General Caching Is Hard: Even with Small Pages

Transcription

1 Algorithmica manuscript No. (will b insrtd by th ditor) Gnral Caching Is Hard: Evn with Small Pags Luká² Folwarczný Ji í Sgall August 1, 2016 Abstract Caching (also known as paging) is a classical problm concrning pag rplacmnt policis in two-lvl mmory systms. Gnral caching is th variant with pags of dirnt sizs and fault costs. Th strong NP-hardnss of its two important cass, th fault modl (ach pag has unit fault cost) and th bit modl (ach pag has th sam fault cost as siz) has bn stablishd, but undr th assumption that thr ar pags as larg as half of th cach siz. W prov that this alrady holds whn pag sizs ar boundd by a small constant: Th bit and fault modls ar strongly NP-complt vn whn pag sizs ar limitd to {1, 2, 3}. Considring only th dcision vrsions of th problms, gnral caching is quivalnt to th unsplittabl ow on a path problm and thrfor our rsults also improv th hardnss rsults about this problm. Kywords Gnral caching Small pags NP-hardnss Unsplittabl Flow on a Path 1 Introduction Caching (also known as uniform caching or paging) is a classical problm in th ara of onlin algorithms and has bn xtnsivly studid sinc 1960s. It modls a two-lvl mmory systm: Thr is th fast mmory of siz C (th cach) and a slow but larg main mmory whr all data rsid. Th problm instanc compriss a squnc of rqusts, ach dmanding a pag from th Partially supportd by th Cntr of Excllnc ITI, projct P202/12/G061 of GA ƒr (J. Sgall) and by th projct S of GA ƒr (L. Folwarczný). Computr Scinc Institut of Charls Univrsity, Faculty of Mathmatics and Physics, Malostranské nám. 25, CZ Praha 1, Pragu, Czch Rpublic. {folwar,sgall}@iuuk.m.cuni.cz

2 2 Luká² Folwarczný, Ji í Sgall main mmory. No cost is incurrd if th rqustd pag is prsnt in th cach (a cach hit). If th rqustd pag is not prsnt in th cach (a cach fault), th pag must b loadd at th fault cost of on; som pag must b victd to mak spac for th nw on whn thr ar alrady C pags in th cach. Th natural objctiv is to vict pags in such a way that th total fault cost is minimizd. For a rfrnc on classical rsults, s Borodin and El-Yaniv [11]. In 1990s, with th advnt of World Wid Wb, a gnralizd variant calld l caching or simply gnral caching was studid [19, 21]. In this stting, ach pag p has its siz(p) and cost(p). It costs cost(p) to load this pag into th cach and th pag occupis siz(p) units of mmory thr. Uniform caching is th spcial cas satisfying siz(p) = cost(p) = 1 for vry pag p. Othr important cass of this gnral modl ar th cost modl (wightd caching): siz(p) = 1 for vry pag p; th bit modl: cost(p) = siz(p) for vry pag p; th fault modl: cost(p) = 1 for vry pag p. Caching, as dscribd so far, rquirs th srvic to load th rqustd pag whn a fault occurs, which is known as caching undr th forcd policy. Allowing th srvic to pay th fault cost without actually loading th rqustd pag to th cach givs anothr usful and studid variant of caching, caching undr th optional policy. Prvious work. In this articl, w considr th problm of nding th optimal srvic in th oin vrsion of caching whn th whol rqust squnc is known in advanc. Uniform caching is solvabl in polynomial tim with a natural algorithm known as Blady's rul [9]. Caching in th cost modl is a spcial cas of th k-srvr problm and is also solvabl in polynomial tim [13]. In lat 1990s, th qustions about th complxity status of gnral caching wr raisd. Th situation was summd up by Albrs t al. [3]: Th hardnss rsults for caching problms ar vry inconclusiv. Th NP-hardnss rsult for th Bit modl uss a rduction from partition, which has psudopolynomial algorithms. Thus a similar algorithm may wll xist for th Bit modl. W do not know whthr computing th optimum in th Fault modl is NP-hard. Thr was no improvmnt until a brakthrough in 2010 whn Chrobak t al. [15] showd that gnral caching is strongly NP-hard, alrady in th cas of th fault modl as wll as in th cas of th bit modl. Gnral caching is usually studid undr th assumption that th largst pag siz is vry small in comparison with th total cach siz, as is for xampl th cas of th aformntiond articl by Albrs t al. [3]. Instancs of caching with pags largr than half of th cach siz (so calld obstacls) ar rquird in th proof givn by Chrobak t al. Thrfor, this hardnss rsult is in fact still quit inconclusiv. Indpndntly, a wak NP-hardnss of th fault modl was provn by Darmann t al. [16]. A substantially simplr proof of th strong NP-hardnss of gnral caching was givn by Bonsma t al. [10]; only pags of sizs in {1, 2, 3} ar ndd in th proof, but thy hav many dirnt costs: This mans that

3 Gnral Caching Is Hard: Evn with Small Pags 3 th costs ar far from th fault modl and in fact thy ar also far from th bit modl. Contribution. W giv a novl proof of strong NP-hardnss for gnral caching which givs th rst hardnss rsult rstrictd to small pags in th fault and bit modls: Thorm 1.1 Gnral caching is strongly NP-hard vn in th cas whn th pag sizs ar limitd to {1, 2, 3}, for both th fault modl and th bit modl, and undr ach of th forcd and optional policis. W also giv a sparat slf-containd vrsion of th proof of th rsult for gnral costs (and sizs {1, 2, 3}). This vrsion is rathr simpl, in particular signicantly simplr than th on givn by Chrobak t al. [15] and at th sam tim it uss only two dirnt costs, which is a simplr structur compard to Bonsma t al. [10]. At th sam tim, th simplid vrsion illustrats som of th main idas of our rduction. Th rductions for th rsult in th fault and bit modls ar signicantly mor involvd and rquir a non-trivial potntialfunction-lik argumnt. Opn problms. W prov that gnral caching with pag sizs {1, 2, 3} is strongly NP-hard whil gnral caching with unit pag sizs is asily polynomially solvabl. Closing th rmaining gap will dnitly contribut to a bttr undrstanding of th problm: Qustion 1.2 Is gnral caching also (strongly) NP-hard whn pag sizs ar limitd to {1, 2}? Can caching with pag sizs {1, 2} b solvd in polynomial tim, at last in th bit or fault modl? In a broadr prspctiv, th complxity status of gnral caching is still far from bing wll undrstood as th bst known rsult on approximation is a 4- approximation du to Bar-Noy t al. [7] and thr is no rsult on th hardnss of approximation. Thrfor, bttr undrstanding of th complxity of gnral caching rmains an important challng. Qustion 1.3 Is thr an algorithm for gnral caching with an approximation ratio bttr than 4, or vn a PTAS? Is gnral caching APX-hard? Rlatd work. In th dcision vrsion, oin gnral caching is quivalnt (togthr with intrval schduling/packing, rsourc allocation and othr problms) to th unsplittabl ow on a path problm (UFPP). This quivalnc dos not translat to th approximation vrsion of th problm, as th ow corrsponds not to th cost of caching, but rathr to th complmntary quantity of th total saving of th cost compard to th trivial srvic that faults on ach rqust. An important paramtr in th world of UFPP is task dnsity, in th languag of caching it is th fault cost dividd by th pag siz. An approximation schm with quasi-polynomial tim complxity whn th ratio of th maximum dnsity to th minimum dnsity is quasi-polynomial was givn by Bansal t al. [6]. Th rst non-trivial instanc of UFPP for which a PTAS was invntd by Batra t al. [8] is th on whn th task dnsitis ar

4 4 Luká² Folwarczný, Ji í Sgall in a constant rang. Th bit modl of caching is quivalnt to th cas whn all task dnsitis ar qual to on. Th bst known approximation for UFPP is (2 + ε)-approximation du to Anagnostopoulos t al. [4]. It is also known that UFPP is not APX-hard, unlss NP DTIME(2 polylog(n) ) [6]. W wish to conclud th ovrviw of rsults with rcnt advancs in randomizd onlin caching (comptitiv analysis against oblivious advrsary is considrd). Classical rsults stat that th lowr bound on th comptitiv ratio for uniform caching is H C (th C-th harmonic numbr) [17] and H C -comptitiv algorithms wr dsignd [20,1]; it has also bn known that for gnral caching an H C -comptitiv algorithm is impossibl [14]. Rcntly, Brodal t al. [12] dvlopd a substantially fastr H C -comptitiv algorithm for uniform caching. Basd on nw mthods for rounding of linar programs, Bansal t al. [5] rcntly invntd an O(log 2 C)-comptitiv algorithm for gnral caching and Adamaszk t al. [2] improvd th algorithm to b O(log C)- comptitiv. Outlin. Th main part of our work a polynomial-tim rduction from indpndnt st to caching in th fault modl undr th optional policy with pag sizs rstrictd to {1, 2, 3} is xplaind in Sction 2 and its validity is provn in Sction 3. In Sction 4, w show how to modify th rduction so that it works for th bit modl as wll. In Sction 5, w show how to obtain th hardnss rsults also for th forcd policy. Finally, w giv a slf-containd prsntation of th simpl proof of strong NP-hardnss for gnral costs (in fact, only two dirnt and polynomial costs ar ndd) in Appndix A. An xtndd abstract of this papr appard in Procdings of th 26th Intrnational Symposium on Algorithms and Computation (ISAAC 2015) [18]. 2 Rduction Th dcision problm IndpndntSt is wll-known to b NP-complt. By 3Caching(forcd) and 3Caching(optional) w dnot th dcision vrsions of caching undr ach policy with pag sizs rstrictd to {1, 2, 3}. Problm: IndpndntSt Instanc: A graph G and a numbr K. Qustion: Is thr an indpndnt st of cardinality K in G? Problm: Instanc: Qustion: 3Caching(policy) A univrs of pags, a squnc of pag rqusts, numbrs C and L. For ach pag p it holds siz(p) {1, 2, 3}. Is thr a srvic undr th policy policy of th rqust squnc using th cach of siz C with a total fault cost of at most L? W dn 3Caching(fault,policy) to b th problm 3Caching(policy) with th additional rquirmnt that pag costs adhr to th fault modl. Th problm 3Caching(bit,policy) is dnd analogously.

5 Gnral Caching Is Hard: Evn with Small Pags 5 In this sction, w dscrib a polynomial-tim rduction from IndpndntSt to 3Caching(fault,optional). Informally, a st of pags of siz two and thr is associatd with ach dg and a pag of siz on is associatd with ach vrtx. Each vrtx-pag is rqustd only twic whil thr ar many rqusts on pags associatd with dgs. Th rqust squnc is dsignd in such a way that th numbr of vrtx-pags that ar cachd btwn th two rqusts in th optimal srvic is qual to th th siz of th maximum indpndnt st. W now show th rqust squnc of caching corrsponding to th graph givn in IndpndntSt with a paramtr H. In th nxt sction, w prov that it is possibl to st a propr valu of H and a propr fault cost limit L such that th rduction bcoms a valid polynomial-tim rduction. Rduction 2.1 Lt G = (V, E) b th instanc of IndpndntSt. Th graph G has n vrtics and m dgs and thr is an arbitrary xd ordr of dgs 1,..., m. Lt H b a paramtr boundd by a polynomial function of n. Th corrsponding instanc I G of 3Caching(fault,optional) is an instanc with th cach siz C = 2 mh + 1 and th total of 6 mh + n pags. Th structur of th pags and th rqusts squnc is dscribd blow. Pags. For ach vrtx v, w hav a vrtx-pag p v of siz on. For ach dg, thr ar 6H dg-pags associatd with it that ar dividd into H groups. Th ith group consists of six pags ā i, α i, a i, b i, β i, b i whr pags α i and β i hav siz thr and th rmaining four pags hav siz two. For a xd dg, lt ā -pags b all pags ā i for i = 1,..., H. Lt also ā-pags b all ā -pags for = 1,..., m. Th rmaining collctions of pags (α -pags, α-pags,... ) ar dnd in a similar fashion. Rqust squnc. Th rqust squnc of I G is organizd in phass and blocks. Thr is on phas for ach vrtx v V, w call such a phas th v-phas. Thr ar xactly two rqusts on ach vrtx-pag p v, on just bfor th bginning of th v-phas and on just aftr th nd of th v-phas; ths rqusts do not blong to any phas. Th ordr of phass is arbitrary. In ach v- phas, thr ar 2H adjacnt blocks associatd with vry dg incidnt with v; th blocks for dirnt incidnt dgs ar ordrd arbitrarily. In addition, thr is on initial block I bfor all phass and on nal block F aftr all phass. Altogthr, thr ar d = 4mH + 2 blocks. Lt = {u, v} b an dg, lt us assum that th u-phas prcds th v-phas. Th blocks associatd with in th u-phas ar dnotd by B1,1, B1,2,..., Bi,1, B i,2,..., B H,1, B H,2, in this ordr, and th blocks in th v- phas ar dnotd by B1,3, B1,4,..., Bi,3, B i,4,..., B H,3, B H,4, in this ordr. An xampl is givn in Fig. 1. Evn though ach block is associatd with som xd dg, it contains on or mor rqusts to th associatd pags for vry dg. In ach block, w procss th dgs in th ordr 1,..., m that was xd abov. Pags associatd with th dg ar rqustd in two rounds. In ach round, w procss groups

6 6 Luká² Folwarczný, Ji í Sgall I u-phas B 1 1,1 B1 1,2 B1 2,1 B1 2,2 p u v-phas B 2 1,1 B2 1,2 B2 2,1 B2 2,2 p v w-phas B 1 1,3 B1 1,4 B1 2,3 B1 2,4B 2 1,3 B2 1,4 B2 2,3 B2 2,4 F p w Fig. 1 An xampl of phass, blocks and rqusts on vrtx-pags for a graph with thr vrtics u, v, w and two dgs 1 = {u, w}, 2 = {v, w} whn H = 2 Tabl 1 Rqusts associatd with an dg Block First round Scond round bfor B i,1 ā i B i,1 ā i, α i b i B i,2 α i, a i b i btwn B i,2 and B i,3 a i b i B i,3 a i b i, β i B i,4 a i β i, b i aftr B i,4 b i 1,..., H in this ordr. Whn procssing th ith group of th dg, w rqust on or mor pags of this group, dpnding on th block w ar in. Tabl 1 dtrmins which pags ar rqustd. Not that th numbr of rqusts to pags of siz two is btwn mh and 2mH pr block; in addition, ach block xcpt for th initial and nal ons contains xactly on rqust to a pag of siz thr. Rduction 2.1 is now complt. An xampl of rqusts on dg-pags associatd with on dg is dpictd in Fig. 2. Not that th ordr of th pags associatd with is th sam in all blocks; mor prcisly, in ach block th rqusts on th pags associatd with form a subsqunc of ā 1 α 1 a 1... ā i α i a i... ā H α H a H b 1 β 1 b 1... b i β i b i... b H β H b H. (1) Prliminaris for th proof. Instad of minimizing th srvic cost, w maximiz th total saving compard to th srvic which dos not us th cach at all. This is clarly quivalnt whn considring th dcision vrsions of th problms. Without loss of gnrality, w assum that any pag is brought into th cach only immdiatly bfor som rqust to that pag and rmovd from th cach only aftr som (possibly dirnt) rqust to that pag; furthrmor, th cach is mpty at th bginning and at th nd. That is, a pag may b in th cach only btwn two conscutiv rqusts to this pag, and ithr it is in th cach for th whol intrval or not at all. At ach tim, thr xists at most on pag of siz thr that can possibly b cachd. This follows sinc ach block contains at most on rqust to a pag of siz thr and ach pag of siz thr is rqustd only twic in two

7 Gnral Caching Is Hard: Evn with Small Pags 7 ā 1 α1 a 1 ā 2 α2 a 2 ā 3 α3 a 3 b 1 β1 b 1 b 2 β 2 b 2 b 3 β3 b 3 B1,1 B1,2 B2,1 B2,2 B3,1 B3,2 B1,3 B1,4 B2,3 B2,4 B3,3 B3,4 Fig. 2 Rqusts on all pags associatd with th dg whn H = 3. Each column rprsnts som block(s). Th labld columns rprsnt th blocks in th hading, th rst column rprsnts vry block bfor B1,1, th middl column rprsnts vry block btwn B3,2 and B 1,3, and th last column rprsnts vry block aftr B 3,4. Th rqusts in on column ar ordrd from top to bottom. conscutiv blocks. Sinc th cach siz C = 2m + 1 is odd, this implis that a srvic of dg-pags is valid if and only if at ach tim, at most mh dgpags ar in th cach. It is convnint to think of th cach as of mh slots for dg-pags. As ach vrtx-pag is rqustd twic, th saving on th n vrtx-pags is at most n. Furthrmor, a vrtx-pag can b cachd if and only if during th phas it nvr happns that at th sam tim all slots for dg-pags ar full and a pag of siz thr is cachd. Lt S B dnot th st of all dg-pags cachd at th bginning of th block B and lt SB b th st of pags in S B associatd with th dg. W us s B = S B and s B = S B for th sizs of th sts. Each dg-pag is rqustd only in a contiguous sgmnt of blocks, onc in ach block. It follows that an dg-pag can possibly b in S B only if it is rqustd both in B and in th prvious block. Th total saving on dg-pags is thn qual to B s B whr th sum is ovr all blocks. In particular, th maximal possibl saving on th dg-pags is (d 1)mH, using th fact that S I is mpty. W shall

8 8 Luká² Folwarczný, Ji í Sgall show that th maximum saving is (d 1)mH + K whr K is th siz of th maximum indpndnt st in G. Almost-fault modl. To undrstand th rduction, w considr what happns if w rlax th rquirmnts of th fault modl and st th cost of ach vrtx-pag to 1/(n + 1) instad of 1 as rquird by th fault modl. In this scnario, th total saving on vrtx-pags is n/(n + 1) < 1 which is lss than a saving achivd by any singl dg-pag. Thrfor, dg-pags must b srvd optimally in th optimal srvic of th whol rqust squnc. In this cas, th rduction works alrady for H = 1. This lads to a quit short proof of th strong NP-hardnss for gnral caching and w giv this proof in Appndix A. Hr, w show just th main idas that ar important also for th dsign of our caching instanc in th fault and bit modls. W rst prov that for ach dg and ach block B I w hav s B = 1 (s Appndix A for dtails). Using this w show blow that for ach dg, at last on of th pags α 1 and β 1 is cachd btwn its two rqusts. This implis that th st of all vrtics v such that p v is cachd btwn its two rqusts is indpndnt. For a contradiction, lt us assum that for som dg, nithr of th pags α1 and β1 is cachd btwn its two rqusts. Bcaus pags α1 and β1 ar forbiddn, thr is b 1 in S B 1,2 and a 1 in S B 1,4. Thus somwhr in th two blocks B1,2 and B1,3, w must switch from caching b 1 to caching a 1. Howvr, this is impossibl, bcaus th ordr of rqusts implis that w would hav to cach both b 1 and a 1 at som momnt, or non of thm would b in B1,3 (s Fig. 3). Howvr, thr is no plac in th cach for such an opration, as s B = 1 for vry and B I and in ach block, th rqusts associatd with any ar not intrlavd with thos associatd to. ā 1 α1 a 1 b 1 β1 B 1,1 B 1,2 B 1,3 B 1,4 b 1 Fig. 3 Pags associatd with on dg whn H = 1 In th fault modl, th corrsponding claim s B = H dos not hold. Instad, w prov that th valu of s B cannot chang much during th srvic and whn w us H larg nough, w still gt a working rduction.

9 Gnral Caching Is Hard: Evn with Small Pags 9 3 Proof of Corrctnss In this sction, w show that th rduction dscribd in th prvious sction is indd a rduction from IndpndntSt st to 3Caching(fault,optional). W prov that thr is an indpndnt st of cardinality K in G if and only if thr is a srvic of th caching instanc I G with th total saving at last (d 1)mH + K. First th asy dirction, which holds for any valu of th paramtr H. Lmma 3.1 Lt G b a graph and I G th corrsponding caching instanc from Rduction 2.1. Suppos that thr is an indpndnt st W of cardinality K in G. Thn thr xists a srvic of I G with th total saving at last (d 1)mH + K. Proof For any dg, dnot = {u, v} so that th u-phas prcds th v- phas. If u W, w kp all ā -pags, b -pags, β -pags and b -pags in th cach from th rst to th last rqust on ach pag, but w do not cach a - pags and α -pags at any tim. Othrwis, w cach all ā -pags, α -pags, a -pags and b -pags, but do not cach b -pags and β -pags at any tim. Fig. 4 shows ths two cass for th rst group of pags. In both cass, at ach tim at most on pag associatd with ach group of ach dg is in th cach and th saving on thos pags is (d 1)mH. W know that th pags t in th cach bcaus of th obsrvations mad in Sction 2. For any v W, w cach p v btwn its two rqusts. To chck that this is a valid srvic, obsrv that if v W, thn during th corrsponding phas no pag of siz thr is cachd. Thus, th pag p v always ts in th cach togthr with at most mh pags of siz two. B 1,1 B 1,2 B 1,3 B 1,4 B 1,1 B 1,2 B 1,3 B 1,4 ā 1 α1 a 1 b 1 β1 b 1 ā 1 α1 a 1 b 1 β1 b 1 Fig. 4 Th two ways of caching in Lmma 3.1 W prov th convrs in a squnc of lmmata. In Sction 4, w will show how to rus th proof for th bit modl. To b abl to do that, w list xplicitly all th assumptions about th caching instanc that ar usd in th following proofs.

10 10 Luká² Folwarczný, Ji í Sgall Proprtis 3.2 Lt T G b an instanc of gnral caching corrsponding to a graph G = (V, E) with n vrtics, m dgs 1,..., m, th sam cach siz and th sam univrs of pags as in Rduction 2.1. Th rqust squnc is again split into phass, on phas for ach vrtx. Each phas is again partitiond into blocks, thr is on initial block I bfor all phass and on nal block F aftr all phass. Thr is th total of d blocks. Th instanc T G is rquird to fulll th following list of proprtis: (a) Each vrtx pag p v is rqustd xactly twic, right bfor th v-phas and right aftr th v-phas. (b) Th total saving achivd on dg-pags is qual to s B (summing ovr all blocks). (c) For ach dg, thr ar xactly H pags associatd with rqustd in I, all th ā -pags, and xactly H pags associatd with rqustd in F, all th b -pags. (d) In ach block, pags associatd with 1 ar rqustd rst, thn pags associatd with 2 ar rqustd and so on up to m. () For ach block B and ach dg, all rqusts on a -pags and b -pags in B prcd all rqusts on ā -pags and b -pags in B. (f) Lt = {u, v} b an dg and p an α -pag or β -pag. Lt B b th rst block and B th last block whr p is rqustd. Thn B and B ar ithr both in th u-phas or both in th v-phas. Furthrmor, no othr pag of siz thr is rqustd in B, B, or any block btwn thm. Lmma 3.3 Th instanc from Rduction 2.1 satiss Proprtis 3.2. Proof All proprtis (a), (b), (c), (d), (f) follow dirctly from Rduction 2.1 and th subsqunt obsrvations. To prov (), rcall that th pags associatd with an dg rqustd in a particular block always follow th ordring (1). W nd to vrify that whn th pag a i is rqustd, no pag ā j for j i is rqustd and that whn th pag b i is rqustd, no ā -pag and no pag b j for j i is rqustd. This can b sn asily whn w xplicitly writ down th rqust squncs for ach kind of block, s Tabl 2. For th following claims, lt T G b an instanc fullling Proprtis 3.2. W x a srvic of T G with th total saving at last (d 1)mH. Lt B b th st of all blocks and B th st of all blocks xcpt for th initial and nal on. For a block B, w dnot th block immdiatly following it by. W dn two usful valus charactrizing th srvic for th block B: δ B = mh s B (th numbr of fr slots for dg-pags at th start of th srvic of th block) and γb = s B s B (th chang of th numbr of slots occupid by pags associatd with aftr rqusts from this block ar srvd). Th rst asy lmma says that only a small numbr of blocks can start with som fr slots in th cach.

11 Gnral Caching Is Hard: Evn with Small Pags 11 Tabl 2 Rqust squncs on all pags associatd with an dg Block bfor B1,1 Bi,1 Bi,2 btwn B H,2 and B 1,3 Bi,3 Bi,4 aftr B H,4 First round Scond round ā 1... ā H a 1... a i 1 ā i α i ā i+1ā i+2... ā H b 1... b i a 1... a i 1 α i a i ā i+1ā i+2... ā H b 1... b i a 1... a H b 1... b H a i... a H b 1... b i 1 b i β i b i+1 b i+2... b H a i... a H b 1... b i 1 β b i i b i+1 b i+2... b H b 1... b H Lmma 3.4 W hav (summing ovr all blocks xcpt for th initial on) δ B n. B B\{I} Proof Th total saving on vrtx-pags is at most n du to th proprty (a). Using th proprty (b), s I = 0, and th dnition of δ B, th total saving on dg-pags is qual to s B = (d 1)mH δ B. B B\{I} B B\{I} Th total saving is assumd to b at last (d 1)mH, thus summing th savings form vrtx- and dg-pags w obtain n + (d 1)mH δ B (d 1)mH. B B\{I} Th claim of th lmma now follows. Th scond lmma stats that th numbr of slots occupid by pags associatd with a givn dg dos not chang much during th whol srvic. Lmma 3.5 For ach dg E, B B Proof Lt us us th notation S k B w shall prov for ach k m B B s k γ B 6n. = S1 B S k B s k B and s k B = S k B. First, 3n. (2)

12 12 Luká² Folwarczný, Ji í Sgall Lt P dnot th st of all blocks B from B satisfying s k B s k B 0 and lt N dnot th st of all th rmaining blocks from B. As a consqunc of th proprty (c), w gt s k I [kh δ I, kh] and [kh δ F, kh]. So w obtain th inquality s k F s k F s k I δ F. (3) W claim s k B s k B δ B for ach B B. W assum for a contradiction s k B s k B > δ B for som block B. W us th proprty (d). Thn aftr procssing th dg k in B, th numbr of dg-pags in th cach is (s B s k B ) + s k B > s B + δ B = mh. But mor than mh dg-pags in th cach mans a contradiction. Th summation ovr all blocks from P and Lmma 3.4 giv us th rst bound B P ( s k s k B Using th fact P N = B and (3), w hav B P ( s k s k B ) + ( s k B N ) δ B n. (4) B P s k B ) = s k F s k I δ F ; togthr with (4) and δ F n which follows by Lmma 3.4, w obtain th scond bound ( ) s k B s k B ( ) s k B s k B + δ F 2n. (5) B N B P Combining th bounds (4) and (5), w prov (2) B B s k s k B = B P ( s k s k B ) ( s k B N s k B ) n + 2n = 3n. For th dg 1, th claim of this lmma is wakr than (2) bcaus γ 1 B = s 1 B s1 B. Proving our lmma for k whn k > 1 is just a mattr of using (2) for k 1 and k togthr with th formula x y x + y : B B γ k B B B s k s k B + B B s k 1 s k 1 B 3n+3n = 6n For th rst of th proof, w st H = 6mn + 3n + 1. This nabls us to show that th xd srvic must cach som of th pags of siz thr. Lmma 3.6 For ach dg E, thr is a block B such that som α -pag or β -pag is in S B and δ B = 0.

13 Gnral Caching Is Hard: Evn with Small Pags 13 Proof Fix an dg = k. For ach block B, w dn ε B = numbr of α -pags and β -pags in S B. Obsrv that du to th proprty (f), ε B is always on or zro. W us a potntial function Φ B = numbr of a -pags and b -pags in S B. Bcaus thr ar only ā-pags in th initial block and only b-pags in th nal block (proprty (c)), w know Φ I = 0 and Φ F H δ F. (6) Now w bound th incras of th potntial function as k 1 Φ Φ B δ B + γ l B + ε B. (7) To justify this bound, w x a block B and look at th cach stat aftr rqusts on dgs 1,..., k 1 ar procssd. How many fr slots thr can b in th cach? Thr ar initial δ B fr slots in th bginning of th block B, and th numbr of fr slots can b furthr incrasd whn th numbr of pags in th cach associatd with 1,..., k 1 dcrass. This incras can b naturally boundd by k 1. Thrfor, th numbr of fr slots in th l=1 γ l B l=1 γ l B. cach is at most δ B + k 1 Bcaus of th proprty (), th numbr of cachd a -pags and b -pags can only incras by using th fr cach spac or caching nw pags instad of α -pags and β -pags. W alrady boundd th numbr of fr slots and ε B is a natural bound for th incras gaind on α -pags and β -pags. Thus, th bound (7) is corrct. Summing (7) ovr all B B, w hav l=1 Φ F Φ I = (Φ Φ B ) ( k 1 ) δ B + γ l B + ε B B B B B l=1 which w combin with (6) into H δ F B B ( k 1 δ B + l=1 and us Lmmata 3.4 and 3.5 to bound ε B as B B ε B H δ F B B γ l B + ε B ( k 1 δ B + H n n (k 1)6n l=1 H 6mn 2n = n + 1. ), γ l B )

14 14 Luká² Folwarczný, Ji í Sgall As thr is at most on pag of siz thr rqustd in ach block (proprty (f)), th inquality ε B n + 1 implis that thr ar at last n + 1 blocks whr an α -pag or a β -pag is cachd. At most n blocks hav δ B non-zro (Lmma 3.4); w ar don. W ar rady to complt th proof of th hardr dirction. Lmma 3.7 Suppos that thr xists a srvic of T G with th total saving of at last (d 1)mH + K. Thn th graph G has an indpndnt st W of cardinality K. Proof Lt W b a st of vrtics such that th corrsponding pag p v is cachd btwn its two rqusts. Thr ar at last K vrtics in W bcaus th maximal saving on dg-pags is (d 1)mH. Considr an arbitrary dg = {u, v}. Du to Lmma 3.6, thr xists a block B such that δ B = 0 and som α -pag or β -pag is cachd in th bginning of th block. Thus th cach of siz 2mH + 1 is full, as it contains mh dg-pags including on pag of siz thr. This block B is ithr in th u-phas or in th v-phas, bcaus of th statmnt of th proprty (f). This mans that at last on of th two pags p u and p v is not cachd btwn its two rqusts, bcaus th cach is full. As a consqunc, th st W is indd indpndnt. Th valu of H was st to 6mn + 3n + 1, thrfor Rduction 2.1 is indd polynomial. Lmmata 3.1, 3.3 and 3.7 togthr imply that thr is an indpndnt st of cardinality K in G if and only if thr is a srvic of th instanc I G with th total saving at last (d 1)mH + K. W showd that th problm 3Caching(fault,optional) is indd strongly NP-hard. 4 Bit Modl In this sction, w show how to modify th proof for th fault modl from th prvious sctions so that it works as a proof for th bit modl as wll. Rduction 4.1 Lt G b a graph and I G th corrsponding instanc of th problm 3Caching(fault,optional) from Rduction 2.1. Thn th modid instanc ĨG is an instanc of 3Caching(bit,optional) with th sam cach siz and th sam st of pags with th sam sizs. Th structur of phass and rqusts on vrtx-pags is also prsrvd. Th blocks from I G ar also usd, but btwn ach pair of conscutiv blocks thr ar v nw blocks insrtd. Lt B and b two conscutiv blocks. Btwn B and w insrt v nw blocks B (1),..., B (5) with th following rqusts B (1) : do not rqust anything; B (2) : rqust all pags of siz two that ar rqustd both in B and ; B (3) : rqust a pag (thr is ithr on or non) of siz thr that is rqustd both in B and ;

15 Gnral Caching Is Hard: Evn with Small Pags 15 B (4) : rqust all pags of siz two that ar rqustd both in B and ; B (5) : do not rqust anything. S Fig. 5 for an xampl. In ach nw block, th ordr of chosn rqusts is th sam as in B (which is th sam as in, as both follow th sam ordring of dgs (1)). Th nw instanc has th total of d = d + 5(d 1) = 6d 5 blocks. This tim w prov that th maximal total saving is ( d 1)mH + K whr K is th cardinality of th maximum indpndnt st in G. a β B a β B B (1) B (2) B (3) B (4) B (5) Fig. 5 Th modication of th instanc for a pag a of siz two and a pag β of siz thr Lmma 4.2 Suppos that th graph G has an indpndnt st W of cardinality K. Thn thr xists a srvic of th modid instanc ĨG with th total saving at last ( d 1)mH + K. Proof W considr th srvic of th original instanc I G dscribd in th proof of Lmma 3.1 and modify it so it bcoms a srvic of th modid instanc. In th nw srvic, vrtx-pags ar srvd th sam way as in th original srvic. Th saving on vrtx-pags is thus again K. For ach pair of conscutiv blocks B and, ach pag kpt in th cach btwn B and in th original srvic is kpt in th cach in th nw srvic for th whol tim btwn B and (it spans ovr svn blocks now). For a pag of siz two, a saving two is achivd thr tims. For a pag of siz thr, a saving thr is incurrd twic. On ach pag in th nw srvic w sav six instad of on. Thrfor, th total saving on dg-pags is 6(d 1)mH = ( d 1)mH. Th total saving is ( d 1)mH + K. Lmma 4.3 Suppos that thr xists a srvic of th modid instanc ĨG with th total saving at last ( d 1)mH + K. Thn th graph G has an indpndnt st W of cardinality K. Proof This lmma is th sam as Lmma 3.7. W just nd to vrify that th modid instanc fullls Proprtis 3.2. To prov that th proprty (b) is prsrvd, w obsrv that ach two conscutiv rqusts on a pag of siz two ar sparatd by xactly on block whr th pag is not rqustd. Consquntly, whn thr is a saving two on a rqust on th pag of siz two, w assign on unit of th saving to th block whr th saving was achivd and on unit of th saving to th prvious block. Similarly, ach pair of conscutiv rqusts on a pag of siz thr is sparatd by xactly two blocks whr th pag is not rqustd. Whn thr

16 16 Luká² Folwarczný, Ji í Sgall is a saving thr on a rqust on th pag of siz thr, w assign on unit of th saving to th block whr th saving was achivd and on unit of th saving to ach of th two prvious blocks. As a consqunc, th total saving gaind on dg-pags may indd b computd as s B. Th proprty (a) is prsrvd bcaus th rqusts on vrtx-pags ar th sam in both instancs. Th proprty (c) is prsrvd bcaus th initial and nal blocks ar th sam in both instancs. Each squnc of rqusts in a block of th modid instanc ĨG is ithr th sam or a subsqunc of th squnc in a block in th original instanc. Thrfor, th proprtis (d), () and (f) ar prsrvd as wll. Lmmata 4.2 and 4.3 imply that w hav a valid polynomial-tim rduction and so th problm 3Caching(bit,optional) is strongly NP-hard. 5 Forcd Policy Thorm 5.1 Both th problm 3Caching(fault,forcd) and th problm 3Caching(bit,forcd) ar strongly NP-hard. Proof For both th fault modl and th bit modl, w show a polynomial-tim rduction from caching undr optional policy to th corrsponding variant of caching with undr forcd policy. Lt us hav an instanc of caching undr th optional policy with th cach siz C and th rqust squnc ρ = r 1... r n ; lt M b th maximal siz of a pag in ρ (in our prvious rductions, M = 3). W crat an instanc of caching undr th forcd policy. Th cach siz is C = C + M. Th rqust squnc is ρ = r 1 q 1 r 2 q 2... r n q n whr q 1,..., q n ar rqusts to n dirnt pags that do not appar in ρ and hav siz M. Th costs of th nw pags ar on in th fault modl and M in th bit modl. W claim that thr is a srvic of th optional instanc with saving S if and only if thr is a srvic of th forcd instanc with saving S. W srv th rqusts on original pags th sam way as in th optional instanc. Th cach is largr by M which is th siz of th largst pag. Thus, pags that wr not loadd into th cach bcaus of th optional policy t in thr; w can load thm and immdiatly vict thm. Nw pags t into th cach as wll and w also load thm and immdiatly vict thm. This way w hav th sam saving as in th optional instanc. W construct a srvic for th optional instanc: For ach i, whn srving r i w considr th victions don whn srving r i and q i of th forcd instanc. If a pag rqustd bfor r i is victd, w vict it as wll. If a pag rqustd by r i is victd, w do not cach it at all. Bcaus th pag rqustd by q i has siz M, th original pags occupy at most C slots in th cach aftr q i is srvd. This way w obtain a srvic of th optional instanc with th sam saving. Using th strong NP-hardnss of th problms 3Caching(fault,optional) and 3Caching(bit,optional) provn in Sctions 2 and 3 and th obsrvation that th rduction prsrvs th maximal siz of a pag, w obtain th strong

17 Gnral Caching Is Hard: Evn with Small Pags 17 NP-hardnss of 3Caching(fault,forcd) and 3Caching(bit,forcd). Acknowldgmnts. W ar gratful to th anonymous rviwrs, in particular for bringing th articls [10, 16] to our attntion and for suggstions that hlpd us to improv th prsntation. Rfrncs 1. Achlioptas, D., Chrobak, M., Noga, J.: Comptitiv analysis of randomizd paging algorithms. Thortical Computr Scinc 234(1-2), (2000). DOI / S (98) URL A prliminary vrsion appard at ESA Adamaszk, A., Czumaj, A., Englrt, M., Räck, H.: An O(log k)-comptitiv algorithm for gnralizd caching. In: Procdings of th 23rd Annual ACM-SIAM Symposium on Discrt Algorithms (SODA), pp (2012). DOI / URL 3. Albrs, S., Arora, S., Khanna, S.: Pag rplacmnt for gnral caching problms. In: Procdings of th 10th Annual ACM-SIAM Symposium on Discrt Algorithms, pp (1999). URL 4. Anagnostopoulos, A., Grandoni, F., Lonardi, S., Wis, A.: A mazing 2+ε approximation for unsplittabl ow on a path. In: C. Chkuri (d.) Procdings of th Twnty-Fifth Annual ACM-SIAM Symposium on Discrt Algorithms, SODA 2014, Portland, Orgon, USA, January 5-7, 2014, pp SIAM (2014). DOI / URL 5. Bansal, N., Buchbindr, N., Naor, J.: Randomizd comptitiv algorithms for gnralizd caching. SIAM Journal on Computing 41(2), (2012). DOI / URL A prliminary vrsion appard at STOC Bansal, N., Chakrabarti, A., Epstin, A., Schibr, B.: A quasi-ptas for unsplittabl ow on lin graphs. In: J.M. Klinbrg (d.) Procdings of th 38th Annual ACM Symposium on Thory of Computing, pp ACM (2006). DOI / URL 7. Bar-Noy, A., Bar-Yhuda, R., Frund, A., Naor, J., Schibr, B.: A unid approach to approximating rsourc allocation and schduling. Journal of th ACM 48(5), (2001). DOI / URL A prliminary vrsion appard at STOC Batra, J., Garg, N., Kumar, A., Mömk, T., Wis, A.: Nw approximation schms for unsplittabl ow on a path. In: P. Indyk (d.) Procdings of th 26th Annual ACM- SIAM Symposium on Discrt Algorithms, pp SIAM (2015). DOI / URL 9. Blady, L.A.: A study of rplacmnt algorithms for a virtual-storag computr. IBM Systms Journal 5(2), (1966). DOI /sj URL org/ /sj Bonsma, P.S., Schulz, J., Wis, A.: A constant-factor approximation algorithm for unsplittabl ow on paths. SIAM Journal on Computing 43(2), (2014). DOI / URL Borodin, A., El-Yaniv, R.: Onlin computation and comptitiv analysis. Cambridg Univrsity Prss (1998) 12. Brodal, G.S., Moruz, G., Ngoscu, A.: Onlinmin: A fast strongly comptitiv randomizd paging algorithm. Thory of Computing Systms 56(1), 2240 (2015). DOI /s y. URL A prliminary vrsion appard at WAOA 2011.

18 18 Luká² Folwarczný, Ji í Sgall 13. Chrobak, M., Karlo, H.J., Payn, T.H., Vishwanathan, S.: Nw rsults on srvr problms. SIAM Journal on Discrt Mathmatics 4(2), (1991). DOI / URL A prliminary vrsion appard at SODA Chrobak, M., Larmor, L.L., Lund, C., Ringold, N.: A bttr lowr bound on th comptitiv ratio of th randomizd 2-srvr problm. Information Procssing Lttrs 63(2), 7983 (1997). DOI /S (97) URL /S (97) Chrobak, M., Wogingr, G.J., Makino, K., Xu, H.: Caching is hard Evn in th fault modl. Algorithmica 63(4), (2012). DOI /s URL A prliminary vrsion appard at ESA Darmann, A., Pfrschy, U., Schaur, J.: Rsourc allocation with tim intrvals. Thortical Computr Scinc 411(49), (2010). DOI /j.tcs URL Fiat, A., Karp, R.M., Luby, M., McGoch, L.A., Slator, D.D., Young, N.E.: Comptitiv paging algorithms. Journal of Algorithms 12(4), (1991). DOI / (91)90041-V. URL A prliminary vrsion appard in Folwarczný, L., Sgall, J.: Gnral caching is hard: Evn with small pags. In: K.M. Elbassioni, K. Makino (ds.) Algorithms and Computation - 26th Intrnational Symposium, ISAAC 2015, Nagoya, Japan, Dcmbr 9-11, 2015, Procdings, Lctur Nots in Computr Scinc, vol. 9472, pp Springr (2015). DOI / _11. URL Irani, S.: Pag rplacmnt with multi-siz pags and applications to wb caching. Algorithmica 33(3), (2002). DOI /s URL doi.org/ /s A prliminary vrsion appard at STOC McGoch, L.A., Slator, D.D.: A strongly comptitiv randomizd paging algorithm. Algorithmica 6(6), (1991). DOI /BF URL org/ /bf A prliminary vrsion appard in Young, N.E.: On-lin l caching. Algorithmica 33(3), (2002). DOI /s URL A prliminary vrsion appard at SODA 1998.

19 Gnral Caching Is Hard: Evn with Small Pags 19 A Th Simpl Proof In this appndix, w prsnt a simpl variant of th proof for th almost-fault modl with two distinct costs. This complts th sktch of th proof prsntd at th nd of Sction 2. W prsnt it with a complt dscription of th simplid rduction, so that it can b rad indpndntly of th rst of th papr. This appndix can thrfor srv as a short proof of th hardnss of gnral caching. Thorm A.1 Gnral caching is strongly NP-hard, vn in th cas whn pag sizs ar limitd to {1, 2, 3} and thr ar only two distinct fault costs. W prov th thorm for th optional policy. It is asy to obtain th thorm also for th forcd policy th sam way as in th proof of Thorm 5.1. Th Rduction Th rduction dscribd hr will b quivalnt to Rduction 2.1 with H = 1 and th fault cost of ach vrtx-pag st to 1/(n + 1). Suppos w hav a graph G = (V, E) with n nods and m dgs. W construct an instanc of gnral caching whos optimal solution ncods a maximum indpndnt st in G. Fix an arbitrary numbring of dgs 1,..., m. Th cach siz is C = 2m + 1. For ach vrtx v, w hav a vrtx-pag p v with siz on and cost 1/(n + 1). For ach dg, w hav six associatd dg-pags a, ā, α, b, b, β ; all hav cost on, pags α, β hav siz thr and th rmaining pags hav siz two. Th rqust squnc is organizd in phass and blocks. Thr is on phas for ach vrtx. In ach phas, thr ar two adjacnt blocks associatd with vry dg incidnt with v; th incidnt dgs ar procssd in an arbitrary ordr. In addition, thr is on initial block I bfor all phass and on nal block F aftr all phass. Altogthr, thr ar d = 4m + 2 blocks. Thr ar four blocks associatd with ach dg ; dnot thm B1, B 2, B3, B 4, in th ordr as thy appar in th rqust squnc. For ach v V, th associatd pag p v is rqustd xactly twic, right bfor th bginning of th v-phas and right aftr th nd of th v-phas; ths rqusts do not blong to any phas. An xampl of th structur of phass and blocks is givn in Fig. 6. I u-phas B 1 1 B 1 2 p u v-phas B 2 1 B 2 2 p v w-phas B 1 3 B 1 4 B 2 3 B 2 4 Fig. 6 An xampl of phass, blocks and rqusts on vrtx-pags for a graph with thr vrtics u, v, w and two dgs 1 = {u, w}, 2 = {v, w} whn H = 2 p w F Evn though ach block is associatd with som xd dg, it contains on or mor rqusts to th associatd pags for vry dg. In ach block, w procss th dgs 1,..., m in this ordr. For ach dg, w mak on or mor rqusts to th associatd pags according to Tabl 3. Figur 7 shows an xampl of th rqusts on dg-pags associatd with on particular dg. Proof of Corrctnss Instad of minimizing th srvic cost, w maximiz th saving compard to th srvic which dos not us th cach at all. This is clarly quivalnt whn considring th dcision vrsion of th problm.

20 20 Luká² Folwarczný, Ji í Sgall Tabl 3 Rqusts associatd with an dg Block bfor B1 B1 B2 btwn B 2 and B 3 B3 B4 aftr B 4 Rqustd pags ā ā, α, b α, a, b a, b a, b, β a, β, b b B1 B2 B3 B4 α a b β b Fig. 7 Rqusts on all pags associatd with th dg. Each column rprsnts som block(s). Th four labld columns rprsnt th blocks in th hading, th rst column rprsnts vry block bfor B1, th middl column rprsnts vry block btwn B 3 and B4, and th last column rprsnts vry block aftr B 4. Th rqusts in on column ar ordrd from top to bottom. Without loss of gnrality, w assum that any pag is brought into th cach only immdiatly bfor a rqust to that pag and rmovd from th cach only immdiatly aftr a rqust to that pag; furthrmor, at th bginning and at th nd th cach is mpty. I.., a pag may b in th cach only btwn two conscutiv rqusts to this pag, and ithr it is in th cach for th whol intrval or not at all. Each pag of siz thr is rqustd only twic in two conscutiv blocks, and ths blocks ar distinct for all pags of siz thr. Thus, a srvic of dg-pags is valid if and only if at ach tim, at most m dg-pags ar in th cach. It is thus convnint to think of th cach as of m slots for dg-pags. Each vrtx-pag is rqustd twic. Thus, th saving on th n vrtx-pags is at most n/(n + 1) < 1. Sinc all dg-pags hav cost on, th optimal srvic must srv thm optimally. Furthrmor, a vrtx-pag can b cachd if and only if during th phas it nvr happns that at th sam tim all slots for dg-pags ar full and a pag of siz thr is cachd. Lt S B dnot th st of all dg-pags cachd at th bginning of th block B and lt s B = S B. Now obsrv that ach dg-pag is rqustd only in a contiguous sgmnt of blocks, onc in ach block. It follows that th total saving on dg-pags is qual to B s B whr th sum is ovr all blocks. In particular, th maximal possibl saving on th dg-pags is (d 1)m, using th fact that S I is mpty. W prov that thr is a srvic with th total saving at last (d 1)m + K/(n + 1) if and only if thr is an indpndnt st of siz K in G. First th asy dirction.

21 Gnral Caching Is Hard: Evn with Small Pags 21 B1 B2 B3 B4 α a b β b B1 B2 B3 B4 α a b β b Fig. 8 Th two ways of caching in Lmma A.2 Lmma A.2 Suppos that G has an indpndnt st W of siz K. Thn thr xists a srvic with th total saving (d 1)m + K/(n + 1). Proof For any, dnot = {u, v} so that u prcds v in th ordring of phass. If u W, w kp ā, b, b and β in th cach from th rst to th last rqust on ach pag, and w do not cach a and α at any tim. Othrwis w cach b, a, ā and α, and do not cach b and β at any tim. In both cass, at ach tim at most on pag associatd with is in th cach and th saving on thos pags is (d 1)m. S Fig. 8 for an illustration. For any v W, w cach p v btwn its two rqusts. To chck that this is a valid srvic, obsrv that if v W, thn during th corrsponding phas no pag of siz thr is cachd. Thus th pag p v always ts in th cach togthr with at most m pags of siz two. Now w prov th convrs in a squnc of claims. Fix a valid srvic with saving at last (d 1)m. For a block B, lt dnot th following block. Claim A.3 For any block B, with th xcption of B = I, w hav s B = m. Proof For ach B I w hav s B m. Bcaus s I = 0, th total saving on dg-pags is B s B (d 1)m. W nd an quality. W now prov that ach dg occupis xactly on slot during th srvic. Claim A.4 For any block B I and for any, S B contains xactly on pag associatd with. Proof Lt us us th notation S k B = S 1 B S k B and S s k B = k B. First, w shall prov for ach k m = k. (8) s k B This is tru for B = F, as only th m dg-pags b can b cachd thr, and by th prvious claim all of thm ar indd cachd. Similarly for B = I (i.., immdiatly following th initial block). If (8) is not tru, thn for som k and B {I, F } w hav s k B < s k. Thn aftr procssing th dg k in th block B w hav in th cach all th pags in (S B \S k B ) S k. Thir numbr is (m s k B ) + s k > m, a contradiction. Th statmnt of th claim is an immdiat consqunc of (8). Claim A.5 For any dg, at last on of th pags α and β is cachd btwn its two rqusts.

22 22 Luká² Folwarczný, Ji í Sgall Proof Assum that non of th two pags is cachd. It follows from th prvious claim that b S B 2, as at this point α and b ar th only pags associatd with that can b cachd. Similarly, a S B 4. It follows that thr xists a block B btwn B1 and B 4 such that S B contains th pag b and S contains th pag a. Howvr, in B, th pag a is rqustd bfor th pag b. Thus at th point btwn th two rqusts, th cach contains two pags associatd with, plus on pag associatd with vry othr dg, th total of m+1 pags, a contradiction. Now w ar rady to complt this dirction. Lmma A.6 Suppos that thr xists a valid srvic with th total saving (d 1)m + K/(n + 1). Thn G has an indpndnt st W of siz K. Proof Lt W b th st of all v such that p v is cachd btwn its two rqusts. Th total saving implis that W = K. Now w claim that W is indpndnt. Suppos not, lt = uv b an dg with u, v W. Thn p u and p v ar cachd in th corrsponding phass. Thus nithr α nor β can b cachd, sinc togthr with othr m 1 rqusts of siz 2 associatd with th rmaining dgs, th cach siz ndd would b 2m + 2. Howvr, this contradicts th last claim. Lmmata A.2 and A.6 togthr show that w constructd a valid polynomial-tim rduction from th problm of indpndnt st to gnral caching. Thrfor, Thorm A.1 is provn.