GAPs: Geospatial Abduction Problems

27 Pages·2011·0.79 MB·English

GAPs: Geospatial Abduction Problems
PAULOSHAKARIAN andV.S.SUBRAHMANIAN,UniversityofMaryland
MARIALUISASAPINO,Universita` diTorino
Therearemanyapplicationswhereweobservevariousphenomenainspace(e.g.,locationsofvictimsofa
serialkiller),andwherewewanttoinfer“partner”locations(e.g.,thelocationwherethekillerlives)that
aregeospatiallyrelatedtotheobservedphenomena.Inthisarticle,wedefinegeospatialabductionproblems
(GAPs for short). We analyze the complexity of GAPs, develop exact and approximate algorithms (often
withapproximationguarantees)fortheseproblemstogetherwithanalysesofthesealgorithms,anddevelop
aprototypeimplementationofourGAPframework. Wedemonstrateaccuracyofouralgorithmsonareal
worlddatasetconsistingofinsurgentIED(improvisedexplosivedevice)attacksagainstU.S.forcesinIraq
(theobservationswerethelocationsoftheattacks,whilethe“partner”locationsweweretryingtoinferwere
thelocationsofIEDweaponscaches).
Categories and Subject Descriptors: I.2.3 [Artificial Intelligence]: Deduction and Theorem Proving—
Nonmonotonic reasoning and belief revision; I.2.3 [Artificial Intelligence]: Problem Solving, Control
Methods, and Search—Heuristic methods; I.2.1 [Artificial Intelligence]: Applications and Expert
Systems—Cartography
GeneralTerms:Theory,Algorithms,Experimentation
AdditionalKeyWordsandPhrases:Abduction,complexityanalysis,heuristicalgorithms
ACMReferenceFormat:
Shakarian,P.,Subrahmanian,V.S.,andSapino,M.L.2011. GAPs: Geospatialabductionproblems. ACM
Trans.Intell.Syst.Technol.3,1,Article7(October2011),27pages.
DOI=10.1145/2036264.2036271http://doi.acm.org/10.1145/2036264.2036271
1. INTRODUCTION
There are numerous applications where we wish to draw geospatial inferences from
observations. Forexample,criminologists[RossmoandRombouts2008;Brantingham
andBrantingham2008]havefoundthattherearespatialrelationshipsbetweenase-
rialkiller’shouse(thegeospatialinferencewewishtomake),andlocationswherethe
crimeswere committed (the observations). A marine archaeologistwho finds parts of
a wrecked ship or its cargo at variouslocations (the observations) is interested in de-
terminingwherethemainportionofthewrecklies(thegeospatialinference). Wildlife
expertsmightfind droppingsof an endangeredspeciessuch as the Malayan sun bear
(observations) and might want to determine where the bear’s den is (the geospatial
inference to be made). In all these cases, we are trying to find a single location that 7
SomeoftheauthorsofthisarticlewerefundedinpartbyAFOSRgrantFA95500610405andAROgrants
W911NF0910206andW911NF0910525.
Author’s addresses: P. Shakarian, U.S. Military Academy, West Point, NY 10996; email:
[email protected]; V. S. Subrahmanian, University of Maryland, College Park, MD 20742; email:
[email protected];M.L.Sapino,UniversitadiTorino,Torino,Italy;email:[email protected]
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ACMTransactionsonIntelligentSystemsandTechnology,Vol.3,No.1,Article7,Publicationdate:October2011.
7:2 P.Shakarianetal.
best explains the observations (or the k locations that best explain the observations).
Therearetwocommonelementsinsuchapplications.
First,thereisasetOofobservationsofthephenomenaunderstudy. Forthesakeof
simplicity,weassumethattheseobservationsarepointswherethephenomenonbeing
studied was known to have been present. Second, there is some domain knowledge
D specifying known relationships between the geospatial location we are trying to
find and the observations. For instance, in the serial killer application, the domain
knowledgemighttellusthatserialkillersusuallyselectlocationsfortheircrimesthat
areatleast1.2kmfromtheirhomesandatmost3kmfromtheirhomes. Inthecaseof
the sun bear, the domain knowledge might state that the sun bear usually prefersto
haveadeninacave,whileinthecaseofthewreck,itmightbeusuallywithinaradius
of10milesoftheartifactsthathavebeenfound.
Thegeospatialabductionproblem(GAPforshort)istheproblemoffindingthemost
likely set of locations that is compatible with the domain knowledge D and that best
“explains” the observations in O. To see why we look for a set of locations, we note
thattheserialkillermightbeusingbothhishomeandhisofficeaslaunchingpadsfor
his attacks. In this case, no single location may best accountfor the observations. In
thispaper,weshowthatmanynaturalproblemsassociatedwithgeospatialabduction
are NP-Complete, which causes us to resort to approximation techniques. We then
showthatcertaingeospatialabductionproblemsreducetoseveralwell-studiedcombi-
natorialproblemsthathave viableapproximationalgorithms. We implementsomeof
themoreviableapproacheswithheuristicssuitableforgeospatialabduction,andtest
themonareal-worlddata-set. Theorganizationandmaincontributionsofthisarticle
areasfollows.
—Section 2 formally defines geospatial abduction problems (GAPs for short), and
Section3analyzestheircomplexity.
—Section 4 develops a “naive” algorithm for a basic geospatial abduction problem
called k-SEP and shows reductions to set-covering, dominating set, and linear-
integer programming that allow well-known algorithms for these problems to be
appliedtoGAPs.
—Section5describestwogreedyalgorithmsfork-SEPandcomparesthemtoareduc-
tiontotheset-coveringproblem.
—Section 6 describes our implementation and shows that our greedy algorithms
outperform the set-covering reduction in a real-world application on identifying
weapons caches associated with Improvised Explosive Device (IED) attacks on US
troopsinIraq. Weshowthatevenifwesimplifyk-SEPtoonlycaseswherek-means
classificationalgorithmswork,ouralgorithmsoutperformthose.
—Section7comparesourapproachwithrelatedwork.
2. GEOSPATIALABDUCTIONPROBLEM(GAP)DEFINITION
Throughout this article, we assume the existence of a finite, 2-dimensional M × N
space S1 for someintegers M,N ≥ 1 called the geospatialuniverse(orjust universe).
Each point p ∈ S is of the form (x,y) where x,y are integers and 0 ≤ x ≤ M and
0 ≤ y ≤ N. We assume that all observations we make occur within space S. We
use the space shown in Figure 1 throughout this article to illustrate the concepts we
introduce. We assume that S has an associated distance function d which assigns a
nonnegativedistancetoanytwopointsandsatisfiestheusualdistanceaxioms.2
1Weuseintegercoordinatesasmostrealworldgeospatialinformationsystems(GIS)systemsusediscrete
spatialrepresentations.
2d(x,x)=0;d(x,y)=d(y,x);d(x,y)+d(y,z)≥d(x,z).
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GAPs: GeospatialAbductionProblems 7:3
Fig.1. Aspace.Reddotsdenoteobservations.Yellowsquaresdenoteinfeasiblelocations.Greenstarsshow
one(0,3)explanation,whilepinktrianglesshowanother(0,3)explanation.
Definition 2.1 Observation. AnobservationOisanyfinitesubsetofS.
ConsiderthegeospatialuniverseshowninFigure1. Intheserialkillerapplication,the
reddotswouldindicatethelocationsofthemurders,whileintheshipwreckexample,
theywouldindicate the locationswhereartifacts werefound. We wish to identifythe
killer’slocation(orthesunkenshiporthesunbear’sden).
Asmentionedearlier,therearemanyconstraintsthatgovernwheresuchlocations
might be. For instance, it is unlikely that the sunbear’s den (or the killer’s house or
office)isinthewater,whilethesunkenshipisunlikelytobeonland.
Definition 2.2 Feasibilitypredicate. AfeasibilitypredicatefeasisafunctionfromS
to{TRUE,FALSE}.
Thus, feas(p) = TRUE means that point p is feasible and must be considered in the
search. Figure 1 denotes infeasible places via a yellow square. Throughoutthis arti-
cle,weassumethatfeasisanarbitrary,butfixedpredicate.3 Further,asfeasisdefined
asafunctionover{TRUE,FALSE},itcan allowforuser inputbased onanalytical pro-
cessescurrentlyin place. For instance,in themilitary, analysts oftencreate“MCOO”
overlayswhere“restrictedterrain”isdeemedinfeasible[USArmy1994]. Wecanalso
easilyexpressfeasibility predicatesinaProlog-stylelanguage;wecaneasilystate (in
the serial killer example) that point p is considered feasible if p is within R units of
distancefromsomeobservationand pisnotin thewater. Likewise,in thecaseofthe
sun bear example,the same languagemightstate that pis consideredfeasible if pis
within R unitsof distance frommarks on trees,within R unitsof scat, and if phas
1 2
some landcover that would allow the bear to hide. A Prolog-style language that can
express such notions of feasibility is the hybrid knowledge base paradigm [Lu et al.
1996]inwhichPrologstylerulescandirectlyinvokeaGISsystem.
Definition 2.3 (α,β)Explanation. Suppose O is a finite set of observations, E is a
finite set of points in S, and α ≥ 0, β > 0 are some real numbers. E is said to be an
(α,β)explanationofO iff:
— p∈E impliesthatfeas(p)=TRUE,thatis,allpointsinE arefeasibleand
—(∀o ∈ O)(∃p ∈ E)α ≤ d(p,o) ≤ β, that is, every observation is neither too close nor
toofarfromsomepointinE.
Thus, an (α,β) explanation is a set of points (e.g., denoting the possible locations
of the home/officeof the serial killer or the possible locations of the bear’s den). Each
3Wealsoassumethroughoutthearticlethatfeasiscomputableinconstanttime.Thisisarealisticassump-
tion,asformostapplications,weassumefeastobeuser-defined. Hence,wecanleverageadata-structure
indexedwiththecoordinatesofStoallowforconstant-timecomputation.
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point must be feasible and every observation must have an analogous point in the
explanationwhichisneithertooclosenortoofar.
Given an (α,β) explanation E, there may be an observation o ∈ O such that there
aretwo(ormore)points p ,p ∈E satisfyingtheconditionsofthesecondbulletabove.
1 2
If E is an explanation for O, a partnering function ℘E is a function from O to E such
that for all o ∈ O, α ≤ d(℘E(o),o) ≤ β. ℘E(o) is said to be o’s partner according to the
partneringfunction℘E. Wenowpresentasimpleexampleof(α,β)explanations.
Example 2.1. ConsidertheobservationsinFigure1andsupposeα =0,β =3. Then
the two green stars denote an (α,β) explanation, that is, the set {(6,6),(12,8)} is a
(0,3) explanation. So is the set of three pink triangles, that is, the set {(5,6),(10,6),
(13,9)}isalsoan(0,3)explanation.
Thebasicproblemthatwewishtosolveinthisarticleisthefollowing.
TheSimple(α,β)ExplanationProblem(SEP)
INPUT: Space S, a set O of observations, a feasibility predicate feas, and numbers
α ≥0,β >0.
OUTPUT:“Yes”ifthereexistsan(α,β)explanationforO —“no”otherwise.
A variant of this problem is the k-SEP problem which requires, in addition, that E
containskelementsorless,fork<|O|. Yetanothervariantoftheproblemtriestofind
anexplanationE thatis“best”accordingtosomecostfunction.
Definition 2.4 Costfunctionχ. Acostfunctionχ isamappingfromexplanationsto
nonnegativereals.
We willassume thatcostfunctionsaredesignedso thatthe smaller the valuethey
return, the more desirable an explanation is. Some example cost functions are given
below. Thesimpleonebelowmerelylooksatthemeandistancesbetweenobservations
andtheirpartners.
Example 2.2 Mean-distance. SupposeS,O,feas,α,β areallgivenandsupposeE is
an (α,β) explanationfor O and ℘E is a partneringfunction. We could initially set the
costofanexplanationE (withrespecttothispartneringfunction)tobe:
χ℘E(E) = (cid:6)o∈O d|(Oo,|℘E(o)).
Suppose ptn(E) is the set of all partner functions for E in the above setting. Then we
cansetthecostofE as:
χmean(E) = inf{χ℘E(E)|℘E ∈ ptn(E)}.
This definition removesreliance on a single partneringfunction as there may be sev-
eralpartneringfunctionsassociatedwithasingleexplanation. Weillustratethisdefi-
nitionusingoursunbearexample.
Example 2.3. Wildlife experts have found droppings and other evidence of the
Malayan sun bear in a given space, S, depicted in Figure 2. Points {o ,o ,o } indi-
1 2 3
cate locations of evidence of the Malayan sun bear (we shall refer to these as set O).
Points {p ,p ,...,p } indicate feasible dwellings for the bear. The concentric rings
1 2 8
around each element of O indicate the distance α = 1.7km and β = 3.7km. The set
{p ,p } is a valid (1.7,3.7) explanation for the set of evidence, O. However, we note
3 6
thatobservation o can be partneredwith either point. If weare looking to minimize
2
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Fig.2. Left: Points {o1,o2,o3} indicate locations of evidence of theMalayan sun bear (we shall refer to
theseassetO).Points{p1,p2,...,p8}indicatefeasibledwellingsforthebear. Theconcentricringsaround
eachelementofO indicatethedistanceα = 1.7kmandβ = 3.7km. Right: Points{p1,p2,p3}arefeasible
forcrime-scenes{o1,o2}. {p1,p2}aresafe-houseswithinadistanceof[1,2]km. fromcrimesceneo1 and
{p2,p3}aresafe-houseswithinadistanceof[1,2]km.fromcrimesceneo2.
distance,wenoticethatd(o ,p )=3kmandd(o ,p )=3.6km;hence, p isthepartner
2 3 2 6 3
foro suchthatthedistanceisminimized.
2
Wenowdefinean“optimal”explanationasonethatminimizescost.
Definition 2.5. Suppose O is a finite set of observations, E is a finite set of points
in S, α ≥ 0, β > 0 are some real numbers, and χ is a cost function. E is said to be
an optimal(α,β) explanation iff E is an (α,β) explanation for O and there is no other
(α,β)explanationE(cid:7) forO suchthatχ(E(cid:7))<χ(E).
Wepresentanexampleofoptimal(α,β)explanationsinthefollowing.
Example 2.4. Consider the sun bear from Example 2.3 whose behavior is depicted
in Figure 2 (left). While {p ,p } is a valid solution for the k-SEP problem (k = 2),
3 6
it does not optimize mean distance. In this case the mean distance would be 3km.
However,thesolution{p ,p }providesamean-distanceof2.8km.
3 7
SupposewearetrackingaserialkillerwhohasstruckatlocationsO={o ,o }. The
1 2
points {p ,p ,p } are feasible locations as safe houses for the killer (partners). This
1 2 3
is depicted in Figure 2 (right). Based on historical data, we know that serial killers
strikesareatleast1kmawayfromasafe-houseandatmost2kmfromthesafehouse
(α = 1, β = 2). Thus, for k = 2, any valid explanation of size 2 provides an optimal
solutionwrtmean-distanceaseveryfeasiblelocationforasafe-houseiswithin2kmof
acrimescene.
Wearenowreadytodefinethecost-basedexplanationproblem.
TheCost-based(α,β)ExplanationProblem.
INPUT: Space S, a set O of observations, a feasibility predicate feas, numbersα ≥ 0,
β >0,acostfunctionχ andarealnumberv >0.
OUTPUT: “Yes” if there exists an (α,β) explanation E for O such that χ(E) ≤ v, “no”
otherwise.
Itiseasytoseethatstandardclassificationproblemslikek-means4canbecaptured
within our framework by simply assuming that α = 0, β > max(M,N)2 and that all
4SeeAlpaydin[2010]forasurveyonclassificationwork.
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7:6 P.Shakarianetal.
points are feasible. In contrast, standard classification algorithms cannot take feasi-
bilityintoaccount,andthisisessentialfortheabovetypesofapplications.
3. COMPLEXITYOFGAPPROBLEMS
SEPcan be easily solvedin PTIME. GivenasetO ofobservations,foreacho ∈ O, let
P = {p ∈ S | feas(p) = TRUE ∧ α ≤ d(p,o) ≤ β}. If P (cid:9)= ∅ for each o, we return
o o
“yes.” We call this algorithm STRAIGHTFORWARD-SEP. Another algorithm would
merely find the set F of all feasible points and return “yes” iff for every observation
o, there is at least one point p ∈ F such that α ≤ d(p,o) ≤ β. In this case, F is the
explanation produced, but it is a very poor explanation. In the serial killer example,
F merelytells thepoliceto search all feasible locationswithouttrying to doanything
intelligent. k-SEPallowstheusertoconstrainthesizeoftheexplanationsothat“short
andsweet”explanationsthataretrulymeaningfulareproduced. Thefollowingresult
states that k-SEP is NP-Complete; the proof is a reduction from Geometric Covering
byDiscs(GCD)[Johnson1982].
THEOREM 3.1. k-SEPisNP-Complete.
In the associated optimization problem with k-SEP, we wish to produce an expla-
nationofminimumcardinality. Notethatminimumcardinalityisacommoncriterion
for parsimony in abduction problems [Reggia and Peng 1990]. We shall refer to this
problem as MINSEP. This problem is obviously NP-hard by Theorem 3.1. We can ad-
justSTRAIGHTFORWARD-SEPtofind asolution toMINSEP by findingthe minimum
hittingsetofthe P ’s.
o
Example 3.1. Consider the serial killer scenario in Example 2.4 and Figure 2
(right). Crime scene (observation) o can be partnered with two possible safe houses
1
{p ,p } and crime scene o can be partnered with {p ,p }. We immediately see that
1 2 2 2 3
the potential safe house located at p is in both sets. Therefore, p is an explana-
2 2
tion for both crime scenes. As this is the only such point, we conclude that {p } is
2
the minimum-sized solution for the SEP problem. However, while it is possible for
STRAIGHTFORWARD-SEP to return this set, there are no assurances it does. As we
saw in Example 2.4, E = {p ,p } is a solution to SEP, although a solution with lower
1 2
cardinality({p })exists. ThisiswhyweintroducetheMINSEPproblem.
2
With the complexity of k-SEP, the following corollary tells us the complexity class
oftheCost-basedExplanationproblem. Weshowthisreductionbysimplysetting the
costfunctionχ(E)=|E|.
COROLLARY 3.1. Cost-basedExplanationisNP-Complete.
As described earlier, MINSEP has the feel of a set-covering problem. Although the
generalizedcost-based explanation cannotbe directly viewedwith a similar intuition
(as the cost maps explanations to reals, not elements of S), there is an important
variantofthe Cost-based problemthatdoes. We introduceweightedSEP, orWT-SEP
here.
WeightedSpatialExplanation(WT-SEP)
INPUT:AspaceS,asetOofobservations,afeasibilitypredicatefeas,numbersα ≥0,
β >0,aweightfunctionc:S →(cid:12),andarealnumberv >0. (cid:2)
OUTPUT:“Yes”ifthereexistsan(α,β)explanationE forOsuchthat c(p)≤v —
p∈E
“no”otherwise.
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In this case, we can easily show NP-Completeness by redu(cid:2)ction from k-SEP, we
simply set the weight for each elementof S to be one, causing c(p) to equal the
p∈E
cardinalityofE.
COROLLARY 3.2. WT-SEPisNP-Complete.
Cost-basedexplanationproblemspresentedinthissectionareverygeneral. Whilethe
complexity results hold for an arbitrary function in a general case, we also consider
specificfunctionsaswell. Inthefollowing,wepresentthetotal-distanceminimization
explanation problem (TD-SEP). This is a problem where we seek to minimize the
sum of distances between observations and their closest partners while imposing a
restrictiononcardinality.
TotalDistanceMinimizationExplanationProblem(TD-SEP)
ForspaceS,letd:S×S →(cid:12)betheEuclideandistancebetweentwopointsinS.
INPUT:AspaceS,asetOofobservations,afeasibilitypredicatefeas,numbersα ≥0,
β >0,positiveintegerk<|O|,andrealnumberv >0.
O(cid:2)UTPUT: “Yes” if there exists an (α,β) explanation E for O such that |E| = k and
oi∈Ominpj∈Ed(oi,pj)≤v,“no”otherwise.
THEOREM 3.2. TD-SEPisNP-Complete.
The NP-hardness of the TD-SEP is based on a reduction from the k-Median Prob-
lem [Papadimitriou 1981]. This particular reduction (details in the appendix) also
illustrates how the k-median problem is a special case of GAPs, but k-median prob-
lems cannot handle arbitrary feasibility predicates of the kind that occur in real-life
geospatialreasoning. Thesameargumentappliestok-meansclassifiersaswell.
4. EXACTALGORITHMFORGAPPROBLEMS
Thissectionpresentsfourexactapproachestosolvek-SEPandWT-SEP.First,wepro-
vide an enumerative approach that exhaustively searches for an explanation. Then,
we show that the problem reduces to set-cover, dominating set, and linear-integer
programming. Existing algorithms for these problems can hence be used directly.
Throughout this section, we shall use the symbols (cid:7) to represent the bound on the
number of partners that can be associated with a single observation and f to repre-
sent the bound on the number of observations supported by a single partner. Note
thatbothvaluesareboundedbyπ(β2−α2),howevertheycanbemuchlessinpractice;
specifically, f isnormallymuchsmallerthan(cid:7).
4.1 NaiveExactAlgorithm
We now show correctness of NAIVE-KSEP-EXACT. This algorithm provides an exact
solution to k-SEP but takes exponential time (in k). The algorithm first identifies a
set L of all elementsof S that could be possible partnersfor O. Then, it considersall
subsets of L of size less than or equal to k. It does this until it identifies one such
subsetasanexplanation.
PROPOSITION 4.1. Ifthereisak-sizedsimple(α,β)explanationforO,thenNAIVE-
KSEP-EXACTreturnsanexplanation.Otherwise,itreturnsNO.
Finally,wehavethecomplexityofthealgorithm.
PROPOSITION 4.2. The complexity of NAIVE-KSEP-EXACT is O((k−11)!(π(β2 −
α2)|O|)(k+1)).
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Algorithm1(NAIVE-KSEP-EXACT)
INPUT:SpaceS,asetOofobservations,afeasibilitypredicatefeas,realnumbersα≥0,β >0,
andnaturalnumberk>0
OUTPUT:SetE ⊆S ofsizek(orless)thatexplainsO
(1) Let Mbeamatrixarrayofpointerstobinarystring{0,1}|O|. Misofthesamedimensions
asS. Eachelementin MisinitializedtoNULL.Foragiven p∈S, M[p]istheplaceinthe
array.
(2) LetL bealistofpointerstobinarystrings. L isinitializedasnull.
(3) Foreacho ∈Odothefollowing
i
(a) Determineallpoints p∈Ssuchthatα≤d(o,p)≤β suchthatfeas(p)=TRUE.
(b) Foreachofthesepoints, p,if M[p]=NULLtheninitializeanewarraywhereonlybiti
issetto1. Thenaddapointerto M[p]inL.
(c) Otherwise,setbitioftheexistingarrayto1.
(4) For any k elements of L (actually the k elements pointed to by elements of L), we shall
designate(cid:9) ,...,(cid:9),...(cid:9) astheelements. Wewillrefertotheithbitofelement(cid:9) as(cid:9)(i).
1 j k j j
(5) Exhaustivelygenerate allpossible(cid:2)combinationsof k elements of L untilonesuch combi-
nationisfoundwhere∀i∈[1,|O|], k ((cid:9)(i))>0
j=1 j
(6) If nosuch combinationis found, return NO. Otherwise, return thefirstcombination that
wasfound.
An exact algorithm for the cost-based explanation problems follows trivially from
theNAIVE-KSEP-EXACTalgorithmbyaddingthestepofcomputingthevalueforχ for
eachcombination. Providedthiscomputationtakesconstanttime,thisdoesnotaffect
the O( 1 (π(β2−α2)|O|)(k+1))runtimeofthatalgorithm.
(k−1)!
4.2 AnExactSet-CoverBasedApproach
We now show that k-SEP polynomially reduces to an instance of the popular set-
coveringproblem[Karp1972]whichallowsustodirectlyapplythewell-knowngreedy
algorithmreviewedinPaschos[1997]. Set-Coverisdefinedasfollows.
TheSet-CoverProblem(Set-Cover)
INPUT: Set of elements, E and a family of subsets of E, F ≡ {S ,...,S }, and
1 max
positiveintegerk.
(cid:3)
OUTPUT:“Yes”ifthereexistsak-sizedsubsetof F, F ,suchthat k {S ∈ F }≡ E.
k i=1 i k
Through a simple modification of NAIVE-KSEP-EXACT, we can take an instance of
k-SEP and produce an instance of Set-Cover. We run the first four steps, which only
takes O((cid:7)·|O|)timebytheproofofProposition4.2.
THEOREM 4.1. k-SEPpolynomiallyreducestoSet-Cover.
Example 4.1. Consider the serial killer scenario in Example 2.4 and Figure 2
(right). Suppose we want to solve this problem as an instance of k-SEP by a reduc-
tion to set-cover. We consider the set of crime-scene locations, O ≡ {o ,o } as the set
1 2
wewishtocover.WeobtainourcoversfromthefirstfourstepsofNAIVE-KSEP-EXACT.
Let us call the result list L. Hence, we can view the values of the elements in L as
the following sets S ≡ {o },S ≡ {o ,o },S ≡ {o }. These correspond with points
1 1 2 1 2 3 2
p ,p ,p respectively. As S coversO, p isanexplanation.
1 2 3 2 2
The traditional approach for approximation of set-cover has a time complexity of
O(|E|·|F|·size),wheresizeisthecardinalityofthelargestsetinthefamily F(i.e. size=
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Algorithm2(NAIVE-KSEP-SC)
INPUT:SpaceS,asetO ofobservations,afeasibilitypredicatefeas, andrealnumbersα ≥ 0,
β >0
OUTPUT:SetE ⊆S thatexplainsO
(1) InitializelistE tonull
(2) Let MbeamatrixarrayofthesamedimensionsasS oflistsofpointersinitializedtonull.
Foragiven p∈S, M[p]istheplaceinthearray.
(3) LetL bealistofpointerstolistsin M,L isinitializedtonull.
(4) LetO(cid:7) beanarrayofBooleansoflength|O|. ∀i∈[1,|O|],initializeO(cid:7)[i]=TRUE. Forsome
elemento ∈O,O(cid:7)[o]isthecorrespondingspaceinthearray.
(5) LetnumObs=|O|
(6) Foreachelemento ∈O,dothefollowing.
(a) Determineallelements p∈S suchthatfeas(p)=TRUEandd(o,p)∈[α,β]
(b) Iftheredoesnotexista p∈S meetingtheabovecriteria,thenterminatetheprogram
andreturnIMPOSSIBLE.
(c) If M[p]=nullthenaddapointerto M[p]toL
(d) Addapointertoo tothelistM[p].
(7) WhilenumObs>0loop
(a) Initializepointercur ptrtonull
(b) Initializeintegercur sizeto0
(c) Foreach ptr∈L,dothefollowing:
(i) Initializeintegerthissizeto0
(ii) Let M[p]betheelementof Mpointedtoby ptr
(iii) Foreachobs ptrinthelistM[p],dothefollowing
A. LetibethecorrespondinglocationinarrayO(cid:7)toobs ptr
B. IfO(cid:7)[i]=TRUE,incrementthissizeby1
(iv) Ifthissize>cursize,setcursize=thissizeandhavecur ptrpointto M[p]
(d) Add ptoE
(e) Foreveryobs ptrinthelistpointedtobycur ptr,dothefollowing:
(i) LetibethecorrespondinglocationinarrayO(cid:7) toobs ptr
(ii) IfO(cid:7)[i],thensetittoFALSEanddecrementnumObsby1
(f) AddthelocationinspaceS pointedtobycur ptrtoE
(8) ReturnE
maxi≤|F||Si|). This approach obtains an approximation ratio of 1 + ln(size) [Paschos
1997]. As f isthequantityofthelargestnumberofobservationssupportedbyasingle
partner, the approximation ratio for k-SEP using a greedy-scheme after a reduction
fromset-coveris 1+ln(f). The NAIVE-KSEP-SC algorithmbelowleveragestheabove
reductiontosolvethek-SEPproblem.
PROPOSITION 4.3. NAIVE-KSEP-SC has a complexity of O((cid:7) · f · |O|2) and an
approximationratioof1+ln(f).
PROPOSITION 4.4. A solution E to NAIVE-KSEP-SC provides a partner to every
observationinO ifapartnerexists–otherwise,itreturnsIMPOSSIBLE.
ThealgorithmNAIVE-KSEP-SCisanaive,straight-forwardapplicationoftheO(|E|·
|F|·size)greedyapproachforset-coveraspresentedinPaschos[1997]. Wenotethatit
ispossibletoimplementaheaptoreducethetime-complexitytoO((cid:7)·f·|O|·lg((cid:7)·|O|)),
avoidingthecostofiteratingthroughallpossiblepartnersintheinner-loop.
ACMTransactionsonIntelligentSystemsandTechnology,Vol.3,No.1,Article7,Publicationdate:October2011.
7:10 P.Shakarianetal.
Algorithm3(KSEP-TO-DOMSET)
INPUT:SpaceS,asetO ofobservations,afeasibilitypredicatefeas, andrealnumbersα ≥ 0,
β >0
OUTPUT:GraphGO foruseinaninstanceofaDomSetproblem
(1) LetGO =(VO,EO)beagraph. SetVO =S and EO =∅.
(2) Let Sbea mappingdefined as S: S → VO. In words, Stakes elements of thespaceand
returnsnodesfrom GO asdefinedinthefirststep. Thismappingdoesnotchangeduring
thecourseofthealgorithm.
(3) Foreacho ∈Odothefollowing
i
(a) Determineallpoints p∈Sthataresuchthatα≤d(o,p)≤β. Callthisset P
i
(b) Forall p∈ P calculatefeas(p). Iffeas(p)=FALSE,remove pfrom P.
i i
(c) LetVi={v∈VO|∃p∈ PisuchthatS(p)=v}.
(d) Add|Pi|newnodestoVO. AddthesenodestoViaswell.
(e) Foreverypairofnodesv1,v2 ∈Vi,addedge(v1,v2)to EO.
(4) Removeallv∈VO wheretheredoesnotexistanv(cid:7) suchthat(v,v(cid:7))∈ EO
(5) Ifany Pi≡∅returnIMPOSSIBLE.OtherwisereturnGO.
In additionto thestraightforwardgreedyalgorithmforset-covering,thereare sev-
eralotheralgorithmsthatprovidedifferenttimecomplexity/approximationratiocom-
binations. However,withareductiontotheset-coveringproblemwemustconsiderthe
resultofLund andYannakakis [1994], whichstates thatset-covercannotbe approxi-
matedwithinaratioc·log(n)foranyc<0.25(wherenisthenumberofsubsetsinthe
family F)unless NP⊆ DTIME[npoly log n].
A reduction to set-covering has the advantage of being straightforward. It also al-
lows us to leverage the wealth of approaches developedfor this well-known problem.
Inthenextsection,weshowthatk-SEPreducestothedominatingsetproblemaswell.
Wethenexplorealternateapproximationtechniquesbasedonthisreduction.
4.3 AnExactDominatingSetBasedApproach
We show below that k-SEP also reduces to the well known dominating set problem
(DomSet) [Garey and Johnson 1979] allowing us to potentially leverage fast algo-
rithmssuch astherandomized-distributedapproximationschemein Jiaetal.[2002].
DomSetisdefinedasfollows.
DominatingSet(DomSet)
INPUT:Graph G =(V,E)andpositiveinteger K ≤|V|.
OUTPUT: “Yes” if there is a subset V(cid:7) ⊂ V such that |V(cid:7)| ≤ K and such that every
vertexv ∈V −V(cid:7) isjoinedtoatleastonememberofV(cid:7) byanedgein E.
As the dominating set problem relies on finding a certain set of nodes in a graph,
then, unsurprisingly, our reduction algorithm, Algorithm 3, takes space S, an
observationset O, feasibility predicate feas, and numbersα,β and returnsgraph GO
basedonthesearguments.
Wenowpresentanexampletoillustratetherelationshipbetweenadominatingset
of size k in GO and a k-sized simple (α,β) explanation for O. The following example
illustratestherelationshipbetweenak-SEPproblemandDomSet.
Example 4.2. Consider the serial killer scenario in Example 2.4, pictured in
Figure 2(right). Supposewe wantto solve this problemas an instance of k-SEP by a
reductionto DomSet. We wantto finda1-sizedsimple (α,β) explanation(safe-house)
ACMTransactionsonIntelligentSystemsandTechnology,Vol.3,No.1,Article7,Publicationdate:October2011.