Stochastic Submodular Maximization Arash Asadpour, Hamid Nazrzadh, and Amin Sabri Stanford Univrsity, Stanford, CA. {asadpour,hamidnz,sabri}@stanford.du Abstract. W study stochastic submodular maximization problm with rspct to a cardinality constraint. Our modl can captur th ffct of uncrtainty in diffrnt problms, such as cascad ffcts in social ntworks, capital budgting, snsor placmnt, tc. W study non-adaptiv and adaptiv policis and giv optimal constant approximation algorithms for both cass. W also bound th adaptivity gap of th problm btwn 1.21 and 1.59. 1 Introduction Th problm of maximizing submodular functions with rspct to known constraints is a vry wll-studid problm in oprations rsarch and computr scinc. A function f : R n R is submodular if for all x, y R n : f(x y) + f(x y) f(x) + f(y) whr x y and x y dnot th componnt-wis maximum and th componntwis minimum of x and y, rspctivly. If f is twic diffrntiabl, thn submodularity is quivalnt to th condition 2 f x i x j 0, whr x i and x j ar any two coordinats of x [17]. On may imagin th function f on th domain of 0/1-vctors as a st function whr f(s) = f(x), x i = 1 whnvr i S, and x i = 0 othrwis. In othr words, a st function f : 2 N R is submodular if for any two substs S, T N: f(s T ) + f(s T ) f(s) + f(t ) A wid rang of optimization problms that aris in th ral world can b modld as maximizing submodular functions with rspct to som (usually cardinality) constraints. On instanc is th problm of viral markting and maximizing influnc through th ntwork [9, 14], whr th goal is to choos an initial activ st of popl, so as to maximiz th sprad of an innovation or bhavior in a social ntwork. It is wll-known that undr many modls of influnc propagation in ntworks (.g. dcrasing cascad modl [9]), th xpctd siz of th final cascad is a submodular function of th st of initially activatd popl. Also, du to som budgt limitations th numbr of popl that w can activat in th bginning is boundd. Hnc, th maximizing influnc problm can b sn as a maximizing submodular function problm subjct to som cardinality constraint.
2 L A TEX styl fil for Lctur Nots in Computr Scinc documntation Anothr xampl is th capital budgting problm which is to find th optimal invstmnt of capital among diffrnt projcts with a limitd budgt. Thr is a st of projcts and on wants to invst on a group of thm that maximizs his xpctd profit whil not xcding his budgt. This problm has bn studid xtnsivly undr various assumptions on th utility of th invstor and dpndncis among th projcts [20, 21, 13, 2]. Naturally, th utility functions ar non-ngativ and monoton. Also, th risk-avrs invstors ar charactrizd by thir submodular utility functions. Thrfor, such invstors nd to solv a submodular maximization problm to find thir bst bt. Yt anothr xampl is th problm of optimal placmnt of snsors for nvironmntal monitoring [11, 12] whr th objctiv is to plac a limitd numbr of snsors in th nvironmnt in ordr to most ffctivly rduc uncrtainty in obsrvations. It is known that th fficincy of a subst of snsors is a submodular st function. For th problm of maximizing submodular st functions subjct to a cardinality constraint, th clbratd rsult of Cornujols t al. [4] and Nmhausr t al. [15] shows that for nonngativ monoton submodular functions th grdy algorithm that at ach stp chooss an lmnt with th maximum marginal valu givs a (1 1 0.632)-approximation of th optimal solution. This problm has also bn studid for mor complicatd domains. In particular, for maximizing a submodular function ovr a matroid (not that th cardinality constraint is a spcial cas of this) a rcnt rsult by Vondrak [19] shows that it is still possibl to gt a 1 1 -approximation. Howvr, in practic on must dal with th stochasticity causd by th uncrtain natur of th problm, th incomplt information about th nvironmnt, tc. For instanc, in viral markting som popl in th initial st might not adopt th bhavior. Anothr xampl is th capital budgting problm whr som projcts takn by an invstor may fail (in th bginning) du to som unxpctd vnts in th markt. Also, in nvironmntal monitoring som snsors might not work proprly bcaus of bad wathr or inconsistnt connctions. All ths possibilitis motivat th problm of stochastic submodular maximization. In th stochastic stting, th outcom of th lmnts in th slctd st ar not known in advanc; whn thy ar pickd, with som known probability thy might rmain in th st or not. On may think of probability p i corrsponding to ach lmnt i, and thn th xpctd valu of slcting a st S will b th xpctd valu of function f ovr st Ŝ drivd from S by rmoving ach lmnt i indpndntly with probability 1 p i. In fact, in this papr w will considr a mor gnral vrsion of this problm in which th stochasticity of th problm convrts th st S into a vctor in th continuous spac via som known probability distributions. For th xact dfinitions s Sction 2. Th main diffrnc btwn th non-stochastic and stochastic problms is that th lattr can bnfit from adaptivity. An adaptiv policy can us th outcom of th stps takn so far to optimiz th dcisions it is going to mak in th futur. On th othr hand, th actions chosn by a non-adaptiv policy ar indpndnt of th outcom of th othr actions. Thrfor, a non-adaptiv pol-
L A TEX styl fil for Lctur Nots in Computr Scinc documntation 3 icy is quivalnt to a prdtrmind subst of lmnts. Although non-adaptiv policis may not prform as wll as adaptiv ons, thy ar particularly usful whn it is difficult or tim-consuming to discovr th outcom of an action. Our Rsults: In Sction 3 w first show that th xpctd valu of a submodular st function in th stochastic stting is still a submodular st function. This immdiatly lads to a (1 1 )-approximation of th optimal non-adaptiv policy. Thn, w considr th adaptiv policy that at ach stp chooss an lmnt with th maximum xpctd marginal valu, conditiond on th outcom of th prvious lmnts. W show that th approximation ratio of this grdy policy with rspct to th optimal adaptiv policy is 1 1. 1 W also giv a lowr bound and an uppr bound on th adaptivity gap of th problm. Th adaptivity gap is dfind as th maximum ratio btwn th xpctd valu of th optimal adaptiv and non-adaptiv policis [5]. As a lowr bound, w prov that th adaptivity gap of stochastic submodular maximization problm is at last 1.21 (s Sction 2.1). On th flip sid, in Sction 4 w show that th adaptivity gap is boundd from abov by 1 1.59, i.. thr xists a non-adaptiv policy 1 which achivs at last 1.59 fraction of th valu of bst adaptiv policy. W also show that a non-adaptiv policy within a factor of ( 1 )2 1 2.51 of th optimum adaptiv policy can b found in polynomial tim. In ordr to prov this bound, w gnraliz som of th tchniqus dvlopd by Vondrák [18]. Ths xtnsions could b of indpndnt intrst. 1.1 Rlatd Work W first brifly ovrviw som parts of th litratur on (non-stochastic) submodular optimization. Thn, w xplain som of th works that study stochastic sttings similar to ours. Cornujols t al. [4] provd that a simpl grdy algorithm givs a (1 1 )- approximation for th problm of maximizing monoton submodular st functions subjct to capacity constraints. Latr, Fig [6] provd that it is not possibl to improv this ratio unlss NP T IME(n O(log log n) ). For maximizing non-monoton submodular, rcntly Fig t al. [7] gav a constant approximation algorithm. Anothr wll-studid submodular maximization problm is th problm of allocating rsourcs to agnts with submodular utilitis, for which svral intrsting approximation algorithms hav bn dvlopd, s [18]. In this papr, w us som of ths tchniqus to bound th adaptivity gap. Gomans and Vondrák [8] considr th problm of stochastic covring. In this problm th goal is to covr all lmnts of a targt st using minimum numbr of substs. Th substs ar random variabls and thir probability distributions ar givn. Thy propos adaptiv and non-adaptiv policis for th problm. Thy also obsrv that th adaptivity gap is not constant. Chan and Farias [3] study a gnralization of th stochastic maximum k- covr problm whr th squnc of lmnts arriv according to a stochastic 1 This is also indpndntly obsrvd by Chan and Farias [3].
4 L A TEX styl fil for Lctur Nots in Computr Scinc documntation procss and utility functions may vary ovr tim. Thy show that undr som conditions, a myopic policy is a 2-approximation of th optimal adaptiv policy. In a rcnt work, Strtr and Golovin [16] study an onlin job schduling problm in a stting that th cost of jobs is givn by a submodular function. Th goal is to covr as many jobs as possibl subjct to a budgt constraints. Thy tak th rgrt minimization approach and prsnt approximat optimal policy. 2 Problm Dfinition W dfin th following abstraction for th stochastic submodular maximization problm. A st A = {X 1,, X n } of indpndnt random variabls is givn. Aftr choosing X i, its actual valu (outcom of an lmnt), dnotd by x i, is discovrd. W assum that x i [0, 1]. Lt S A b a subst of variabls. Also, lt vctor s = < ˆx 1,, ˆx n > dnot th ralization of st S, whr ˆx i = x i for i S and ˆx i = 0 for i / S. Th valu obtaind by choosing th st S aftr th ralization is qual to f(s), whr f : [0, 1] N R + is a submodular function. Lt g i b th probability distribution of random variabl X i. For vry subst S A, it dfins a probability masur g S : [0, 1] n R, which rprsnts th probability dnsity function of obsrving s whil slcting S: g S (ds) = g i (dx i ) x ds i S Also, g S (ds) is dfind to b zro if thr xist i such that s i 0, i / S. Now dfin function F g : [0, 1] n R + as th xpctd valu obtaind by choosing st S, i.., F g (S) = f(s)g S (ds) (1) s [0,1] n Our goal is to choos a st S of siz at most k which maximizs F g (S). max F g(s). S A: S k For simplicity, w assum on cannot choos an lmnt of A multipl tims. 2 For this problm, w study two typs of policis: adaptiv and non-adaptiv. A non-adaptiv policy is rprsntd by a fixd subst of A. An adaptiv policy uss th ralizd valu of th prviously chosn lmnts to dtrmin th nxt lmnt in th subst. In ordr to compar th valu of ths optimal policis, w study th adaptivity gap of th problm. Th adaptivity gap is dfind as th ratio btwn th xpctd valu of optimal adaptiv and non-adaptiv policis. 2 This assumption is not ncssary for our rsults, and is mad for sak of simplicity. On can crat k indpndnt copis of ach random variabl to simulat multiplicity. This is in contrast with th st covr problm whr allowing to chos multipl copis of an lmnt significantly rducs th adaptivity gap [8].
L A TEX styl fil for Lctur Nots in Computr Scinc documntation 5 On issu that ariss in th algorithmic discussions of this papr and many of th rlatd works is computing th valu of functions similar to F g. W assum that w ar givn an oracl which computs ths valus up to a dsird dgr of accuracy. In fact, it can b shown that in many intrsting cass such an oracl can b built fficintly. Two most important cass ar whn th probability distribution functions (g j s) ar constant Lipschitz continuous, or whn thir support is a polynomial siz st of discrt valus. Thrfor, from now on, all of our rsults will involv an arbitrary small rror trm of ɛ that w will not mntion xplicitly. In th nxt sction, w illustrat th problm by giving an xampl. W also prsnt a non-adaptiv and an adaptiv policy for this xampl. 2.1 An Exampl: Stochastic Maximum k-covr A spcial cas of stochastic submodular maximization is th stochastic maximum k-covr problm. Givn a collction F of substs of {1, 2,, n}, th max k-covr problm is dfind as finding k substs from F such that thir union has th maximum cardinality [6]. In th stochastic vrsion, th subst that an lmnt of F would covr bcoms known aftr choosing th lmnt. In this sction, w dfin an instanc of this problm. W also us this xampl to giv a lowr bound on th adaptivity gap. Th instanc w considr in this sction is as follows: A ground st G = {1, 2,, 2n} and a collction F = {C 1, C 2,, C 2n } of its substs ar givn. For 1 i n, C i = {1, 2,, n} with probability 1 n and is th mpty st with probability 1 1 n. For n + 1 i 2n, C i = {i} with probability 1 and is th mpty st with th rmaining probability. Th goal is to covr th maximum numbr of lmnts in G by slcting at most n substs in C. Lmma 1. For larg nough valus of n, th optimal non-adaptiv policy is to slct S = {C 1, C 2,, C n }. Also, th xpctd valu of this policy is n(1 1 ). Proof. Considr a subst S slctd by a non-adaptiv policy. Lt q th fraction of lmnts of S that ar in {C 1, C 2,, C n }, i.., q = S {C 1, C 2,, C n } /n. Such a policy covrs th lmnts of {1, 2,, n} with probability 1 (1 1 n )nq. Also, in xpctation, S covrs at most n (1 q) lmnts from st {n+1,, 2n}. Thrfor, th xpctd numbr of covrd lmnts is n(1 (1 1 n )nq ) + n (1 q) W can approximat th xprssion abov by n(1 + 1 ( 1 )q q ) with arbitrary high prcision for larg nough n. This xprssion is incrasing in q. Thrfor, for th optimum non-adaptiv policy w hav q = 1 or quivalntly S = {C 1, C 2,, C n }. Now considr th following adaptiv policy that at ach stp chooss th lmnts with maximum marginal valu: At stp i, 1 i n, choos st C i until
6 L A TEX styl fil for Lctur Nots in Computr Scinc documntation on of ths sts covrs {1,, n}. Aftr that, pick a st from {C n+1,, C 2n } until th numbr of chosn sts rachs n. Th following lmma givs a lowr bound on th numbr of lmnts covrd by th adaptiv policy. Lmma 2. For larg nough n, th xpctd numbr of lmnts covrd by th adaptiv policy dscrib abov is n(1 1 + 1 2 ). Proof. Th probability that C i covrs th first n lmnts is 1 n (1 1 n )i 1. If C i covrs th first n lmnts, th policy will choos n i substs from C n+1,, C 2n, ach covrs a singl lmnt with probability 1. Thrfor, th xpctd numbr of covrd lmnts is: n [ 1 n (1 1 n )i 1 (n + (n i) 1 )] = (1 + 1 n ) (1 1 n )i 1 1 n i(1 1 n n )i 1 i=1 i=1 (1 + 1 )n(1 1 ) 1 [n 2 (1 (1 1n n )n ) n 2 (1 1n ] )n n [(1 1 2 ) 1 (1 2 ] ) i=1 = n(1 1 + 1 2 ). which complts th proof of th lmma. By combining th rsults of Lmmas 1 and 2 w hav th corollary blow. Corollary 3. Th adaptivity gap of stochastic maximum k-covr is at last: 1 1 + 2 1 1 > 1.21 3 Nar-Optimal Non-Adaptiv and Adaptiv Policis In this sction w first prsnt non-adaptiv policy for th stochastic submodular maximization problm. Latr, w giv an adaptiv policy. A non-adaptiv policy is rprsntd by a fixd subst S A. Th xpctd valu of th policy is qual to F g (S). Thrfor, finding th optimal non-adaptiv policy is quivalnt to finding st S which maximizs F g (S). Not that th maximum k-covr problm is a spcial cas of our problm. Thrfor, it is not possibl to find an approximation ratio bttr than 1 1 for th optimal adaptiv policy unlss NP T IME(n O(log log n) ) [6]. In this sction, w show that thr xists a policy that is implmntabl in polynomial tim and its valu is within a 1 1 ratio of th optimal non-adaptiv policy. For th as of notation, whn it is clar from th contxt, w us F (S) instad of F g (S). Not that F (S) is a convx combination of a st of monoton submodular functions. Thrfor, w hav th following lmma.
L A TEX styl fil for Lctur Nots in Computr Scinc documntation 7 Lmma 4. Th function F (S) is monoton and submodular in S. Submodularity of F, immdiatly lads to th following rsult [4, 15]. Corollary 5. Considr th non-adaptiv grdy policy that at ach stp chooss th lmnt with maximum marginal incras in valu. Th approximation ratio of this policy with rspct to th optimal non-adaptiv policy is at last 1 1. Now w prsnt an adaptiv grdy policy with approximation ratio 1 1, with rspct to th optimal adaptiv policy. It is asy to s that finding maximum k-covr can b rducd to dsigning an adaptiv policy. Thrfor, it is not possibl to improv this ratio unlss NP T IME(n O(log log n) ). Thorm 6. Considr th adaptiv grdy policy that at ach stp slcts an lmnt with th maximum marginal valu, conditiond on th ralizd valu of th prviously chosn lmnts. Th approximation ratio of th adaptiv grdy policy with rspct to th optimal adaptiv policy is 1 1. Bfor stating th proof, w dscrib som notations. For 1 i k, lt S i b th st of lmnts chosn by th grdy adaptiv policy up to (and including) stp i. Dfin S 0 to b th mpty st. Also, lt s i dnot th ralization of S i. Th adaptiv grdy policy at ach stp i chooss an lmnt in argmax j A\Si 1 E[F (S i 1 j) s i 1 ] Proof. Th proof prsntd hr is similar to th proof of Klinbrg t al.[10] for submodular st functions. Lt T j b th st chosn by th optimal adaptiv policy up to stp j. Also, dnot th xpctd marginal valu of th i th lmnt chosn by th grdy policy by i, i.., i = E[F (S i ) s i 1 ] f(s i 1 ) = E[F (S i ) F (S i 1 ) s i 1 ] Considr a ralization s i of S i. Bcaus th ralization of ach lmnt of T j is indpndnt from othr lmnts, and f is submodular, w can writ F (T j S i s i ) as th sum of a st of monoton submodular functions. Thrfor, F (T j S i s i ) is monoton submodular with rspct to j. Hnc, for T = T k w hav: E[F (T ) s i ] E[F (T S i ) s i ] E[F (S i ) + k(f (T 1 S i ) F (S i )) s i ] Bcaus i E[F (T 1 S i ) F (S i ) s i ] w gt, E[F (T ) s i ] E[F (S i ) + k i+1 s i ] Sinc th inquality abov holds for vry history, adding up all such inqualitis, for all i, 0 i k 1, w hav: E[F (T )] E[F (S i )] + ke[ i+1 ] = E[ 1 + + i ] + ke[ i+1 ]
8 L A TEX styl fil for Lctur Nots in Computr Scinc documntation W multiply th i th inquality, 0 i k 1, by (1 1 k )k 1 i, and add thm all up. Th sum of th cofficints of E[F (T )] is qual to k 1 (1 1 k 1 k )k 1 i = (1 1 k )i = 1 (1 1 k )k 1 (1 1 k ) = k(1 (1 1 k )k ) (2) i=0 i=0 On th right hand sid, th sum of th cofficint of E[ i ], 1 i k, is qual to k(1 1 k 1 k )k i + (1 1 k )k 1 j = k(1 1 k )k i + j=i Thrfor, by inqualitis (2) and (3) w gt E[F (T )] (1 (1 1 k )k ) k i 1 j=0 (1 1 k )j = k(1 1 k )k i + k(1 (1 1 k )k i ) = k (3) k E[ i ] = (1 (1 1 k )k )E[F (S k )] i=1 Thrfor, th approximation ratio of th grdy policy is at last 1 1. It is asy to s from th proof abov that if at vry stp, a policy chooss an lmnt which is an α approximation of th maximum marginal valu, thn it achivs approximation ratio 1 ( 1 )α. 4 Adaptivity Gap: An Uppr Bound of 1.59 Our concrn in this sction will b to st an uppr bound for th adaptivity gap. In othr words, w want to hav a lowr bound on th approximation ratio of non-adaptiv policis against th bst adaptiv policy. W stablish such a bound through th following thorm: Thorm 7. Thr xists a non-adaptiv policy that approximats th optimal adaptiv policy within a factor of 1 1 1.59. Morovr, Thr xists a polynomial tim non-adaptiv policy with th approximation ratio at last ( 1)2 1 2.51Ṫh proof of th abov thorm is inspird by th tchniqus in Sction 3.5 of [18]. For th sak of consistncy, w will us th sam notation as [18] whrvr possibl. W gnraliz ths tchniqus by xtnding th domain of th function F g to ral vctors. W will dfin a function f + which sts an uppr bound on th prformanc of all adaptiv policis and also lis within a constant factor (at most 1 ) of th maximum valu of F g. As w will s, this implis that for vry adaptiv policy Adapt thr xists a non-adaptiv policy which gains at last a
L A TEX styl fil for Lctur Nots in Computr Scinc documntation 9 fraction of 1 of th xpctd valu gaind by Adapt. Also, Corollary 5 shows that th grdy non-adaptiv policy approximats th optimal non-adaptiv by a factor of 1 1. Hnc, it will b within a factor of ( )2 of th bst adaptiv policy. Hr coms our basic obsrvation about adaptiv policis. Considr an arbitrary adaptiv policy Adapt. Any such policy dcids to choos a squnc of lmnts, whr th dcision about which lmnt to choos at any stp might dpnd on th ralizd valus of th prviously chosn lmnts. Thrfor, for any ralization of outcoms (i.. ralizd valus of lmnts) a distribution on th squnc of lmnts will b implid by Adapt 3. This distribution corrsponds to what Adapt dos if it obsrvs that spcific ralization. Any adaptiv policy can b dscribd by a (possibly randomizd) dcision tr in which at ach stp an lmnt is bing addd to th currnt slction. Du to th constraints of th problm, th hight of th tr is k. Each path from th root to a laf of this tr corrsponds to a subst with k lmnts and occurs with som crtain probability. Clarly, ths probabilitis sum up to on. Lt y i b th probability that lmnt i is chosn by Adapt. Also, lt β s b th probability dnsity function for th outcom s. Thn, w hav th following proprtis: 1. β s = 1. s 2. s : β s 0. 3. i, dx i : β s ds = y i g i (x i )dx i. s,s i dx i Th first two proprtis hold bcaus β dfins a probability masur on th spac of all outcoms. Th third proprty is du to th fact that th lft hand sid is in fact computing th probability that th lmnt i is chosn and its obsrvd valu is x i. Now, w ar rady to dfin th function f + : [0, 1] n R which stablishs an uppr bound on th prformanc of any adaptiv policy. Th dfinition of f + is motivatd by th abov obsrvation about all th possibl outcoms of an arbitrary adaptiv policy. It can b sn as th gnralization of function f + in [18] to th continuous domain. For any vctor y [0, 1] n w dfin f + (y) as follows: { } sup α α s f(s) s whr th suprmum is takn ovr all probability masurs α dfind on [0, 1] n for which i, dx i : α s ds = y i g i (x i )dx i. s,s i dx i Now, w can bound th prformanc of Adapt using function f +. Considr all possibl ralizations of lmnts undr Adapt. Lt y i and β s b dfind as 3 Th rason that w mntiond a distribution (and not just a spcific squnc) is that Adapt may b a randomizd policy by itslf. But as w will s, it dos not affct our argumnts.
10 L A TEX styl fil for Lctur Nots in Computr Scinc documntation bfor. Thn, th xpctd valu of Adapt is s β sf(s). On th othr hand, by th construction of β, it is on of th possibilitis that can b usd as α in th sup trm in th dfinition of f + (y). Thrfor, th prformanc of th policy is boundd by f + (y) and w hav th following lmma. Lmma 8. Th xpctd valu of th adaptiv policy Adapt is at most f + (y). Now, lt P k b th matroid polytop {v : 1 v k, v 0}. Any valid policy is limitd to slct at most k lmnts, i.., y P k. Thrfor, th dsird uppr bound on th optimal adaptiv policy can b obtaind as a corollary of Lmma 8. Corollary 9. Th xpctd valu of th optimal adaptiv policy is boundd from abov by max y Pk {f + (y)}. W also dfin th xtnsion F g : [0, 1] n R. For any vctor y [0, 1] n, F g (y) is th xpctd valu of its ralizd outcom ŷ whn y i is st to b 1 with probability y i and 0 othrwis. Mor formally, if N = {1, 2,, n} thn F g (y) = E[F g (ŷ)] = R N y i (1 y i )F g (R). To complt th proof of Thorm 7 w nd to prov that thr xists a 0/1 vctor w P k such that th valu of F g (w) is a good approximation of th optimum valu of f +. W will do that in two stps. First, w show that for any vctor y, th valus of F g (y) and f + (y) ar within a constant of ach othr. Thn, in Lmma 11 w will show that thr xists a propr 0/1 vctor w so that F g (w) F g (y). Rmmbr that in fact F g (w) = F g (S) for th subst S corrsponding to ntris qual to 1 in w. It provs that th ratio of th bst non-adaptiv and adaptiv policy is at last 1 1. For th scond part of th rsult w show that a 0/1 vctor w such that F g (w ) (1 1 )F g(w) can b found in polynomial tim which provids an fficint way to find a non-adaptiv policy within a factor (1 1 )2 of th optimal adaptiv policy. Th following lmma provs that th dfind xtnsion f + cannot b too far from F. Lmma 10. For any monoton submodular function f and any vctor y, F g (y) (1 1 )f + (y). Proof. Th proof might b viwd as of a gnralization of th proof of Lmmas 3.7 and 3.8 in [18] to th continuous spac. W dfin an auxiliary function f : [0, 1] n R as th following: f (y) = inf {f(z) + g j (s)(f(z s (j)) f(z))ds j, z j N s j>z j whr z s (j) is th vctor z with its j-th ntry changd to s j whnvr s j > z j. i R i / R
L A TEX styl fil for Lctur Nots in Computr Scinc documntation 11 W prov that for any vctor y, F g (y) f + (y) f (y). Th first inquality follows dirctly from th dfinition of F g and f +. To prov th scond inquality, not that for any fasibl masur α and any vctor z, α s f(s)ds α s [f(z) + (f(z s (j)) f(z))]ds s s j N f(z) + g j (s)(f(z s (j)) f(z))ds j. j N s j>z j Th first inquality abov holds du to submodularity of f and th scond on is a consqunc of th dfinition of α. Also, obsrv that by plugging α = β in this inquality w hav f + (y) f (y). Now, it is nough to prov that for all y, F g (y) (1 1 )f (y). Similar to th proof of Lmma 3.8 [18], for ach j w dfin a Poisson clock C j with rat y j. W start with a vctor z = 0. Onc clock C j snds a signal, a random variabl x is producd from distribution g j. Thn, if z j < x, th valu of z j will chang to x. By abus of notation, w dnot this nw valu by z x (j) and th valu of vctor z at tim t by z(t). On can obsrv that E[f(z(1))] F g (y), using monotonicity of f. On th othr hand, E[f(z(t + dt)) f(z(t)) z(t) = z] = y j dt[ g j (x)(f(z x (j)) f(z))dx]. j x>z j But th R.H.S. is at last (f (y) f(z))dt, by th dfinition of f. Thrfor, th following bound can b drivd on th drivativ of E[f(z(t))]: 1 dt E[f(z(t + dt)) f(z) z(t) = z] (f (y) f(z))dt d dt E[f(z(t))] (f (y) E[f(z(t))])dt. Solving th diffrntial quation abov, shows that E[f(z(t)) (1 t )f (y). Combining this with th fact that f + (y) f (y) and also that E[f(z(1)] F g (y) complts th proof of lmma. Th nxt lmma shows how to round th vctor y to a propr 0/1 vctor w. Lmma 11. Thr xists a 0/1 vctor w P k such that y : F g (w) F g (y). Proof. Th ssntial rounding tool for th proof is pipag rounding introducd by [1]. In ordr to b abl to us pipag rounding w nd to prov som convxity proprty on F g. Dfin F y ij = F g(y ij (λ)) whr y ij (λ) is a vctor obtaind by adding λ to y i, subtracting λ from y j and laving all othr ntris of y unchangd. First, w show that F y ij is a convx function of λ. For any y, th function F g(y) can b writtn as blow. F g (y) = (1 y k ) [(1 y i )(1 y j )F g (R) + R N\{i,j} k R y k k / R {i,j} (1 y i )y j F g (R + j) + y i (1 y j )F g (R + i) + y i y j F g (R + i + j)].
12 L A TEX styl fil for Lctur Nots in Computr Scinc documntation Hnc, w can writ th scond drivativ of F y ij in an xplicit form: 2 F y ij λ 2 = R N\{i,j} k R y k k / R {i,j} (1 y k ) [ F g (R) + F g (R + i) + F g (R + j) F g (R + i + j)], which is clarly non-ngativ du to submodularity of F g. As a rsult of convxity of F y ij, for any vctor y P k, th main rsult of [1] nsurs that Pipag rounding yilds a 0/1 vctor w insid P k such that F g (w) F g (y). Hnc, thr xists such a vctor w for which F g (w) F g (y) holds for all y P k. Now, w ar rady to prov Thorm 7. Proof. [Thorm 7]. Lmma 8 shows that OPT = max y Pk f + (y) is an upprbound on th prformanc of th bst adaptiv policy. But from Lmma 10 w know that thr xists a vctor y such that F g (y ) (1 1 )OPT. On th othr hand, Lmma 11 implis that thr xists a 0/1 vctor w P such that F g (w) F g (y ) and hnc, F g (w) (1 1 )OPT. Notic that F g(w) is in fact th xpctd valu gaind by a non-adaptiv policy that slcts th st S corrsponding to th vctor w. Also, du to Corollary 5 grdy non-adaptiv policy obtains a valu at last (1 1 )F g(w) that will b at last (1 1 )2 OPT. Rfrncs 1. A. A. Agv and M. Sviridnko. Pipag rounding: A nw mthod of constructing algorithms with provn prformanc guarant. J. Comb. Optim., 8(3):307 328, 2004. 2. S. Ahmd and A. Atamtürk. Maximizing a class of submodular utility functions. Rsarch Rport BCOL.08.02, IEOR, Univrsity of California-Brkly, March 2008. 3. C. Chan and V. Farias. Stochastic dpltion problms: Effctiv myopic policis for a class of dynamic optimization problms. manuscript, 2008. 4. G. Cornujols, M. Fishr, and G. Nmhausr. Location of bank accounts to optimiz float. Managmnt Scinc, 23:789 810, 1977. 5. B. C. Dan, M. X. Gomans, and J. Vondrák. Approximating th stochastic knapsack problm: Th bnfit of adaptivity. In FOCS, pags 208 217, 2004. 6. U. Fig. A thrshold of ln for approximating st covr. J. ACM, 45(4):634 652, 1998. 7. U. Fig, V. S. Mirrokni, and J. Vondrák. Maximizing non-monoton submodular functions. In FOCS, pags 461 471, 2007. 8. M. X. Gomans and J. Vondrák. Stochastic covring and adaptivity. In LATIN, pags 532 543, 2006. 9. D. Kmp, J. M. Klinbrg, and É. Tardos. Maximizing th sprad of influnc through a social ntwork. In L. Gtoor, T. E. Snator, P. Domingos, and C. Faloutsos, ditors, KDD, pags 137 146. ACM, 2003.
L A TEX styl fil for Lctur Nots in Computr Scinc documntation 13 10. J. M. Klinbrg, C. H. Papadimitriou, and P. Raghavan. Sgmntation problms. J. ACM, 51(2):263 280, 2004. 11. A. Kraus and C. Gustrin. Nar-optimal nonmyopic valu of information in modls. In AAAI, pags 324 331, 2005. 12. A. Kraus and C. Gustrin. Nar-optimal obsrvation slction using submodular functions. In AAAI, pags 1650 1654. AAAI Prss, 2007. 13. A. Mhrz and Z. Sinuany-Strn. Rsourc allocation to intrrlatd risky projcts using a multiattribut utility function. Managmnt Scinc, (29):490439, 1983. 14. E. Mossl and S. Roch. On th submodularity of influnc in social ntworks. In Approx, pags 128 134, 2007. 15. G. Nmhausr, L. Wolsy, and M. Fishr. An analysis of th approximations for maximizing submodular st functions. Mathmatical Programming, 14:265 294, 1978. 16. M. Strtr and D. Golovin. An onlin algorithm for maximizing submodular functions. Tch Rport CMU-CS-07-171, 2008. 17. D. M. Topkis. Minimizing a submodular function on a lattic. Oprations Rsarch, 26(2):305 321, 1978. 18. J. Vondrák. Submodularity in combinatorial optimization. PhD thsis, Charls Univrsity, Pragu, 2007. 19. J. Vondrák. Optimal approximation for th submodular wlfar problm in th valu oracl modl. In STOC, 2008. 20. H. Wingartnr. Mathmatical Programming and th Analysis of Capital Budgting Problms. Prntic-Hall, 1963. 21. H. Wingartnr. Capital budgting of intrrlatd projcts: Survy and synthsis. Managmnt Scinc, 1966. This articl was procssd using th L A TEX macro packag with LLNCS styl