglesolution
applications with practical significance, ranging from
on some instances, especially when the time bound
computationalbiologyandgeneticstoschedulingtasks
was small. Thus motivated, this article will demon-
and coding networks. One established and efficient
strate that the issue is rooted in the underlying AND/
classofalgorithmsforsolvingtheseproblemsexactly
ORsearchspace.
isdepth-firstBranchandBoundoverAND/ORsearch
These search spaces were originally introduced to
spaces.Developedinthepastdecadewithintheprob-
graphical models to facilitate problem decomposi-
abilistic reasoning and constraint communities, these
tion during search (e.g. [5]) and can be explored by
methods are effective because they use sophisticated
any search strategy. When traversed depth-first, how-
lower bound schemes such as soft arc-consistency
ever,allbutonedecomposedsubproblemwillbefully
[15] or the mini-bucket heuristic [6,16], because they
solved before a single overall solution can be com-
posed,voidingthealgorithm’sanytimecharacteristics.
*Correspondingauthor:4099DonaldBrenHall,Dept.ofCom-
Weobservethatundercertainconditions,namelyif
puterScience,UniversityofCalifornia,Irvine,CA92697,U.S.A.
E-mail:[email protected],Fax:[phone]. only one of the decomposed subproblems is “hard”,
AICommunications
ISSN0921-7126,IOSPress.Allrightsreserved
2 L.OttenandR.Dechter/AnytimeAND/ORDepth-firstSearchforCombinatorialOptimization
thisadverseeffectcanbemitigatedbyprocessingsub- specttosuccessorordering.However,itwasonlypre-
problems in a suitable order, which we demonstrate sented in a general, non-optimization OR search con-
empirically. textforconstraintsatisfactionproblems.
This article’s main contribution is a new Branch Connections can also be made to recent contribu-
and Bound scheme over AND/OR search spaces, tions in the area of distributed, multi-agent constraint
called Breadth-Rotating AND/OR Branch and Bound optimization, where algorithms like NCBB [3] and
(BRAOBB)thataddressestheanytimeissueinaprinci- BnB-ADOPT [23] organize agents along a pseudo
pledway,whilemaintainingthefavorablecomplexity tree-likestructure,therebyobtainingsolutionstoinde-
guarantees of depth-first search. The algorithm com- pendentsubproblemsinparallel.
binesdepth-firstandbreadth-firstexplorationbyperi- Finally,mostdirectlyrelatedtotheobjectiveofthis
odically rotating over the different subproblems, each workistheconceptoflocalsearch,whichcanbeseen
ofwhichisprocesseddepth-first. as specifically targeting anytime performance. How-
Experimental evaluation is conducted on a variety ever,itdiffersfromAOBBandtheproposedBRAOBB
ofbenchmarkdomains,includinghaplotypecomputa- inthatitcannotproveoptimalityofthesolutionsitre-
tion problems in genetic pedigrees, random grid net- turns. We include the state-of-the-art stochastic local
works,andproteinside-chainpredictioninstances.We searchsolverGLS+inourempiricalevaluation[9].
compareBRAOBBagainstoneofthebestvariantsof
PaperOutline
AND/OR branch and Bound search, AOBB [16], and
Theremainderofthisarticleisstructuredasfollows:
againstan“adhoc”fixthatwesuggest–thelatteral-
Section2introducestheunderlyingconceptsofAND/
gorithm relies on a heuristic to quickly find a solu-
OR Branch and Bound. Section 3 identifies the cen-
tiontoeachsubproblembeforerevertingtodepth-first
tralconflictbetweenproblemdecompositionandany-
search.Wefurthermorecompareagainstastate-of-the-
artstochasticlocalsearchsolver,whichisspecifically timeperformanceandprovidesempiricalresultswhere
targeted at anytime performance but cannot provide thelatteriscompromised.ThenewalgorithmBreadth-
anyproofofoptimality[9]. RotatingAOBBisproposedinSection4anditstheo-
Theempiricalresultsdemonstratesuperioranytime retical properties are analyzed. Section 5 presents ex-
behavior of BRAOBB, especially over problematic haustiveexperimentalresultsandanalysisusingawide
cases where standard AOBB and its ad hoc fix fail, rangeofexampleproblemsaswellassummarystatis-
including several very hard instances from the 2010 tics across more than 500 instances. Section 6 con-
UAIApproximateInferenceChallengethatweremade cludes.
available and three weighted constraint satisfaction
probleminstancesthatareknowntobeverycomplex.
We also show how combining local search and ex- 2. Background
haustive AND/OR search lets us enjoy the benefits of
both approaches. Notably, a solver based on this con- We consider a MPE (most probable explanation,
ceptrecentlywonallthreecategories(20seconds,20 sometimesalsocalledMAP,maximumaposteriorias-
minutes, and 1 hour) in the MPE track of the PAS- signment)problemoveragraphicalmodel,definedby
CAL 2011 Inference Challenge [8], the successor to
the2010UAIChallenge.
RelatedWork
Theworkpresentedhereisfocusedonoptimization
problems defined over graphical models. As such our
resultsarealsorelevantforrelatedschemeslikerecur-
sive conditioning [4] and value elimination [2] in the
areaofprobabilisticreasoning,orBTD(Backtracking
Tree Decomposition [10]) in constraint optimization.
Infact,thepresentedconceptscarryovertocombina-
torialAND/ORsearchspacesingeneral. (a) (b) (c)
Second, we note Interleaved Depth-First Search
Fig.1.(a)Exampleprimalgraphofagraphicalmodelwithsixvari-
[18],whichusesasimilarideaofinterleavedprocess- ables,(b)itsinducedgraphalongorderingd=A,B,C,D,E,F,
ingofdifferentbranchestomitigatemistakeswithre- and(c)acorrespondingpseudotree.
L.OttenandR.Dechter/AnytimeAND/ORDepth-firstSearchforCombinatorialOptimization 3
(a) (b)
Fig.2.(a)AND/ORsearchtreeand(b)context-minimalAND/ORsearchgraphcorrespondingtothepseudotreeinFigure1(c).
thetuple(X,F,D,max,Q).F={f1,...,fr}isaset EXAMPLE 2. A pseudo tree for the example in Fig-
of functions over variables X = {X1,...,Xn} with ure 1(a) is shown in Figure 1(c), corresponding to
discretedomainsD={D1,...,Dn},weaimtocom- the induced graph in Figure 1(b) along ordering d =
pute max Q f , the probability of the most likely A,B,C,D,E,F.NotehowB hastwochildren,cap-
X i i
assignment.Anothercloselyrelatedcombinatorialop- turingthefactthatthetwosubproblemsoverC,Dand
timization problem is the weighted constraint prob- E,F,respectively,areindependentonceAandBhave
lem, where we aim to minimize the sum of all costs, beeninstantiated.
i.e.computemin P f [21].Thesetaskshavemany
X i i
practicalapplicationsbutareknowntobeNP-hard. 2.1. AND/ORSearchTrees
The set of function scopes implies a primal graph
and, given an ordering of the variables, an induced Given a graphical model instance with variables
graph (where, from last to first, each node’s earlier X and functions F , its primal graph (X,E), and a
neighborsareconnected)withacertaininducedwidth, pseudo tree T, the associated AND/OR search tree
the maximum number of earlier neighbors over all consists of alternating levels of OR and AND nodes
nodes[20,12]. [5].Itsstructureisbasedontheunderlyingpseudotree
T :therootoftheAND/ORsearchtreeisanORnode
EXAMPLE 1. Figure 1(a) depicts the (primal) graph
of an example graphical model with six variables, A labeledwiththerootofT .ThechildrenofanORnode
throughF.Theinducedgraphfortheexampleproblem hXiiareANDnodeslabeledwithassignmentshXi,xji
thatareconsistentwiththeassignmentsalongthepath
along ordering d = A,B,C,D,E,F is depicted in
Figure 1(b), with two new induced edges, (B,C) and from the root; the children of an AND node hXi,xji
(B,E).Itsinducedwidthis2. are OR nodes labeled with the children of Xi in T ,
representingconditionallyindependentsubproblems.
Differentorderingswillvaryintheirinducedwidth;
findinganorderingofminimalinducedwidthisknown EXAMPLE 3. Figure 2(a) shows the AND/OR search
tobeequallyNP-hard.Inpracticeheuristicslikemin- tree resulting from the primal graph in Figure 1(a)
fillormindegreehaveproventoproducereasonableap- when guided by the pseudo tree in Figure 1(c). Note
proximations[14,13]. thattheANDnodesforBhavetwochildreneach,rep-
TheconceptofAND/ORsearchspaceshasrecently resentingindependentsubtreesrootedatC andE,re-
been introduced to graphical models to better capture spectively,therebycapturingproblemdecomposition.
thestructureoftheunderlyinggraphduringsearch[5].
In general, given a pseudo tree T of height h, the
Thesearchspaceisdefinedusingapseudotreeofthe
size of the AND/OR search tree based on T is O(n·
graph, which captures problem decomposition as fol- kh),wherekboundsthedomainsizeofvariables[5].
lows:
DEFINITION 1. Apseudotreeofanundirectedgraph 2.2. AND/ORSearchGraphs
G = (X,E) is a directed, rooted tree T = (X,E′),
suchthateveryarcofGnotincludedinE′ isaback- Identical subproblems, identified by their context
arcinT ,namelyitconnectsanodeinT toanancestor (thepartialinstantiationthatseparatesthesubproblem
inT .ThearcsinE′maynotallbeincludedinE. from the rest of the network), can be merged, yield-
4 L.OttenandR.Dechter/AnytimeAND/ORDepth-firstSearchforCombinatorialOptimization
inganAND/ORsearchgraph[5].Mergingallcontext- Algorithm1AND/ORBranchandBound(AOBB)
mergeablenodesyieldsthecontext-minimalAND/OR
Given: Graphical model (X,F,D,max,Q) and pseudo
sAeNarDch/OgRraspeha.rcIht gwraasphshhoawssniztehaOt(tnh·ekwco∗)n,tewxht-emreinwim∗aisl Outptrueet:TcowsitthofrooopttiXmaolsolution
theinducedwidthoftheproblemgraphalongadepth- 1: OPEN←{hX0i}
firsttraversalofT [5]. 2: whileOPEN6=∅
3: n←top(OPEN) //topnodefromstack,depth-first
EXAMPLE4. Figure2(b)displaysthecontext-minimal 4: ifcheckpruning(n)=success
AND/OR graph obtained when applying full caching 5: prune(n) //performpruning
totheAND/ORsearchtreeinFigure2(a).Inparticu- 6: elseifcachelookup(n)=success
lar,theORnodesforD (withcontext{B,C})andF 7: readcache(n) //retrievecachedvalue
(context{B,E})havetwoedges converging fromthe 8: elseifn=hXiiisORnode
ANDlevelabovethem,signifyingcaching(namely,the 9: forxj ∈Di
assignmentofAdoesnotmatter). 10: createANDchildhXi,xji
11: addhXi,xjitotopofOPEN
GivenanAND/ORsearchspaceST,asolutionsub- 12: elseifn=hXi,xjiisANDnode
treeSolST isatreesuchthat(1)itcontainstherootof 13: forYr ∈childrenT(Xi)
ST ;(2)ifanonterminalANDnoden∈ST isinSolST 14: generateORnodehYri
thenallitschildrenareinSolST ;(3)ifanonterminal 15: addhYritotopofOPEN
OR node n∈ST is in SolST then exactly one of its 16: ifchildren(n)=∅ //nisleaf
childrenisinSol . 17: propagate(n) //upwardsinsearchspace
ST 18: return value(hX0i) //rootnodehasoptimalsolution
2.3. WeightedAND/ORSearchSpaces
rent best solution to the subproblem rooted at each
GivenanAND/ORsearchgraph,eachedgefroman node),untilsearchterminatesandtheoptimalsolution
OR node X to an AND node x can be annotated by hasbeenfound[16].
i i
weightsderivedfromthesetofcostfunctionsF inthe Algorithm 1 shows pseudo code for AOBB: Start-
graphicalmodel:theweightl(Xi,xi)isthecombina- ing with just the root node hX0i on the stack, the al-
tionofallcostfunctionswhosescopeincludesX and gorithmiterativelytakesthetopnodenfromthestack
i
is fully assigned along the path from the root to x , (line3).Lines4–7trytoprunethesubproblembelow
i
evaluated at the values along this path. Furthermore, n (by comparing a heuristic estimate of n against the
each node n in the AND/OR search graph can be as- currentlowerbound)andcheckthecachetoseeifthe
sociatedwithavaluev(n),capturingtheoptimalsolu- subproblem below n has previously been solved (de-
tioncosttothesubproblemrootedatn,subjecttothe tailsin[16]).Ifneitheroftheseissuccessful,thealgo-
current variable instantiation along the path from the rithm generates the children of n (if any) and pushes
rootton.v(n)canbecomputedrecursivelyusingthe thembackontothestack(8–15).Ifnisaterminalnode
valuesofn’ssuccessors[5]. inthesearchspace(itwaspruned,itssolutionretrieved
fromcache,orthecorrespondingX isaleafinT)its
i
2.4. AND/ORBranchandBound value is propagated upwards in the search space, to-
wards the root node (16–17). When the stack eventu-
AND/OR Branch and Bound (AOBB) is a state- allybecomesempty,thevalueoftherootnodehX0iis
of-the-artalgorithmforsolvingoptimizationproblems returnedasthesolutiontotheproblem(18).
such as max-product over graphical models [16,17]. We will use AOBB to denote the algorithm above
Assuming a maximization query, AOBB traverses the in its specific graphical models context as well as a
weightedcontext-minimalAND/ORgraphinadepth- generic name for any depth-first Branch and Bound
first manner while keeping track of the current lower schemeoveranAND/ORsearchspace.
boundonthemaximalsolutioncost.Anodenwillbe
pruned if this lower bound exceeds a heuristic upper 2.5. Mini-BucketHeuristics
bound on the solution to the subproblem below n (cf.
Section 2.5). The algorithm interleaves forward node The heuristic h(n) that we use in our experiments
expansion with a backward cost revision or propaga- isthemini-bucketheuristic.Itisbasedonmini-bucket
tion step that updates node values (capturing the cur- elimination, which is an approximate variant of vari-
L.OttenandR.Dechter/AnytimeAND/ORDepth-firstSearchforCombinatorialOptimization 5
ableeliminationandcomputesapproximationstorea-
pedigree30x1, i10 (n=1289 k=5 w=21 h=108)
soningproblemsovergraphicalmodels[6].Acontrol
−137.0
parameteriallowsatrade-offbetweenaccuracyofthe y)−137.5
heuristicanditstimeandspacerequirements–higher bilit−138.0
a
values of i yield a more accurate heuristic but take ob−138.5
moretimeandspacetocompute.Itwasshownthatthe g(pr−139.0
Lo−139.5
intermediate functions generated by the mini-bucket increasing
−140.0 decreasing
algorithm MBE(i) can be used to derive a heuristic −140.5
functionthatisadmissible,namelyinamaximization 100 101 102 103 104
Search time in seconds
context it overestimates the optimal cost solution to a
pedigree41x1, i7 (n=1062 k=5 w=33 h=100)
subproblemintheAND/ORsearchgraph[11]. −120
−122
y)
bilit−124
3. AnytimeBehaviorversusProblem ba−126
o
DecompositioninAND/ORSearch og(pr−−113208
L
−132 increasing
decreasing
As a depth-first branch and bound scheme one −134
wouldexpectAOBBtoquicklyproduceanon-optimal 100 101 102 103 104
Search time in seconds
solutionandthengraduallyimproveuponit,maintain-
pedigree34x2, i15 (n=2320 k=5 w=31 h=102)
ing the current best one throughout the search. How-
−222
everthisabilityiscompromisedinthecontextofAND/
y)−224
ORsearch. bilit−226
Specifically, in AND/OR search spaces depth-first ba
o−228
traversal of a set of independent subproblems will pr
g(−230
solvetocompletionallbutonesubproblembeforethe Lo
−232 increasing
lastoneisevenconsidered.Asaconsequence,thefirst decreasing
−234
generated overall non-optimal solution contains con- 100 101 102 103 104
ditionallyoptimalsolutionstoallsubproblemsbutthe Search time in seconds
lastone.Furthermore,dependingontheproblemstruc-
Fig.3.ImpactofsubproblemorderingonAOBB.Specifiedforeach
ture and the complexity of the independent subprob-
network:numberofvariablesn,max.domainsizek,inducedwidth
lems, the time to return even this first non-optimal w along the chosen ordering, height of the corresponding pseudo
overallsolutioncanbesignificant,practicallynegating treeh.Thedashedgraylineindicatestheoptimalsolutionvalue.
theanytimebehaviorofdepth-firstsearch(DFS).
(in the general description of AOBB the subproblem
orderingisleftunspecified).
3.1. SubproblemOrdering
Figure 3 contrasts the anytime behavior of AOBB
using this “increasing” subproblem order against the
In certain cases, the above suggests a simple rem-
inverseone(“decreasing”)byplottingthesolutioncost
edy: if decomposition yields only one large subprob-
generatedasafunctionoftimeontwoexampleprob-
lem and several smaller ones, the latter can be solved lems(thedashedhorizontallineistheoptimumcost);
depth-first in relatively little time, to be then com- all other aspects of the algorithm remain constant.
bined with the incrementally improving solutions of Pedigree30x1 inparticularfeaturesexactlyonesingle
the larger subproblem. Thus for anytime behavior an complexsubproblemandanumberofrelativelysimple
AOBB algorithm would need to process independent ones; in this case processing subproblems by increas-
subproblemsfrom“easy”to“hard”. ing induced width right away produces a non-optimal
To demonstrate the practical impact of subproblem solutionthatimprovesrapidly.Theinverseorderyields
orderings, we use a simple heuristic that takes the the first solution only after about 90 minutes – the
induced width as a measure of subproblem hardness one complex subproblem has been fully solved and
(motivated by its exponential role in the asymptotic the overall solution is already optimal. Pedigree41x1
complexity),i.e.wemodifyAOBBsuchthatsubprob- hasasimilarlyadvantageousstructureandthusyields
lemswithsmallerinducedwidthwillbeprocessedfirst similar results – with the distinction that the inverse
6 L.OttenandR.Dechter/AnytimeAND/ORDepth-firstSearchforCombinatorialOptimization
subproblemorderdoesnotproduceanysolutionatall 4. Breadth-RotatingAOBB
within24hours.
In case of pedigree34x2, however, decomposition In the following we develop a new search scheme
yields two complex subproblems: the increasing sub- called Breadth-Rotating AND/OR Branch and Bound
problem order still outperforms its inverse, yet it re- (BRAOBB) that addresses the issue of anytime per-
turns the initial solution only after about 1,000 sec- formance over AND/OR search spaces. It combines
onds.Infact,nopossiblesubproblemorderingcanlead depth-first exploration with the notion of “rotating”
toacceptable anytimebehavior inthiscaseduetothe through different subproblems in a breadth-first man-
structureofsubproblems,clearlyhighlightingthelim- ner.Namely,nodeexpansionstilloccursdepth-firstas
itsofthisapproach. instandardAOBB,butthealgorithmtakesturnsinpro-
Independent ofanytime behavior,wepointoutthat cessingsubproblems,eachuptoagivennumberofop-
incorporating different subproblem orderings impacts erationsatatime,round-robinstyle.
thealgorithm’soverallefficiency(i.e.,thetimetofind Tomotivatethisapproach,consideragainthataso-
and prove an optimal solution): knowing the solution lution is represented by a solution tree over an AND/
to one subproblem can aid the pruning of Branch and ORsearchspace,guidedbyapseudotree.ApureDFS
Bound in the next one to varying degrees. However, scheme will construct the different branches of a so-
this issue has not been treated systematically in the lution tree one by one, ensuring optimality for each
literature for graphical models, with sporadic exper- branch before moving to the next. To restore anytime
iments also suggesting an easy-to-hard order, using behavior,weinsteadaimtodevelopallbranchesofthe
some heuristic to determine subproblem complexity solution tree “simultaneously”, which we achieve by
[16].Thisgeneralproblemisoutsidethescopeofthe rotatingthroughthem.
presentpaper,however.
4.1. SubproblemRotation
3.2. GreedySubproblemDive
More systematically, the algorithm maintains a list
Another relatively straightforward remedy that can ofcurrentlyopensubproblemsandrepeatsthefollow-
inghigh-levelstepsuntilcompletion:
be viewed as an “ad hoc” fix is the following: Every
time decomposition is encountered within the search 1. Move to next open subproblem P in a breadth-
space, we will try to greedily find a single initial so- firstfashion.
lutiontoeachindependentsubproblembeforesucces- 2. ProcessP depth-first,untileither:
sively solving each of them to completion depth-first,
(a) P issolvedoptimally,
through normal AOBB. To obtain this initial solution
(b) P decomposesintochildsubproblems,or
the algorithm can perform a greedy “dive” into each
(c) apredefinedthresholdnumberofoperations
subproblem by only considering one value for each
isreached.
variable along the path (in case of the mini-bucket
heuristic, it is easy to see that this is equivalent to a The threshold in (c) is needed to ensure the algo-
forwardpassoverthebucketstructure[11]). rithmdoesnotgetstuckinonelargesubproblemwhere
Clearly,thechoiceofthedivepathiscrucialforthe theothertwoconditions,(a)and(b),donotoccurfor
algorithm’s performance. Namely, if the chosen path a long time. Furthermore, in order to focus on a sin-
leadstoadeadend(zeroprobability),thedivewillbe gle solution tree at a time, a subproblem is only con-
futileandnotyieldasubproblemsolution.Thisagain sidered “open” if it does not currently have any child
negates the desired anytime behavior, since the sub- subproblems,asillustratedbelow.
problem for which the dive failed will not be recon-
sidereduntilthenormaldepth-firstAOBBphase.And 4.2. AlgorithmPseudoCode
infactexperiments inSection 5 willdemonstrate that
theresultingperformancedependsheavilyonthequal- Algorithm2givesmoredetailedpseudocodeforthe
ity of the heuristic, which often prevents satisfactory scheme(withsomedetailsfromstandardAOBBomit-
anytimebehavior.Inthenextsectionwewilltherefore ted,cf.Algorithm1and[16]).Thekeyelementliesin
propose a new search strategy that addresses the any- rotating over the different subproblems of the search
timeissueoverAND/ORsearchspacesinaprincipled space; by organizing these in a global first-in-first-
manner. outqueue(GLOBAL),weemulatebreadth-firstexplo-
L.OttenandR.Dechter/AnytimeAND/ORDepth-firstSearchforCombinatorialOptimization 7
Algorithm2Breadth-RotatingAOBB
Given: Graphical model (X,F,D,max,Q) and pseudo
treeT withrootXo,rotationthresholdZ
Output: costofoptimalsolution
1: ROOT←{hX0i} //generaterootsubproblem
2: pushROOTtoendofGLOBAL
3: whileGLOBAL6=∅
4: LOCAL←front(GLOBAL) //nextsubproblem
5: forz←1toZoruntilLOCAL=∅
oruntilchildSubprob(LOCAL)6=∅
6: n←top(LOCAL) //nextnodeinsubproblem
(a)Expansionofnodes1–12
7: ... //cachingandpruningasinAOBB
8: ifn=hXiiisORnode
9: forxj ∈Di
10: createANDchildhXi,xji
11: addhXi,xjitotopofLOCAL
12: elseifn=hXi,xjiisANDnode
13: Y1,...,Ym ←childrenT(Xi)
14: generateORchildrenhY1i,...,hYri
15: ifm=1 //nodecomposition
16: pushhY1itotopofLOCAL
17: elseifm>1 //problemdecomposition
18: forr←1tom
19: NEW←{hYri} //newchildsubproblem
20: pushNEWtobackofGLOBAL (b)Expansionofnodes13–31
21: ifchildren(n)=∅ //nisleaf
22: propagate(n) //upwardsinsearchspace
23: ifLOCAL6=∅ //subproblemnotyetsolved
24: pushLOCALtoendofGLOBAL
25: return value(hX0i) //rootnodehasoptimalsolution
rationacrossthedifferentbranchesofthesolutiontree.
The input parameter Z gives the rotation threshold.
Eachsubproblemisitselfexploreddepth-first(viaalo-
callast-in-first-outstackofnodes,LOCAL);whenever
a new level of decomposition is encountered, as cap-
tured by the pseudo tree, the resulting child subprob- (c)Expansionofnodes32–44
lemsarepushedtotheendoftheglobalqueue.Finally,
Fig.4.BRAOBBexploration(Z=2)atdifferentstages.Nodesare
subproblemsareonlyconsideredintherotationifthey
numberedinorderoftheirexpansion.
don’tcurrentlyhaveanychildsubproblems.
tothequeue.(3)RotatetosubproblemP1andexpand
4.3. ExampleExecution hCi and hC,0i. (4) Rotate to subproblem P2. Expand
hEi and hE,0i. (5) Rotate to subproblem P0 but skip
Figure 4 demonstrates the scheme’s application it at this point, since its child subproblems P1 and P2
(Z =2) to the AND/OR search graph in Figure 2(b) are still open. (6) Rotation moves to subproblem P1.
(assuming no pruning). Part (a) shows the first 12 ExpandhDiandhD,0i,discoveraleafandpropagate.
nodesexpandedduringthefirstseveniterationsofthe (7)RotatetosubproblemP2,expandhFiandhF,0i–
outer while loop as follows: (1) Taking the overall which,asaleaf,ispropagatedtoyieldthefirstoverall
problemassubproblemP0,expandhAiandhA,0ibe- solution.
forereachingthethresholdZ=2.(2)Withnodecom- Figures 4(b) and (c) illustrate how the search then
position so far rotation returns to subproblem P0. Ex- proceedstotaketurnssolvingsubproblemsP1andP2
pandhBiandhB,0i,yieldingsubproblemsP1andP2 to completion (nodes 13–22) before reopening sub-
rooted at hCi and hEi, respectively, which are added problemP0.Expansion23yieldstwonewindependent
8 L.OttenandR.Dechter/AnytimeAND/ORDepth-firstSearchforCombinatorialOptimization
subproblems P3 and P4; their solution is depicted by Itisworthpointingoutthattheseworst-casebounds
nodes24–41.AfterthatsubproblemP0getsreopened, are often very loose, because the Branch and Bound
where expanding nodes 42–44 again yields two new schemeistypicallyveryefficientandpruneslargeparts
subproblemsP5andP6,andsoforth. of the search space. In particular, we observe that in
practicethepruningkeepsthecachetablesfromreach-
ingtheirworst-caseexponentialsize.
4.4. AnalysisofBreadth-RotatingAOBB
4.4.2. SignificanceofZ
TherotationthresholdZ actsasasafeguardagainst
In this Section we analyze the Breadth-Rotating
overlylargesubproblems,thattakealongtimetosolve
AOBBalgorithmanditspropertiesandcontrastitwith
optimally (condition (a), Section 4.1) or where recur-
standardAOBB.
sive decomposition does not occur for a long time
4.4.1. Correctness,Completeness,andComplexity (condition(b)).Being“stuck”inthiswaycouldagain
Recall that a heuristic function is said to be admis- impair anytime performance, which is why we limit
sibleifitneverunderestimates(inamaximizationsce- thenumberofnodeexpansionsbeforeenforcingaro-
nario) the cost of the optimal solution to a given sub- tation.AsweseeinSection5,however,practicalprob-
problem. The mini-bucket heuristic satisfies this re- lemstypicallyexhibitfrequentsubproblembranching,
quirement[11].Furtherrecallthatndenotesthenum- soarelativelylargethresholdofZ = 1000orsimilar
berofproblemvariables,kthemaximumdomainsize, issufficient,ifrarelyreached.
htheheightoftheguidingpseudotreeT withinduced
4.4.3. MaximumQueueSize
widthandw∗.
It is easy to see that the maximum number of en-
THEOREM1. Breadth-RotatingAOBBiscompleteand triesintheGLOBALqueueisdependentonthenum-
correct assuming an admissible heuristic. Further- ber of branchings in the solution tree, corresponding
more, when searching an AND/OR search tree (i.e., to pseudo tree nodes with more than one successor,
withoutcachingofredundantsubproblems),BRAOBB since that is where the algorithm generates new child
has time complexity O(n · kh) and space complexity subproblems(lines17–20,Algorithm2).Inparticular,
O(n). When searching the context-minimal AND/OR decomposition does not occur along the chains in the
searchgraph(withfullcaching),timeandspacecom- pseudotree,i.e.,pathswherenonode(besidestheend
plexityareO(n·kw∗). points) has outdegree greater than 1. The number of
queue entries is thus bounded by the number of max-
Proof. Because of the heuristic’s admissibility, a sub- imal chains in the pseudo tree. We can thus state the
spaceisprunedonlyifitprovablycannotyieldabet- following:
ter solution than what is already known at this point.
THEOREM 2. When exploring an AND/OR search
Just like standard AOBB the search also remains sys-
space using a guiding pseudo tree P with l leaves,
tematicandallsolutiontreesareconsidered;thealgo-
the number of subproblems in the GLOBAL queue of
rithmisguaranteedtoeventuallyterminateandreturn
BRAOBBisboundedby2l−1.
theoptimalsolutiontotheproblem.
BRAOBB explores the same underlying AND/OR Proof. Since T is a tree with l leaves, there can be at
search space as standard AOBB, hence its asymptotic most l −1 branchings (nodes with outdegree greater
time complexity remains unchanged, i.e. exponential than1)inT toyieldtheseleaves.Eachsuchbranching
in h for tree and exponential in w∗ for graph search. sits at the end of one maximal chain. Together with
SpacecomplexityforAND/ORgraphsearchisdomi- the leaf chains, we obtain an upper bound of 2l − 1
natedbythecachingandthusalsoremainsunchanged maximalchains.
exponentialinw∗.
In case of tree search, recall that subproblems with 4.4.4. ComparisonwithStandardAOBB
childsubproblemsarenotprocessedfurther.Therefore WeexpecttheanytimeperformanceofBRAOBBto
every variable will appear in at most one subproblem be robust with respect to different subproblem order-
at any given time. And since each subproblem is pro- ings,sincethealgorithmisnotforcedto“commit”to
cesseddepth-first,i.e.inlinearspace,thespaceacross asinglesubproblem–whichweidentifiedasthemain
allsubproblemsisalsolinearinn. reason for the poor anytime behavior of plain AOBB
L.OttenandR.Dechter/AnytimeAND/ORDepth-firstSearchforCombinatorialOptimization 9
inSection3.1.Wewillconfirmthisexperimentallyin anytime performance and neither the dive extension
Section5. nor BRAOBB can provide significant improvements.
TheactualnumberofnodesexploredbyBRAOBB Hencewealsocreatedadditionalversionsofeachnet-
might differ from plain AOBB (for both graph and work with two or three identical copies connected at
tree search), since the pruning behavior of the algo- the root (thus ensuring the presence of more than one
rithm can be impacted by the order in which nodes complex subproblem), signified by the “x2” and “x3”
are explored and subproblem solutions produced: On suffix, respectively. This yields a total of 57 pedigree
theonehand,solvingasubproblemtocompletionbe- (eachrunwiththreedifferentheuristicstrengths),150
fore processing the next (in AOBB) might allow the grid, 24 mastermind, and 198 protein prediction in-
algorithm to calculate a tighter upper bound using stancesandresultinginover90,000CPUhoursworth
this optimal solution, resulting in better pruning. On ofexperiments.
theotherhand,exploringsubproblemsconcurrentlyin Wepresentdetailedperformanceresultsforarepre-
BRAOBBmightleadtoatighteroveralllowerbound sentativesubsetoftheprobleminstancesinSection5.1
through combining solutions across subproblems as beforeSection5.2compilessummarystatisticsacross
theyarediscovered(inananytimefashion). all 543 problem instances. Section 5.3 evaluates how
our proposed exhaustive search method can benefit
fromlimitedlocal search.Section 5.4 presents results
5. EmpiricalEvaluation on two additional very challenging problem domains,
protein-protein interaction from the UAI 2010 Chal-
Tovalidateandcomparetheperformanceofthevar- lenge and CELAR radio link frequency assignments.
ious schemes we recorded their anytime behavior on Finally, Section 5.5 analyzes a number of algorithm
a variety of problem instances using a common vari- parametersempirically.
able ordering and mini-bucket heuristic for each in-
stance (24 hour time limit); unless noted otherwise 5.1. DetailedPerformanceAnalysis
subproblems were ordered by increasing width (cf.
Section 3.1). We ran “plain” AOBB, AOBB with the Figure 5 shows anytime profiles for some of the
diveextension(cf.Section3.2),andBreadth-Rotating more interesting initial problem instances (“x1” suf-
AOBB as presented in Section 4; we also included fix),whileFigure6presentsresultsonasubsetofprob-
OR Branch and Bound (without problem decomposi- leminstanceswithmorethanonecomplexsubproblem
tion) as a baseline. In addition, we ran an advanced (“x2”and“x3”suffix).Foreveryprobleminstance,the
stochasticlocalsearch(SLS)algorithm[9],bothonits plottitlespecifiesnumberofvariablesn,max.domain
ownandasainitializationstepforourownexhaustive sizek,inducedwidthwalongthechosenordering,and
search;inparticularweconsidertheGLS+implemen- height of the corresponding pseudo tree h. If known,
tationfrom[9],forwhichsourcecodeispubliclyavail- theoptimalsolutionvalueisindicatedbyagraydashed
able.Notethatasanincompletesearchscheme,itdoes horizontal line. The title of each plot also notes the
not provide a proof of optimality and always runs for mini-bucket i-bound; this was typically chosen to fit
thefull24hoursinourexperiments.Allalgorithmsare a 1GB memory limit, except for pedigree instances,
implemented in C++ and were run on 2.67 GHz Intel wherethreedifferentheuristicstrengths(i=7,10,15)
XeonCPUswith2GBofRAMpercore. wereappliedforeachinstance.
Our initial test set (instance name suffix “x1”) is InFigure5aswellas6wenotethatORBranchand
comprisedof19geneticlinkagepedigreeproblems,50 Bound finds an early lower bound in some cases, but
randomlygeneratedgridnetworks,8mastermindgame generally provides little improvement over time and
instances(allpartoftheUAI2008evaluation1)aswell never gets close to the optimum. The dive extension
as 66 protein side-chain prediction problems (taken showsacceptableanytimebehavioronlyonhalfthein-
from [22]). However, several of these instances are stances shown, confirming our conjecture that its per-
relatively simple or have only one complex subprob- formance depends solely on the success of the initial
lem, which renders them less interesting for the pur- dive – if misguided by the heuristic, the anytime be-
poseofthiswork.Namely,plainAOBB(withsubprob- haviorispredictablyasbadas,orevenslightlyworse
lemsorderedbyincreasingwidth)alreadyyieldsgood thantheplainscheme.
TheproposedBRAOBB,ontheotherhand,exhibits
1http://graphmod.ics.uci.edu/uai08/ impressiveanytimeperformanceinbothFigures5and
10 L.OttenandR.Dechter/AnytimeAND/ORDepth-firstSearchforCombinatorialOptimization
75-25-1x1, i20 (n=624 k=2 w=38 h=111) pedigree19x1, i10 (n=793 k=5 w=26 h=95)
−16.5
bility)−17.0 bility)−100
a a−105
b b
pro−17.5 pro
g( or g(−110 or
Lo−18.0 pdliavein Lo−115 pdliavein
rotate rotate
100 101 102 103 104 100 101 102 103 104
Search time in seconds Search time in seconds
pedigree31x1, i7 (n=1183 k=5 w=30 h=85) pdb1bgfx1, i4 (n=112 k=81 w=14 h=44)
−130
bability)−−115400 bability)−−5408 oprlain
g(pro−160 or g(pro−52 droivtaete
Lo plain Lo
−170 dive −54
rotate
−180
100 101 102 103 104 100 101 102 103 104
Search time in seconds Search time in seconds
Fig.5.AnytimeprofilesofplainAOBB(“plain”),AOBBwithsubproblemdive(“dive”),Breadth-RotatingAOBB(“rotate”),andORBranch
andBound(“or”)onselectedinstancesfromtheinitialsetofproblems(“x1”suffix;1grid,2pedigree,1side-chainprediction).
6,oftenbyalargemargin;inallbutonecasethefirst Furthermore, we see that plain AOBB sometimes
solution is produced more or less instantly, even on has a slight edge in terms of proving optimality,
pedigree51x3(Fig.6),where“plain”and“dive”donot e.g. 172 proved optimal at 10 seconds versus 153
returnanythingwithin24hours.Notethatmastermind forBRAOBB,confirmingthatexploringsubproblems
instances arehighlydeterministicand theinitialsolu- concurrently can slightly impair the pruning (cf. Sec-
tionisalreadyoptimal–butagainBRAOBBreturnsit tion 4.4). Again, as an incomplete solver local search
first. provesnooptimalityatall.
We observe that SLS does very well on the side-
5.2. SummaryStatistics chainpredictionnetworks–theseproblemshaveonly
a few hundred variables but a large max. domain
Table1summarizestheentiresetofexperimentsby size of 81. On the other problem classes, however,
showing,atdifferentpointsoftime,thenumberofin- withthousandsofvariablesandsmallermax.domains,
stances for which any solution was found, for which BRAOBBshowsbetterperformance,inparticularwith
the optimal solution was found, and for which opti- respect to finding the optimal solution. The reason
malitywasproven(i.e.thealgorithmterminated).The for this lies in the heuristic used by AOBB: mini-
resultsconfirmthatBRAOBByieldssuperioranytime bucket space complexity is O(nki) – large domain
performance:forexample,within1seconditprovides size bounds k thus necessitate a significantly lower i-
aninitialsolutionon502instances(outof543),com- bound(i=3incaseoftheside-chainpredictionprob-
paredtojust232forplainAOBB,339forthediveex- lems),whichleadstofarlessaccurateheuristics.
tension,and424forlocalsearch;performanceremains
superiortotheotherAOBBversionsandverycompet- 5.3. CombiningLocalandExhaustiveSearch
itivewithlocalsearchforhighertimebounds.
WealsonotethatBRAOBBfindstheoptimalsolu- Looking again at Table 1, we notice that SLS can
tion quicker than the other schemes (with the excep- sometimes find solutions more quickly than any kind
tion of local search on side-chain prediction), e.g. for ofexhaustivesearch–inparticularithasfoundasolu-
overall266instancesafter10seconds(versus220for tionforall543problemsafter10seconds(versus514
plain).Similarly,inthefull24hours,BRAOBBfound instancesforBRAOBBandjust293forplainAOBB).
the optimal solution to 495 instances, versus 486 for Asoutlinedabove,however,SLSisoftenquicklyout-
plainandjust316forlocalsearch. performed by AOBB and BRAOBB in terms of find-