Maximizing Conjunctive Views in Deletion Propagation

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1 Maximizing Conjunctiv Viws in Dltion Propagation Bnny Kimlfld Jan Vondrá Ryan Williams IBM Rsarch Almadn San Jos, CA 9510, USA {imlfld, jvondra, ABSTRACT In dltion propagation, tupls from th databas ar dltd in ordr to rflct th dltion of a tupl from th viw. Such an opration may rsult in th (oftn ncssary) dltion of additional tupls from th viw, bsids th intntionally dltd on. Th complxity of dltion propagation is studid, whr th viw is dfind by a conjunctiv qury (CQ), and th goal is to maximiz th numbr of tupls that rmain in th viw. Bunman t al. showd that for som simpl CQs, this problm can b solvd by a trivial algorithm. This papr idntifis additional cass of CQs whr th trivial algorithm succds, and in contrast, it provs that for som othr CQs th problm is NP-hard to approximat bttr than som constant ratio. In fact, this papr shows that among th CQs without slf joins, th hard CQs ar xactly th ons that th trivial algorithm fails on. In othr words, for vry CQ without slf joins, dltion propagation is ithr APX-hard or solvabl by th trivial algorithm. Th papr thn prsnts approximation algorithms for crtain CQs whr dltion propagation is APX-hard. Spcifically, two constant-ratio (and polynomial-tim) approximation algorithms ar givn for th class of star CQs without slf joins. Th first algorithm is a grdy algorithm, and th scond is basd on randomizd rounding of a linar program. Whil th first algorithm is mor fficint, th scond on has a bttr approximation ratio. Furthrmor, th scond algorithm can b xtndd to a significant gnralization of star CQs. Finally, th papr shows that slf joins can hav a major ngativ ffct on th approximability of th problm. Catgoris and Subjct Dscriptors: H. [Databas Managmnt]: Systms, Databas Administration Gnral Trms: Algorithms, Thory Kywords: Dltion propagation, dichotomy, approximation Prmission to ma digital or hard copis of all or part of this wor for prsonal or classroom us is grantd without f providd that copis ar not mad or distributd for profit or commrcial advantag and that copis bar this notic and th full citation on th first pag. To copy othrwis, to rpublish, to post on srvrs or to rdistribut to lists, rquirs prior spcific prmission and/or a f. PODS 11, Jun 13 15, 011, Athns, Grc. Copyright 011 ACM /11/06...$ INTRODUCTION Classical in databas managmnt is th viw updat problm [, 8, 9, 11, 15, 0]: translat an updat opration on th viw to an updat of th sourc databas, so that th updat on th viw is proprly rflctd. A spcial cas of this problm is that of dltion propagation in rlational databass: givn an undsird tupl in th viw (dfind by som monotonic qury), dlt som tupls from th bas rlations (whr such tupls ar rfrrd to as facts), so that th undsird tupl disappars from th viw, but th othr tupls in th viw rmain. Th databas rsulting from this dltion of tupls is said to b sid-ffct fr [4], whr th sid ffct is th st of dltd viw tupls that ar diffrnt from th undsird on. Vry oftn, a solution that is sid-ffct fr dos not xist, and hnc, th tas is rlaxd to that of minimizing th sid ffct [4,8]. That is, th problm is to dlt tupls from th sourc rlations so that th undsird tupl disappars from th viw, and th maximum possibl numbr of othr tupls rmain in th viw. Though viw updat is a classical databas problm, rcnt applications provid a rnwd motivation for it. Spcifically, viw updat naturally ariss whn dbugging Information Extraction (IE) programs, which can b highly complicatd [3]. As a concrt xampl, th MIDAS systm [1] xtracts basic rlations from multipl (publicly availabl) financial data sourcs, som of which ar smistructurd or just txt, and intgrats thm into composit ntitis, vnts and rlationships. Naturally, this tas is rror pron, and du to th magnitud and complxity of MIDAS, it is practically impossibl to rach complt prcision. An rronous conclusion, though, can b obsrvd by a usr viwing th final output of MIDAS, and such a conclusion is lily to b th rsult of rrors or ambiguity in th bas rlations. Whn th intgration qury is tan as th viw dfinition, dltion propagation bcoms th tas of suggsting tupls to b dltd from th bas rlations for liminating th rronous conclusion, whil minimizing th ffct on th rmaining conclusions. Furthrmor, liminating tupls from th bas rlations may itslf ntail dltion propagation, sinc ths tupls ar typically xtractd by consulting xtrnal (possibly unclan) data sourcs [3, 5]. Anothr motivation ariss in businss rorganization, and th spcific application is that of liminating undsird lins, such as btwn a spcific mploy and a spcific customr. This problm is nicly illustratd by th following simpl xampl by Cui and Widom [10] (also rfrncd and usd by Bunman t al. [4]). Lt GroupUsr(group, usr) and GroupFil(group, fil) b two rlations rprsnting mm-

2 brships of usrs in groups and accss prmissions of groups to fils. A usr u can accss th fil f if u blongs to a group that can accss f; that is, thr is som g, such that GroupUsr(g, u) and GroupFil(g, f). Suppos that w want to rstrict th accss of a spcific usr to a spcific fil, by liminating som usr-group or group-fil pairings. Furthrmor, w would li to do so in a way that a maximum numbr of usr-fil accss prmissions rmain. This is xactly th dltion-propagation problm, whr th viw is dfind by th following conjunctiv qury (CQ): Accss(y 1, y ) : GroupUsr(x, y 1 ), GroupFil(x, y ) (1) A formal dfinition of a CQ is givn in Sction. For som of th abov applications of dltion propagation (.g., IE), th involvd quris can b much mor complicatd than CQs. Nvrthlss, w bliv that undrstanding dltion propagation for th basic class of CQs is an ssntial stp towards practical algorithms for ths applications. Hnc, th focus of this papr is on th complxity of dltion propagation, whr th viw is dfind by a CQ. Formally, for a CQ Q (th viw dfinition), th input for th dltion-propagation problm consists of a databas instanc I and a tupl a Q(I). A solution is an subinstanc J of I (i.., J is obtaind from I by dlting facts) such that a / Q(J), and an optimal solution maximizs th numbr of tupls in Q(J). Th sid ffct is (Q(I) \ {a}) \ Q(J). Bunman t al. [4] idntify som classs of CQs (.g., projction-fr CQs) for which a straightforward algorithm producs an optimal solution in polynomial tim; w rcall this algorithm in Sction 3 and call it th trivial algorithm. But thos tractabl classs ar highly rstrictd, as Bunman t al. show that vn for a CQ as simpl as th abov Accss(y 1, y ), tsting whthr thr is a solution that is sid-ffct fr is NP-complt. Thrfor, finding an optimal solution minimizing th sid ffct is NP-hard for Accss(y 1, y ). Of cours, on can oftn sttl for a solution that is just approximatly optimal, such as a solution that has a sid ffct that is at most twic as larg as th minimum. Nvrthlss, as notd by Bunman t al. [4], approximating th minimum sid ffct is still hard, sinc such an approximation can b usd to tst for th xistnc of a sid-ffct fr solution (bcaus an approximatly optimal solution must b sid-ffct fr whn a sid-ffct-fr solution xists). In spit of th abov hardnss in dltion propagation, w still want to dsign algorithms (at last for important classs of CQs) with provabl guarants on th quality of a solution. W achiv that by slightly twaing th optimization masur: instad of trying to minimiz th sid ffct (i.., th cardinality of (Q(I) \ {a}) \ Q(J)), our goal is to maximiz th numbr of rmaining tupls (i.., th cardinality of Q(J)). W dnot this problm by MaxDP Q. Of cours, for finding an optimal solution, this maximization problm is th sam as th original minimization problm. But in trms of approximation, th pictur drastically changs. For xampl, w show that for vry CQ Q without slf joins, MaxDP Q is approximabl by a constant factor that dpnds only on Q. Mor spcifically, for vry CQ Q without slf joins thr is a constant α Q (0, 1) (lowr boundd by th rciprocal of th arity of Q), such that th solution J producd by th trivial algorithm satisfis Q(J) α Q Q(J ) for all solutions J. In Sction 4, w formulat th had-domination proprty of CQs. W prov that whn a CQ Q without slf joins has this proprty, MaxDP Q is solvd optimally (and in polynomial tim) by th trivial algorithm. Unfortunatly, for many othr CQs thr is a limit to th xtnt to which MaxDP Q can b approximatd. Particularly, w considr th slf-join-fr CQs, and show a rmarabl phnomnon (Thorm 4.8): for thos CQs that do not hav th had domination proprty, not only is th trivial algorithm suboptimal for MaxDP Q, this problm is hard to solv optimally, and vn hard to approximat bttr than som constant ratio. Mor prcisly, w show th following dichotomy. For vry CQ Q without slf joins, on of th following holds: Q has th had-domination proprty (thus, th trivial algorithm optimally solvs MaxDP Q in polynomial tim). Thr is a constant α Q (0, 1), such that MaxDP Q is NP-hard to α Q-approximat. Th proof of this dichotomy is nontrivial. Rgarding th constant α Q, w show that it is ncssary to hav it dpndnt on Q, sinc thr is no global constant α (0, 1) that wors for all th CQs Q (without slf joins); howvr, w show that such a global constant xists for a larg class of CQs (that contains th acyclic CQs [3, 1]). Th abov dichotomy also holds for th aformntiond problms of Bunman t al. [4]. That is, for th problm of dtrmining whthr thr is a sid-ffct-fr solution, and for th problm of finding an optimal solution, th following holds whn thr ar no slf joins: if th CQ has th had-domination proprty, thn th problms ar solvabl by th trivial algorithm; othrwis, ths problms ar NP-hard. So far, th only algorithm w hav considrd for dltion propagation is th trivial on. Rcall that for vry CQ Q without slf joins, th trivial algorithm is a constant-factor approximation (with a constant dpnding on Q). In Sction 5, w dvis approximation algorithms with much bttr ratios, for th important class of star CQs without slf joins 1 (which gnraliz th quris that ar procssd by th Fagin algorithm [13] and th thrshold algorithm [14]). W first giv an algorithm that provids a 1 -approximation for MaxDP Q, whr Q is a star CQ without slf joins. This algorithm, which w call th grdy algorithm, is basd on purly combinatorial argumnts. This algorithm is vry simpl and has a highly dsirabl complxity: its running tim is polynomial not just undr th standard data complxity, but also undr qury-and-data complxity (which mans that th worst-cas cost of this algorithm is significantly smallr than that of valuating th CQ ovr th databas instanc). W thn prsnt an approximation for MaxDP Q that is basd on randomizd rounding of a linar program (LP). This randomizd algorithm is mor complicatd than th grdy algorithm. W formulat th problm as an intgr linar program, which is rlaxd to an ordinary linar program by mans of randomizd rounding that translats th rsulting LP solution into a probabilistic choic of tupls to dlt. Th LP-basd algorithm trminats in polynomial tim, but unli th grdy algorithm, th polynomial is only undr data complxity (as it ssntially rquirs th 1 S Sction 5 for th xact dfinition of a star CQ. A discussion on randomnss in optimization is givn in Sction 5.

3 valuation of th CQ). Howvr, th LP-basd algorithm has important two advantags ovr th grdy on. Th first advantag of th LP-basd algorithm is that its approximation ratio is highr: 1 1 (roughly, 0.63) rathr than 1 (which is shown to b tight for th grdy algorithm). Th scond advantag is that th LP-basd algo- rithm can b xtndd to a significant gnralization of star CQs. Roughly spaing, in this gnralization (which w formally dfin in Sction 5) th star rstriction is limitd just to th xistntial variabls. As w dmonstrat, th grdy algorithm inhrntly fails on CQs of this gnralization. Th LP-basd approach alon is not nough for this gnralization, but intrstingly, th trivial algorithm handls th cass whr th LP fails. Hnc, our gnralizd algorithm tas th bst solution from thos rturnd by th LP-basd algorithm and th trivial algorithm. Finally, in Sction 6 w show that slf joins in CQs introduc inhrnt hardnss. Rcall that for vry CQ Q without slf joins, MaxDP Q can b fficintly α Q-approximatd by th trivial algorithm, whr α Q is boundd by th rciprocal of th arity of Q. W show that this rsult dos not xtnd to CQs with slf joins. Spcifically, th trivial algorithm fails to giv th dsird approximation, and furthrmor, w show an infinit st of CQs Q, such that th achivabl approximation ratio for MaxDP Q is xponntially smallr than th rciprocal of th arity of Q. In addition, w show a CQ Q with slf joins, such that Q has th had-domination proprty and yt th trivial algorithm is sub-optimal; furthrmor, MaxDP Q is hard to approximat bttr than som constant ratio. Not that th wor rportd hr considrs a basic and rstrictd cas of th viw updat problm: dltion propagation for conjunctiv viws, with th goal of prsrving as many tupls of th viw as possibl. W found that vn in this basic cas, approximation is a nontrivial topic. W bliv that th insights and tchniqus drawn from this wor will b hlpful in th xploration of additional aspcts of viw updat, and dltion propagation in particular, li thos studid in th litratur. For xampl, dltion propagation has bn xplord undr th goal of minimizing th sourc sid ffct [4,8], namly, finding a solution with a minimal numbr of missing facts. Cong t al. [8] also studid th complxity of dltion propagation (with th goal of finding an optimal solution) in th prsnc of y constraints. Naturally, updat oprations othr than dltion (.g., insrtion and rplacmnt) hav also bn invstigatd [], and spcially in th prsnc of functional dpndncis [9, 11, 0]. Th wor of Cosmadais and Papadimitriou [9] is distinguishd by thir rquirmnt for a viw to p intact a complmnt viw (which has also bn xplord mor rcntly by Lchtnbörgr and Vossn []) that, intuitivly, contains th information ignord by th viw. Du to a lac of spac, som of th proofs ar prsntd in th xtndd vrsion of this papr [1].. PRELIMINARIES.1 Schmas, Instancs, and CQs W fix an infinit st Const of constants. W usually dnot constants by lowrcas lttrs from th bginning of th Latin alphabt (.g., a, b and c). A schma is a finit squnc R = R 1,..., R m of distinct rlation symbols, whr ach R i has an arity r i > 0. An instanc I (ovr R) is a squnc R1, I..., Rm, I such that ach Ri I is a finit rlation of arity r i ovr Const (i.., Ri I is a finit subst of Const r i ). W may abus this notation and us R i to dnot both th rlation symbol and th rlation Ri I that intrprts it. If c Const r i, thn R i (c) is calld a fact, and it is a fact of th instanc I if c Ri I. Notationally, w viw an instanc as th st of its facts (hnc, w may writ R(c) I to say that R(c) is a fact of I). If I and J ar two instancs ovr R = R 1,..., R m, thn J is a sub-instanc of I, dnotd J I, if Ri J Ri I for all i = 1,..., m. W fix an infinit st Var of variabls, and assum that Const and Var ar disjoint sts. W usually dnot variabls by lowrcas lttrs from th nd of th Latin alphabt (.g., x, y and z). W us th Datalog styl for dnoting a conjunctiv qury (abbrv. CQ): a CQ ovr a schma R is an xprssion of th form Q(y) : ϕ(x, y, c) whr x and y ar tupls of variabls (from Var), c is a tupl of constants (from Const), and ϕ(x, y, c) is a conjunction of atomic formulas R i(x, y, c) ovr R; an atomic formula is also calld an atom. W may writ just Q(y) or Q if ϕ(x, y, c) is irrlvant. W dnot by atoms(q) th st of atoms of Q. W usually writ ϕ(x, y, c) by simply listing atoms(q). W ma th rquirmnt that ach variabl occurs at most onc in x and y, and no variabl occurs in both x and y. Furthrmor, w rquir vry variabl of y to occur (at last onc) in ϕ(x, y, c). Whn w mntion a CQ Q, w usually avoid spcifying th undrlying schma R, and rathr assum that this schma is th on that consists of th rlation symbols (with th propr aritis) that appar in Q. Whn w want to rfr to that schma, w dnot it by schma(q). Multipl occurrncs of th sam rlation symbol in Q form a slf join; hnc, whn w say that Q has no slf joins w man that vry rlation symbol appars at most onc in ϕ(x, y, c). Lt Q(y) : ϕ(x, y, c) b a CQ. A variabl of x is calld an xistntial variabl of Q, and a variabl of y is calld a had variabl of Q. W us Var (Q) and Var h (Q) to dnot th sts of xistntial variabls and had variabls of Q, rspctivly. Similarly, if φ is an atom of Q, thn Var (φ) and Var h (φ) dnot th st of xistntial variabls and had variabls, rspctivly, that occur in φ. W dnot by Var(Q) and Var(φ) th unions Var (Q) Var h (Q) and Var (φ) Var h (φ), rspctivly. A join variabl of Q is a variabl that occurs in two or mor atoms of Q (not that a join variabl can b an xistntial variabl or a had variabl). Finally, th arity of Q, dnotd arity(q), is th lngth of th tupl y. Exampl.1. An important CQ in this wor is th CQ Q, which is th sam as th CQ Accss(y 1, y ) dfind in (1) (and discussd in th introduction), up to rnaming of rlation symbols. Q (y 1, y ) : R 1 (x, y 1 ), R (x, y ) () Hr and latr, in our xampls R i and R j ar assumd to b diffrnt symbols whn i j. Th atoms of Q ar φ 1 = R 1(x, y 1) and φ = R (x, y ). Not that Q has no slf joins (but it would hav a slf join if w rplacd th symbol R with R 1 ). Thr is only on xistntial variabl in Q, namly x, and th two had variabls ar y 1 and y.

4 Hnc Var (Q) = {x} and Var h (Q) = {y 1, y }. Furthrmor, Var (φ 1 ) = {x} and Var h (φ 1 ) = {y 1 }. Finally, not that x is th singl join variabl of Q. W gnraliz th notation Q to Q, for all positiv intgrs, whr th CQ Q is dfind as follows. Q (y 1,..., y ) : R 1 (x, y 1 ),..., R (x, y ) (3) Not that schma(q ) consists of th (distinct) binary rlations R 1,..., R. Obsrv that th CQ Q from Exampl.1 dos not contain any constants. An xampl of a qury with th constant Emma is th following. Q(y 1, y ) : R 1 (x, y 1 ), R (x, y, Emma) Considr th CQ Q(y), and lt I b an instanc ovr schma(q). An assignmnt for Q is a mapping µ : Var(Q) Const. For an assignmnt µ for Q, th tupl µ(y) is th on obtaind from y by rplacing vry had variabl y with th constant µ(y). Similarly, for an atom φ atoms(q), th fact µ(φ) is th on obtaind from φ by rplacing vry variabl z with th constant µ(z). A match for Q in I is an assignmnt µ for Q, such that µ(φ) is a fact of I for all atoms φ atoms(q). If µ is a match for Q in I, thn µ(y) is calld an answr (for Q in I). Th rsult of valuating Q ovr I, dnotd Q(I), is th st of all th answrs for Q in I.. Dltion Propagation Lt Q b a CQ. Th problm of maximizing th viw in dltion propagation, with Q as th viw dfinition, is dnotd by MaxDP Q and is dfind as follows. Th input consists of an instanc I ovr schma(q), and a tupl a Q(I). A solution (for I and a) is an instanc J I, such that a / Q(J). Th goal is to find an optimal solution, which is a solution J that maximizs Q(J) ; that is, J is such that Q(J) Q(K) for all solutions K. As w discuss latr, finding an optimal solution for th problm MaxDP Q may b intractabl. Oftn, though, w can sttl for approximations, which w naturally dfin as follows. For a numbr α [0, 1], a solution J is α-optimal if Q(J) α Q(K) for all solutions K. An α-approximation for MaxDP Q is an algorithm that, givn I and a, always rturns an α-optimal solution. 3. THE TRIVIAL ALGORITHM In this sction, w rcall a straightforward algorithm of Bunman t al. [4], which thy gav for proving th tractability of finding a solution J with a minimum-siz sid ffct (rcall that th sid ffct is th st (Q(I) \ {a}) \ Q(J)), in th cas whr thr is no projction (i.., Q has no xistntial variabls); hr, w trivially xtnd that algorithm to gnral CQs. Considr a CQ Q(y), and suppos that I and a ar input for MaxDP Q. Lt φ b an atom of Q. A φ-fact of I is a fact f I that is qual to µ(φ) for som assignmnt µ for Q (not that µ is not ncssarily a match for Q in I); furthrmor, if thr is such µ that satisfis µ(y) = a (in addition to µ(φ) = f), thn w say that f is consistnt with a. Th trivial algorithm for gnrating a solution for I and a is as follows. For ach φ atoms(q), w obtain from I th sub-instanc J φ by rmoving all of th φ-facts that ar consistnt with a. Thn, w rturn th J φ that maximizs Q(J φ ). Psudo-cod for Trivial Q is shown in Figur 1. Algorithm Trivial Q (I, a) 1: J : for all φ atoms(q) do 3: J φ I 4: rmov from J φ all φ-facts consistnt with a 5: if Q(J φ ) > Q(J) thn 6: J J φ 7: rturn J Figur 1: Th trivial algorithm Exampl 3.1. Considr th following CQ Q 3, which is a spcial cas of (3). Q 3(y 1, y, y 3) : R 1(x, y 1), R (x, y ), R 3(x, y 3). Figur shows an instanc I 3 ovr schma(q 3), and lt a b th tupl (,, ). Lt us show how th trivial algorithm oprats on I 3 and a. For i {1,, 3}, lt φ i b th atom R i (x, y i ). Thn atoms(q 3) = {φ 1, φ, φ 3 }. For i {1,, 3}, th φ i -facts ar thos that blong to th rlation R i, and th ons that ar consistnt with a ar thos of th form R i(j, ); th solution J φi is thrfor obtaind from I 3 by rmoving from R i all th facts xcpt for R i(i, ). Thus, w hav Q 3(J φ1 ) = {(,, )}, Q 3(J φ ) = {(,, )}, and Q 3(J φ3 ) = {(,, )}. It follows that Q 3(J φi ) = 1 for all i {1,, 3}, and thrfor, th trivial algorithm can rturn any J φi (dpnding on th travrsal ordr ovr th atoms). Th following (straightforward) proposition stats th corrctnss and fficincy of th trivial algorithm. Unlss statd othrwis, in this papr w us data complxity [6] for analyzing th complxity of dltion propagation and algorithms throf; this mans that th CQ Q is hld fixd, and th input consists of th instanc I and th tupl a. Proposition 3.. Trivial Q rturns a solution, and trminats in polynomial tim. Nxt, w show that in th cas whr Q has no slf joins, th trivial algorithm approximats MaxDP Q within a constant ratio that dpnds only on Q. (A discussion on CQs with slf joins is in Sction 6.) 3.1 Approximation Th following proposition shows that, if Q is a CQ without slf joins, thn th trivial algorithm is a constant-factor approximation for MaxDP Q (whr th constant dpnds on Q). Th proof is fairly straightforward. Proposition 3.3. If Q is a CQ without slf joins, thn Trivial Q is a 1 -approximation for MaxDP Q, whr = min{arity(q), atoms(q) }. Nxt, w will show that 1 is also a tight lowr bound on th approximation ratio of th trivial algorithm, for som quris without slf joins. Mor prcisly, w will show that for all natural numbrs, th CQ Q (Exampl.1), which satisfis arity(q ) = atoms(q) =, is such that th trivial

5 R 1 R R Figur : Instanc I 3 ovr schma(q 3) R(y 1, x 1 ) S(x 1, x ) U(y, y 3, x 4) R(y 1, x 1 ) S(x 1, x ) T (x, y, x 3 ) T (x, y, x 3 ) U(y, y 3, x 4) V (y 1, y ) Figur 3: Th graphs G (Q) and G (Q ) for th CQs Q and Q of Exampl 4. algorithm rturns no bttr than a 1 -approximation. So, lt b a natural numbr. W will construct an instanc I and a tupl a Q (I ) that raliz th 1 ratio. Each of th rlations R i contains th tupls (1, ),..., (, ); in addition, ach rlation R i contains th tupl (i, ). As an xampl, Figur shows I 3. Th tupl a is (i.., a tupl that consists of diamonds). To s that I and a ar as dsird, lt J b th subinstanc of I that is obtaind by rmoving (i, ) from ach rlation R i. Thn Q (J) contains tupls b 1,..., b, whr ach b i is th tupl that compriss only s, xcpt for th ith lmnt that is. Not that J is optimal, sinc Q (I ) = Q (J) {a}. Nvrthlss, th trivial algorithm will rmov from on of th rlations, say R i, all th tupls of th form (x, ), as dscribd for = 3 in Exampl 3.1. By doing so, th trivial algorithm producs a sub-instanc J, such that Q (J ) contains xactly on tupl, thus no bttr than a 1 -approximation. In th nxt sction w will charactriz th CQs, among thos without slf joins, for which th trivial algorithm is optimal. Furthrmor, w will show that (in th absnc of slf joins) th trivial algorithm is optimal for MaxDP Q prcisly for thos CQs Q with a tractabl MaxDP Q ; for th rmaining CQs Q, MaxDP Q is hard, and vn hard to approximat bttr than som constant ratio. 4. DICHOTOMY In this sction, w dfin th had-domination proprty of a CQ, and show that if a CQ Q without slf joins has this proprty, thn th trivial algorithm (optimally) solvs MaxDP Q. Furthrmor, w show that this proprty xactly capturs th tractability of MaxDP Q, in th sns that in th absnc of this proprty, MaxDP Q is not only intractabl, but actually cannot b approximatd bttr than som constant ratio (it is APX-hard). 4.1 Had Domination Lt Q b a CQ. Th xistntial-connctivity graph of Q, dnotd G (Q), is th undirctd graph that has atoms(q) as th st of nods, and that has an dg {φ 1, φ } whnvr φ 1 and φ hav at last on xistntial variabl in common (that is, Var (φ 1 ) Var (φ ) ). Lt φ b an atom of Q, and lt P b a st of atoms of Q. W say that P is haddominatd by φ if φ contains all th had variabls that occur in P (i.., Var h (φ ) Var h (φ) for all φ P ). Dfinition 4.1. (Had Domination) A CQ Q has th had-domination proprty if for vry connctd componnt P of G (Q) thr is an atom φ atoms(q), such that P is had-dominatd by φ. As an xampl, for th CQ Q dfind in (3), th graph G (Q ) is a cliqu ovr atoms(q ), and so it has only on connctd componnt. For > 1, non of th atoms of Q contains all th had variabls y i, and hnc, Q dos not hav th had-domination proprty. (Not that Q 1 has th had-domination proprty, as dos vry CQ with only on atom or only on had variabl.) Anothr xampl follows. Exampl 4.. Considr th CQ Q dfind by Q(y 1, y, y 3) : R(y 1, x 1), S(x 1, x ), T (x, y, x 3), U(y, y 3, x 4 ). Th lft sid of Figur 3 shows th graph G (Q). Not that G (Q) has two connctd componnts: th first componnt is {R(y 1, x 1), S(x 1, x ), T (x, y, x 3)} and th scond is {U(y, y 3, x 4)}. Th CQ Q dos not hav th haddomination proprty, sinc th first connctd componnt is not had-dominatd by any atom of Q (sinc no atom contains both y 1 and y ). Now, considr th following CQ Q, which is obtaind from Q by adding an atom V (y 1, y ). Q (y 1, y, y 3) : R(y 1, x 1), S(x 1, x ), T (x, y, x 3), U(y, y 3, x 4 ), V (y 1, y ) Th graph G (Q ) is shown on th right of Figur 3. Not that th atom V (y 1, y ) had-dominats th connctd componnt {R(y 1, x 1 ), S(x 1, x ), T (x, y, x 3 )}; hnc, Q has th had-domination proprty. 4. Optimality of th Trivial Algorithm Th following thorm stats that if a CQ Q without slf joins has th had-domination proprty, thn th trivial algorithm is optimal for MaxDP Q. Thorm 4.3. Lt Q b a CQ without slf joins. If Q has th had-domination proprty, thn Trivial Q optimally solvs MaxDP Q. Proof. Lt Q b a CQ without slf joins, such that Q has th had-domination proprty. Considr th input I and a for MaxDP Q. Lt J b any solution (.g., an optimal solution). Lt f b a fact in I \ J. Not that such a fact f must xist, sinc a Q(I) and a / Q(J). Assum, w.l.o.g., that a Q(J {f}); othrwis, f can b addd to J without losing any tupl from Q(J), and thn w can choos anothr f, and so on. Lt φ b th atom of Q, such that f is a φ-fact. Lt P f b th connctd componnt of φ in G (Q), and lt γ b an atom of Q such that γ dominats P f. Such an atom γ xists, sinc Q has th had-domination proprty. Considr th solution J γ that th trivial algorithm constructs (by rmoving from I all th γ-facts that ar consistnt with a). W will prov th thorm by showing that Q(J) Q(J γ) (and hnc, Q(J) Q(J γ) ). If J J γ, thn th claim is obvious (sinc Q is a monotonic qury). Othrwis, lt g b a fact in J \ J γ. Thn g is a γ-fact that is consistnt with a, sinc g / J γ. W will prov that thr

6 is no match µ for Q in J, such that µ(γ) = g. Sinc Q has no slf joins, this would man that g is actually uslss for producing Q(J). As a rsult, by rpating this argumnt for all g J \ J γ w will gt that for J = J J γ it holds that Q(J) = Q(J ); and sinc Q(J ) Q(J γ ) (as J J γ and Q is monotonic), it holds that Q(J) Q(J γ ). Suppos, by way of contradiction, that µ g is a match for Q in J, such that µ g(γ) = g. W will show that a Q(J), and thrby obtain a contradiction to th fact that J is a solution. Lt y b th tupl of had variabls of Q. Rcall that a Q(J {f}), and lt µ f b a match for Q in J {f} with µ f (y) = a. Not that µ f (φ) = f. (Rmmbr that Q has no slf joins.) Obsrv that µ f and µ g agr on all th had variabls of P φ, du to th fact that γ contains all ths variabls, and du to th fact that g is consistnt with a. Also not that no xistntial variabl occurs both in P φ and outsid P φ, sinc P φ is a connctd componnt. It thus follows that µ f and µ g agr on all th variabls that occur both insid P φ and outsid P φ. W now construct a match µ for Q in J, as follows: for ach z Var(Q) w dfin µ(z) = µ g(z) if z appars in P φ, and othrwis µ(z) = µ f (z). Rcall that φ is in P φ, and hnc, µ(φ) = µ g (φ); thus, µ(φ) blongs to J (and in particular, µ(φ) f). It follows that µ is indd a match for Q in J. To complt th proof, w will show that µ(y) = a. Suppos that y = (y 1,..., y ) and a = (a 1,..., a ). Lt i b an indx in {1,..., }. If y i appars in P φ, thn µ(y i) = a i, sinc µ(y i ) = µ g (y i ), µ g (γ) = g, and g is consistnt with a. If y i dos not appar in P φ, thn µ(y i ) = a i again, sinc µ(y i ) = µ f (y i ) and µ f (y) = a. W conclud that µ(y) = a, and hnc, a Q(J), as claimd. As a simpl consqunc of Thorm 4.3, if Q is a CQ without slf joins, and vry join variabl of Q is a had variabl, thn th trivial algorithm optimally solvs MaxDP Q (sinc thn G (Q) has no dgs). Actually, w can show that if vry had variabl is a join variabl, thn this statmnt is tru vn without rquiring lac of slf joins. Howvr, in Sction 6 w show that Thorm 4.3 is not corrct for all th CQs that hav slf joins. Spcifically, w show an xampl of a CQ Q with slf joins, such that Q has th haddomination proprty, and yt th trivial algorithm dos not optimally solv MaxDP Q ; vn mor, for that Q w show that MaxDP Q is hard to approximat bttr than som constant ratio. 4.3 Hardnss Th following thorm 3 stats that if a CQ Q without slf joins dos not hav th had-domination proprty, thn in contrast to Thorm 4.3, MaxDP Q is hard, and vn hard to approximat bttr than som constant ratio. In Sction w discuss th proof. Thorm 4.4. Assum P NP, and lt Q b a CQ without slf joins. If Q dos not hav th had-domination proprty, thn thr is a constant α Q (0, 1), such that MaxDP Q cannot b α Q-approximatd in polynomial tim. 3 This rsult was found aftr th submission of th rviwd vrsion of this papr, and was addd (with th consnt of th program committ) aftr th papr was accptd for publication. In th rviwd vrsion, a similar rsult was shown for a mor rstrictd class of CQs without slf joins. On may wondr whthr, in Thorm 4.4, it is ncssary to hav α Q dpnding on Q. In othr words, dos Thorm 4.4 hold for a global α that is applicabl to all th CQs without slf joins? Th following thorm shows that th answr is positiv if w ar rstrictd to th class of acyclic CQs [3, 1] and th class of CQs ovr binary 4 rlation symbols. Th proof is, ssntially, through insights on th rductions usd for proving Thorm 4.4 (s Sction 4.3.1) in ths spcial cass. Thorm 4.5. Thr is a constant α (0, 1), such th following holds for vry CQ Q without slf joins, providd that Q is acyclic or schma(q) consists of binary rlation symbols. If Q dos not hav th had-domination proprty, thn it is NP-hard to α-approximat MaxDP Q. Howvr, in contrast to Thorm 4.5, th following proposition shows that no such global α xists for th class of all th CQs without slf joins. Proposition 4.6. For all natural numbrs > 1, thr xists a CQ Q with th following proprtis. 1. Q has no slf joins.. Q dos not hav th had-domination proprty. 3. Trivial Q is a (1 1 )-approximation for MaxDP Q. Proof. W dfin Q as follows. Th schma of Q has rlation symbols R 1,..., R, whr ach R i is -ary. Dfin Q (y 1,..., y ) : φ 1, φ,..., φ whr, for i = 1,...,, th atom φ i is givn by φ i = R i(x, y 1,..., y i 1, y i+1,..., y ). Not that φ i contains all th variabls of Q, xcpt for y i. Clarly, Q satisfis Proprtis 1 and (i.., Q has no slf joins and it violats th had-domination proprty). It is lft to prov Proprty 3. Lt I and a b input for MaxDP Q. Lt i and j b such that 1 i < j, and lt f i and f j b a φ i -fact and a φ j -fact, rspctivly, that ar consistnt with a (whr consistncy is dfind in Sction 3). Th only answr of Q(I) that agrs with both f i and f j on th had variabls is a. Thrfor, among th answrs in Q(I) \ {a}, thos that rquir f i ar disjoint from thos that rquir f j. Hnc, if w ta th bst J φ constructd by Trivial Q (Figur 1), it must b th cas that Q(J φ ) misss at most 1 of th tupls in Q(I) \ {a}. In particular, th bst J φ is a (1 1 )-approximation, as claimd. Nxt, w discuss th proof of Thorm Proving Thorm 4.4 Th full proof of Thorm 4.4 is in th xtndd vrsion of this papr [1]. Hr, w giv a brif ovrviw of th proof. Our first stp is to show hardnss of Q (dfind in ()), which is a spcial cas of a CQ that dos not hav th haddomination proprty. Bunman t al. [4] showd that dciding whthr thr is a solution that is sid-ffct fr is NP-complt for Q. To show that, thy dscribd a rduction from non-mixd 3-satisfiability, which is th problm of dciding on th satisfiability of a formula in 3-CNF, 4 This thorm furthr xtnds to th class of CQs whr ach atom has at most two influntial variabls, whr an influntial variabl is ithr a join or a had variabl

7 whr no claus contains both ngatd and non-ngatd variabls (that is, ach claus is th disjunction of thr litrals, whr th thr ar ithr all ngativ or all positiv). Guruswami [17] showd a constant-factor bound on th approximability of non-mixd 3-satisfiability. Howvr, w cannot simply combin th rduction of Bunman t al. and th rsult of Guruswami, sinc that rduction dos not prsrv approximation (or PTAS). Nvrthlss, w prov inapproximability by combining that rduction with a mor rcnt rsult of Guruswami and Khot [18] showing constant-factor inapproximability for non-mixd 3-satisfiability in th cas whr ach variabl occurs in at most fiv clauss. Thus, w gt th following. Lmma 4.7. Thr is a constant α (0, 1), such that MaxDP Q is NP-hard to α-approximat. Nxt, w fix a CQ Q, such that Q has nithr slf joins nor th had-domination proprty. Our goal is to prov that MaxDP Q is hard to approximat within som factor α Q. W fix a componnt P of G (Q), such that P is haddominatd by non of th atoms of Q. W call two variabls y and y atomic nighbors if Q has an atom that contains both y and y. An important ida in th proof is to distinguish btwn two cass. Th first cas is whr two of th had variabls of P ar not atomic nighbors. Th scond cas is th complmnt: vry two had variabls of P ar atomic nighbors. In th first cas, suppos that y 1 and y ar had variabls of P that ar not atomic nighbors. W show a (fairly simpl) rduction from MaxDP Q to MaxDP Q whr, roughly spaing, th variabl y i (i = 1, ) of Q simulats th variabl y i of Q. In th scond cas, w again show a rduction from th problm MaxDP Q to MaxDP Q, and w again find y 1 and y in Q that can simulat y 1 and y, rspctivly, in Q ; but hr this tas is much mor subtl. In ssnc, for th proof to wor w nd to b abl to assum that in a solution, it is not worth to rmov any φ-fact if φ contains both y 1 and y. To do that, w choos y 1 and y carfully, and w handl diffrntly thos φ that ar insid P and thos that ar outsid P. By handling φ w ssntially augmnt th constructd instanc I (ovr schma(q)) with additional tupls; this is also subtl, sinc w nd to ma sur that not too many answrs ar addd to Q(I), or ls w can asily los th (rough) prsrvation of th approximation ratio in th rduction. Ndd for facing th last problm is th fact that vry two had variabls in P ar atomic nighbors. 4.4 Dichotomy W summariz this sction with th following dichotomy 3 that is obtaind by combining Thorm 4.3 and 4.4. Thorm 4.8 (Dichotomy). For a CQ Q without slf joins, on of th following holds. 1. Th trivial algorithm optimally solvs MaxDP Q in polynomial tim.. Thr is a constant α Q (0, 1), such that it is NP-hard to α Q -approximat MaxDP Q. Morovr, 1 holds if and only if Q has th had-domination proprty. A proof similar to (actually, simplr than) that of Thorm 4.8 givs a similar dichotomy for th problm of tsting whthr thr is a solution that is sid-ffct fr (i.., a solution J such that Q(J) = Q(I)\{a}), which has bn studid by Bunman t al. [4]. Spcifically, for a CQ Q without slf joins, tsting whthr thr is a sid-ffct-fr solution is in polynomial tim whn Q has th had-domination proprty (du to Thorm 4.3), and is NP-complt othrwis. 5. APPROXIMATIONS FOR STAR CQS A CQ is a star CQ if vry join variabl occurs in vry atom; in othr words, a CQ Q is a star CQ if thr is a st Z Var(Q), such that Z = Var(φ 1 ) Var(φ ) whnvr φ 1, φ Q and φ 1 φ (not that Z can b mpty). As an xampl, vry Q (dfind in (3)) is a star CQ (with Z = {x}). Furthrmor, vry CQ with two or fwr atoms is a star CQ. An additional xampl is th following. Q 1 (y 1, y, y 3 ) : R(x 1, y 1 ), S(y 1, x 1, y ), T (x 3, y 1, x 1, x 1 ) Not that in Q 1, th join variabls ar x 1 and y 1, and thy indd occur in ach of th thr atoms. In this sction, w prsnt approximation algorithms for th problms MaxDP Q, whr Q is a star CQ without slf joins. Th following corollary of Thorm 4.5 (for th cas of acyclic CQs) shows that star CQs ar intractabl to approximat within som factor α, xcpt for trivial cass. Corollary 5.1. Thr is a numbr α (0, 1), such that th following holds for all star CQs Q without slf joins. If vry join variabl is a had variabl, or on of th atoms contains all th had variabls, thn Trivial Q optimally solvs MaxDP Q ; othrwis, α-approximating MaxDP Q is NP-hard. Th constant factor α that w found for th hardnss part Thorm 5.1 is fairly clos to 1 (it is largr than 0.9), so this rsult dos not prclud good approximations (though it dos prclud PTAS algorithms). Rcall from Proposition 3.3 that for vry CQ Q without slf joins, MaxDP Q is 1 -approximatd using th trivial algorithm, whr = min{arity(q), atoms(q) }. In this sction, w giv constantfactor approximation algorithms for MaxDP Q, whr th factor dos not dpnd on Q, assuming that Q is a star CQ without slf joins. Towards th nd of this sction, w will show how to xtnd on of th approximations to a significant gnralization of star CQs. 5.1 A Grdy Algorithm W now prsnt a 1 -approximation for MaxDP Q, for th cas whr Q is a star CQ without slf joins. W first considr th spcial cas whr Q is th CQ Q (for som > 0). Latr on, w will discuss th xtnsion of th algorithm to th gnral cas. Fix a natural numbr > 0. Th goal is to approximat MaxDP Q. W call th approximation algorithm w prsnt hr th grdy approximation, and dnot it by Grdy. Figur 4 shows psudo-cod for th algorithm. Th input includs an instanc I and a tupl a = (a 1,..., a ), and th algorithm rturns a solution J. W us th following notation. For a tupl d = (d 1,..., d ), w call a constant b Const a d-joining constant if R i(b, d i) is a fact of I for all i = 1,..., (i.., thr is a match µ for Q in I, such that µ(y) = d and µ(x) = b).

8 Algorithm Grdy (I, a) 1: lt a = (a 1,..., a ) : J I 3: for all a-joining constants b do 4: if xists j whr R j (b, c) I for som c a j thn 5: i a minimal j with R j(b, c) I for som c a j 6: ls 7: i an arbitrary numbr in {1,..., } 8: dlt R i (b, a i ) from J 9: rturn J Figur 4: Grdy approximation for MaxDP Q Th algorithm Grdy starts with J = I. It travrss ovr all th a-joining constants b. For ach b, th fact R i (b, a i ) is dltd from J, whr i is chosn to b th minimal j such that I contains a fact R j (b, c) for som c a j; if no such j xists, thn i is chosn arbitrarily among {1,..., }. Th following lmma stats that th algorithm rturns a solution. Th proof is immdiat from th fact that for ach a-joining constant b, th rturnd instanc J misss R i (b, a i ) for som (actually, for xactly on) i {1,..., }. Lmma 5.. Grdy (I, a) rturns a solution. Nxt, w show that Grdy is a -approximation for MaxDP Q. Fix th input I for th algorithm, and lt J b th rturnd sub-instanc of I. W will show that Q (J) Q (I) \ {a} /. This implis that Grdy is a 1 -approximation rgardlss of th prformanc of an optimal algorithm, sinc Q (J ) Q (I) \ {a} holds for vry solution J. Morovr, this implis that thr is always a solution that rtains at last half of th original (non-a) tupls of Q (I). To show that Q (J) Q (I) \ {a} /, w us th following counting argumnt. W map th surviving answrs to dltd answrs in a way that ach dltd answr is th imag of som surviving answr. Sinc no surviving answr has two imags, this will imply that th numbr of surviving answrs is at last as larg as th numbr of dltd answrs. Th intuitiv rason for th xistnc of such a mapping is that w try to dlt only facts R i (b, a i ) whr thr is an altrnativ fact R i (b, c) for th sam joining constant b; th fact R i (b, c) will prsrv som answrs to account for thos that hav bn dltd. Formally, w dfin a function Ψ : Q (J) Const, and show that Ψ is onto (Q (I) \ {a}) \ Q (J). Th function Ψ is dfind as follows. For a tupl c Q (J), whr c = (c 1,..., c ), th tupl Ψ(c) is obtaind from c by choosing th minimal i with c i a i, and rplacing that occurrnc of c i with a i. For xampl, for = 4 and a = (,,, ), th tupl Ψ(,,, ) is (,,, ). Lmma 5.3. Ψ is onto (Q (I) \ {a}) \ Q (J). Proof. Lt d = (d 1,..., d ) b a tupl of Q (I), such that d is nithr a nor in Q (J). W nd to show that thr is som c Q (J), such that Ψ(c) = d. Lt b b a... n R R 1 1 c 1 c... c n a 1 a Figur 5: An instanc I whr Grdy (I, a) provids a ( )-approximation n d-joining constant (b xists sinc d Q (I)). Th fact that d / Q (J) implis that b is an a-joining constant, and that whn b was visitd w dltd on of th R i(b, d i). Considr such an indx i, and not that d i = a i follows from th dfinition of th algorithm. Now, d a implis that, in th itration of b, th condition of Lin 4 is tru, and hnc, th dltd fact R i (b, d i ) satisfis that i is th minimal j whr R i(b, c) I for som c a i. Hnc, w must hav d j = a j for all j < i (and rcall that w also hav d i = a i). Lt c a i b a constant such that R i (b, c) I, and lt c b obtaind from d by rplacing th ith lmnt, d i, with c. W hav that c Q (J) and, morovr, Ψ(c) = d, as rquird. From Lmmas 5. and 5.3 w gt th following thorm. Thorm 5.4. Grdy is a 1 -approximation for th problm MaxDP Q. Nxt, w show that in th worst cas, Grdy indd givs just 1 -approximation. Our xampl is for =, and it is dpictd in Figur 5. Th rlation R 1 contains th tupls (i, a 1) and (i, ), for all i {1,..., n}. Not that th visual position of nods on th dgs that corrspond to R 1 is in opposition to th ordr of th valus in R 1 (i.., th dg from to corrsponds to th fact R 1 (, )). Th rlation R contains th tupls (i, a ) and (i, c i ) for all i {1,..., n}. Th grdy algorithm will rmov all th facts R 1(i, a 1), gnrating a sub-instanc J with Q (J) = n + 1. On th othr hand, on could rmov all th facts R (i, a ), and thn gt a solution J, such that Q (J ) = n. Hnc, th approximation ratio is at most n Finally, w discuss th gnralization of th grdy algorithm to gnral star CQs without slf joins. Lt Q b a star CQ without slf joins. Suppos that Q has atoms. W rduc MaxDP Q to MaxDP Q in an approximationprsrving mannr. Spcifically, givn th input I and a for MaxDP Q, w gnrat an instanc I ovr schma(q ) and a tupl a, and apply Grdy (I, a ) to gt a solution J for I and a. Finally, w transform J into a solution J for I and a. This rduction is fairly straightforward, and th dtails ar in th xtndd vrsion of this papr [1]. In conclusion, w gt th following thorm. Thorm 5.5. Lt Q b a star CQ without slf joins. Thr is a 1 -approximation for MaxDP Q, whr th running tim is polynomial undr qury-and-data complxity. Rcall that qury-and-data complxity mans that th running tim is masurd as if th qury Q is givn as part of

9 th input (in addition to I and a), and is not fixd. Th polynomial running tim undr qury-and-data complxity is du to th fact that both Grdy and th rduction from MaxDP Q to MaxDP Q ta polynomial tim undr qury-and-data complxity. In contrast, rcall that th trivial algorithm is polynomial tim only undr data complxity (sinc it rquirs th valuation of Q ovr th J φ ). 5. A Randomizd-Rounding Algorithm In this sction, w dscrib a randomizd-rounding algorithm for approximating MaxDP Q whn Q is a star CQ without slf joins. W will show that this algorithm givs an approximation ratio that is highr than Grdy, namly, 1 1 (which is, roughly, 0.63) instad of 1 ; th running tim is still polynomial, but it is not as fast as Grdy. In particular, th algorithm w dscrib hr will trminat in polynomial tim undr (th usual) data complxity, but not undr qury-and-data complxity. At th nd of this sction, w will show that this algorithm can b usd to approximat CQs from a class that significantly gnralizs th star CQs without slf joins. Mor prcisly, th algorithm w prsnt is a randomizd (1 1 )-approximation, whr a randomizd α-approximation for MaxDP Q is a randomizd algorithm that, givn I and a, rturns a solution J such that th xpctd Q(J) is at last α Q(K) for all solutions K. Put diffrntly, if J opt is an optimal solution, thn E[ Q(J) ] α Q(J opt ). This is a standard notion of randomizd optimization (.g., [16, 19]). Such a randomizd algorithm can b asily transformd into a randomizd algorithm that rturns an α-approximation, whr α is arbitrarily clos to (1 1 ), and th rror probability is arbitrarily small. W furthr not that th algo- rithm w prsnt hr can b drandomizd into a dtrministic (i.., ordinary) (1 1 )-approximation, using th pipag-rounding tchniqu of Calinscu t al. [5]; howvr, that drandomization is byond th scop of this papr. Our algorithm uss idas from th framwor of submodular maximization subjct to a matroid constraint [5, 6]. In fact, our problm can b formally rducd to th problm of maximizing a monoton submodular function subjct to a matroid constraint, for which a (1 1 )-approximation has bn dvlopd rcntly [6]. Howvr, it is simplr (and maximiz X d Q (I) Y d subjct to d Q (I) Y d X b JC (a) b JC (d) X Xi b = 1 i=1 b JC (a), i [] 0 X b i 1 d Q (I) 0 Y d 1 X i []\(d a) Figur 6: Th program LP(I, a) X b i Algorithm RRLP (I) 1: lt a = (a 1,..., a ) : J I 3: solv LP(I, a) 4: for all a-joining constants b do 5: indpndntly pic a random i [] with probability X b j for j 6: dlt R i (b, a i ) from J 7: rturn J Figur 7: Randomizd rounding for MaxDP Q mor instructiv) to giv a slf-containd dscription, which is what w do in th rmaindr of this sction. As w did in th prvious sction, w will first dscrib th algorithm for th spcial CQs Q (dfind in (3)), and considr gnral star CQs latr. W fix th input I and a = (a 1,..., a ). Rcall that for a tupl d = (d 1,..., d ), a d- joining constant is a constant b Const such that R i (b, d i ) I for all i = 1,...,. W ma th assumption that for ach fact R i(b, d) of I, th constant b is an a-joining constant. Thr is no loss of gnrality in maing this assumption, sinc thr is no rason to dlt a fact R i (b, d) if b is not an a-joining constant; hnc, th xistnc of such R i (b, d) can only incras th approximation ratio that w achiv by th algorithm. For a tupl d Q (I), w dnot by JC (d) th st of all th d-joining constants. Not that our assumption abov implis that JC (d) JC (a) for all d Q (I). Lt us first formulat MaxDP Q as an intgr linar program (LP). For vry a-joining constant b and indx i {1,..., }, w hav th variabl Xi b that gts valus in {0, 1}. W intrprt Xi b = 1 as saying that th fact R i (b, a i ) should b rmovd. So, w will hav th following constraints which nsur that th rsulting sub-instanc is a solution. Not that [] is a shorthand notation for {1,..., }. b JC (a) X Xi b = 1 i=1 b JC (a), i [] X b i {0, 1} Nxt, w construct th objctiv function. Suppos that d = (d 1,..., d ) is a tupl in Q (I). W dfin th variabl Y d that gts th valu 1 if d survivs (i.., blongs to Q (J)), and 0 othrwis. For that, w dnot by d a th st of indics i [], such that d i = a i. For xampl, if a = (,,, ) and d = (,,, ), thn d a = {1, 4}. So, w hav th following constraint: d Q (I) 0 Y d = X X 1 A b JC (d) i []\(d a) Not that P i []\(d a) Xb i 1 mans that th fact R i(b, a i) that w dlt is such that a i d i, and hnc, non of th R j(b, d j) is dltd (and thn d survivs). Finally, th goal is to maximiz th sum of th Y d. Figur 6 shows th program LP(I, a), which is th LP rlaxation of th intgr LP. X b i

10 Th algorithm for MaxDP Q, calld RRLP (I, a), is dscribd by th psudo-cod of Figur 7. Th algorithm first solvs LP(I, a), and as a rsult, gts an optimal (fractional) assignmnt for ach Xi b and Y d. Li Grdy (I, a), this algorithm constructs a sub-instanc J of I and rturns J in th last lin. Still similarly to Grdy (I, a), for ach a-joining constant b, th algorithm slcts an indx i [] and dlts from J th fact R i(i, a i). Th diffrnc btwn th algorithms is in th way i is chosn. Hr, w apply th standard action in randomizd rounding, namly, i is picd randomly and indpndntly from [], whr th probability of th indx j is Xj b. Not that th constraints of LP(I, a) nsur that, for a spcific b, th Xi b constitut a probability distribution ovr []. Nxt, w prov th corrctnss of RRLP (I, a). Th following lmma shows that th algorithm rturns a solution, and that its running tim is polynomial. Th proof is straightforward. In particular, as notd bfor Lmma 5. about Grdy, for ach a-joining constant b th rturnd instanc J misss R i(b, a i) for som (actually, for xactly on) i []. Lmma 5.6. RRLP (I, a) rturns a solution, and trminats in polynomial tim. Nxt, w show that RRLP (I) is a randomizd (1 1 )- approximation. Th proof is basd on th following lmma, which stats that a tupl d Q(I) survivs with a probability of at last (1 1 )Y d. Lmma 5.7. Considr an xcution of RRLP (I, a) that rsults in a random solution J, and lt d Q (I). Pr (d Q (J)) (1 1 )Y d. Basd on Lmma 5.7, w nxt prov that RRLP (I) is indd a randomizd (1 1 )-approximation. Thorm 5.8. Considr an xcution of RRLP (I, a) that rsults in a random solution J, and lt J opt b an optimal solution. Thn E[ Q (J) ] (1 1 ) Q (J opt). Proof. Considr th xcution of LP(I, a) in Lin 3 of RRLP (I, a). Lt M b th sum P d Q (I) Y d. W.l.o.g., w can assum that in J opt it holds that for all a-joining constants b, xactly on R i(b, a i) is missing. Thn J opt dfins a solution for LP(I, a), and thrfor, M Q (J opt). On th othr hand, from Lmma 5.7 and th linarity of xpctation w hav that E[ Q (J) ] = X Pr(d Q (J)) d Q (I) (1 1 ) X d Q (I) Y d = (1 1 )M Thrfor, E[ Q (J) ] (1 1 ) Q (J opt ), as claimd. W now considr gnral star CQs without slf joins. As w notd bfor Thorm 5.5, thr is a simpl approximationprsrving (and polynomial-tim) rduction from th problm MaxDP Q, whr Q is a star CQ without slf joins, to MaxDP Q, whr = atoms(q). Hnc, Thorm 5.8 immdiatly implis th following rsult. Thorm 5.9. Lt Q b a star CQ without slf joins. Thr is a randomizd (1 1 )-approximation for MaxDP Q, with a polynomial running tim. 5.3 Byond Star CQs In this sction, w xtnd Thorm 5.9 to th class of xistntially star CQs, which gnralizs th class of star CQs without slf joins. Intuitivly, in an xistntially star CQ th star rquirmnt is rstrictd to th xistntial variabls of th qury. Mor formally, lt Q b a CQ. W dnot by Var (Q) th st of xistntial join variabls of Q. W say that Q is xistntially star if for vry atom φ of Q, ithr vry variabl of Var (Q) occurs in Q, or non of thm dos. Exampl Considr th following CQ: Q(y 1, y, y 3, y 4 ) : R(x 1, y 1, y ), S(x 1, y, y 3 ), T (y 3, y 1, x ) Th CQ Q is xistntially star, sinc Var (Q) = {x 1}, and vry atom ithr contains x 1 or not. Actually, by th sam argumnt, vry CQ that has at most on xistntial join variabl is xistntially star. Nxt, w giv a short ovrviw of how w handl xistntially star CQs without slf joins. W dnot by Q th CQ that compriss all th atoms φ of Q having Var (Q) Var(φ). Not that if Q is xistntially star, thn Q is also xistntially star. Also obsrv that th arity of Q can b strictly largr than that of Q. As an xampl, for th CQ Q of Exampl 5.10, th CQ Q is: Q (y 1, y, y 3 ) : R(x 1, y 1, y ), S(x 1, y, y 3 ) Not that th arity of Q is 4, whil that of Q is 3. Lt Q b an xistntially-star CQ Q without slf joins. Our gnral approach to approximating MaxDP Q is as follows. Instad of approximating MaxDP Q, w approximat MaxDP Q ; furthrmor, instad of using th input tupl a of MaxDP Q, for MaxDP Q w us th rstriction of a to th had variabls of Q. Finally, w viw vry occurrnc of a had join variabl as a distinct had variabl, and thus assum that vry join variabl is xistntial. Hnc, w trat Q as if it is a star CQ, and thn w apply Thorm 5.9, to gt a solution J. Thr ar two main problms with th abov approach. First, by rstricting to Q and ignoring th fact that had variabls can b join variabls, whn solving MaxDP Q w may nd up saving tupls of Q (J) that do not giv ris to any tupl of Q(J), whil liminating tupls of Q (I) that giv ris to multipl tupls of Q(I). To handl that, w considr again th program LP(I, a) of Figur 6, and obsrv that w can assign a wight w(d) to ach variabl Y d. That is, th objctiv function can b as follows. X maximiz w(d) Y d d Q (I) Indd, w ar abl to handl th first problm by using propr wights w(d). Lt us call th final algorithm that rducs MaxDP Q to MaxDP Q and uss wights as dscribd abov th xtndd RRLP. Unfortunatly, th xtndd RRLP dos not guarant a propr approximation, du to a scond problm, which is th following. By rstricting to Q instad of Q, w ignor th φ-facts of I for φ atoms(q) \ atoms(q ). But it may still b th cas that dlting facts among thos φ-facts is ncssary to obtain a propr approximat solution. As an xampl, considr th following xistntially-star CQ. Q(y 1, y, y 3 ) : R 1 (x, y 1 ), R (x, y ), R 3 (y, y 3 )