On a Local Protocol for Concurrent File Transfers

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1 On a Local Protocol for Concurrent Fle Transfers MohammadTagh Hajaghay Dep. of Computer Scence Unversty of Maryland College Park, MD hajagha@cs.umd.edu Roht Khandekar IBM T.J. Watson Research Center 19 Skylne Drve Hawthorne, NY rohtk@us.bm.com Vahd Laghat Dep. of Computer Scence Unversty of Maryland College Park, MD vlaghat@cs.umd.edu Guy Kortsarz Dep. of Computer Scence Rutgers Unversty-Camden Camden, NJ guyk@camden.rutgers.edu ABSTRACT We study a very natural local protocol for a fle transfer problem. Consder a scenaro where several fles, whch may have vared szes and get created over a perod of tme, are to be transferred between pars of hosts n a dstrbuted envronment. Our protocol assumes that whle executng the fle transfers, an ndvdual host does not use any global knowledge; and smply subdvdes ts I/O resources equally among all the actve fle transfers at that host at any pont n tme. Ths protocol s motvated by ts smplcty of use and ts applcatons to schedulng map-reduce workloads. Here we study the problem of decdng the start tmes of ndvdual fle transfers to optmze QoS metrcs lke average completon tme or MakeSpan. To begn wth, we show that these problems are NP-hard. We next argue that the ablty of schedulng multple concurrent fle transfers at a host makes our protocol stronger than prevously studed protocols that schedule a sequence of matchngs, n whch no two actve fle transfers share a host at any tme. We then generalze the approach of Queyranne and Svrdenko (J. Schedulng, 00) and Gandh et al. (ACM T. Algorthms, 008) that relates the MakeSpan and completon tme objectves and present constant factor approxmaton algorthms. Categores and Subject Descrptors F.. [Theory of Computaton]: Analyss of Algorthms and Problem ComplextyNon-numercal Algorthms and Problems; G.. Supported n part by NSF CAREER award and Google Faculty Research Award. Part of ths work was done whle the authors were meetng at DI- MACS. We would lke to thank DIMACS for hosptalty. Supported n part by NSF grant number Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. SPAA 11, June 4 6, 011, San Jose, Calforna, USA. Copyrght 011 ACM /11/06...$ [Mathematcs of Computng]: Dscrete Mathematcs graph theory, network problems General Terms Theory Keywords local protocol, schedulng, fle transfer, average completon tme, MakeSpan 1. INTRODUCTION 1.1 Motvaton Today s technologes have enabled resources for complng enormous amounts of data, often beyond the capacty of ndvdual dsks, and too large for processng wth a sngle CPU. Such data s then naturally stored across a cluster of compute hosts and s shared and processed usng dstrbuted computng envronments. One dstrbuted computng paradgm whch has attracted a lot of nterest recently s Map-Reduce [5]. A typcal Map-Reduce job has three phases. The Map phase processes data, often avalable on local dsks, block-by-block and for each such block produces (key, value)-pars that are wrtten to the local dsk. In a typcal mplementaton, these pars are organzed nto multple fles where each fle corresponds to a specfc key. The Shuffle phase transfers these fles contanng the (key, value)-pars from the hosts that run Map tasks to the hosts that run Reduce tasks. Fnally, the Reduce phase apples some functon on all the values correspondng to each ndvdual key and computes an output. The Shuffle phase nvolves the transfer of several fles across multple machnes and can be qute I/O ntensve. Ths phase can be naturally modeled as a fle transfer problem. Let us denote the set of hosts partcpatng n a Map- Reduce computaton by V. The Map tasks may be present on two or more of these hosts. As a Map task fnshes, the data produced by t becomes avalable for the transfer. Note that the amount of data produced by dfferent Map tasks to be consumed by dfferent Reduce tasks can be qute dfferent. Let edge e {u, v} represent the data or a fle to be transferred from a host u runnng a Map task to a host v runnng a Reduce task. Snce we do not dstngush between the overheads caused by ncomng or outgong transfers, we model the transfers by undrected edges. Let r(e) denote the tme

2 at whch fle e becomes avalable for transfer, namely ts release tme. Let l(e) denote the sze of fle e. Although the exact values of r(e) and l(e) are not known a-pror, we assume that they can be estmated by some proflng mechansm. Once a fle transfer has started, t cannot be preempted and t contnues tll completon. Ths assumpton s mportant to elmnate the necessty to keep state and ensure fault tolerance. We next try to model the rates at whch multple actve fle transfers that share a host can proceed concurrently. The actve fle transfers orgnatng or termnatng at a common host share the dsk and other I/O resources of that host. Thus f the number actve fle transfers at a host goes up, the rate of an ndvdual fle transfer goes down. To keep the model smple whle capturng the essence of ths resource sharng, we assume that f there are n actve fle transfers at a host, each fle transfer can take place at a rate no more than 1/n tmes the rate on a dedcated host. Suppose at some pont whle fle e {u, v} s beng transferred, there are a total of n u actve fle transfers at host u and a total of n v actve fle transfers at host v. We assume that the effectve rate at whch e gets transferred s gven by the mnmum of the rates t can get at the two end hosts: mn{1/n u, 1/n v}. Of course, ths rate may change over a perod of tme, snce n u or n v may change wth tme. The scheduler for the Shuffle phase, thus has to decde the start tmes for the ndvdual fles. Once the start tmes are fxed, the fles get transferred at rates gven by the above model. There are several useful objectve functons a scheduler may try to optmze. One such objectve functon s the average completon tme. Ths MnSum objectve gves an estmate on how soon the transfers fnsh so that the Reduce tasks can start. Another objectve functon s the MakeSpan. Ths MnMax objectve captures the fnsh tme of the last fle transfer whch, n turn, s a useful lower bound on the completon tme of the overall Map-Reduce job. 1. Problem Formulaton A fle transfer model s a trple (G, l, r) where G (V, E) s an undrected mult-graph. The vertces V represent the hosts (or compute nodes) and the edges E denote the fles to be transferred between the hosts. Assume that hosts are homogeneous and have dentcal processng capablty of 1. For a vertex v V, let E v denote the edges adjacent to v. The functons l : E N and r : E N denote the length and the release tme of a fle. The unform transfer model s a transfer model where the length of all fles are 1,.e., e E, l(e) 1. The zero-release transfer model s a transfer model where all the fles are avalable from the start,.e., e E, r(e) 0. A schedule S defnes a functon s S : E R + {0}, where s S(e) s the startng tme for the edge e. Once a fle e s started to be transferred, t cannot be preempted and contnues untl t completes at tme f S(e). The rate at whch the fle s transferred however can vary over tme and depends on how loaded the hosts u and v are at a partcular tme durng ts transfer. More precsely, for a tme t, let n u(t) and n v(t) denote the total number of fle transfers actve at tme t nvolvng hosts u and v respectvely. The effectve processng e gets at tme t s the mnmum of the two processng capabltes at the end-ponts: mn{1/n u(t), 1/n v(t)}. Ths denotes the unts of fle transferred per unt tme at tme t. The transfer of fle e contnues tll t gets a total processng of l(e). There are two dfferent cost functons whch we lke to mnmze. In the MakeSpan verson of the problem, the cost of a schedule S s the fnshng tme of the last transfer (job),.e., MAX(S) max e E{f S(e)}. In the MnSum verson, the cost of a schedule S s the sum of fnshng tmes for all the fle transfers,.e., SUM(S) e E fs(e). Note that the MnSum crteron corresponds to mnmzng the average fnshng tme of all edges. For any nstance of the problem, consder opt ms as a schedule wth the mnmum MakeSpan cost of OPT ms MAX(opt ms). Smlarly, consder opt sum as a schedule wth the MnSum cost of OPT sum SUM(opt sum). We may omt the ms and sum ndexes f they are clear from the context. 1.3 Our Model vs. the Non-Concurrent Model The dstngushng feature of our model s that t allows multple actve fle transfers at a host. If at each round only a matchng can be scheduled, the model s called a non-concurrent model. Here we gve smple examples n whch our concurrent fle transfer model gves better values of MakeSpan or average completon tme by a constant factor than the non-concurrent model. Let G K 3 be a trangle and let r(e) 0 and l(e) 1 for all e G. If we start transferrng all three fles at tme 0, each fle receves a rate of 1/ and completes at tme, gvng MakeSpan of. If however we nsst that no two actve fles can share a host, the best way to schedule these edges s one-by-one, gvng a MakeSpan of 3. Moreover, the MakeSpan problem wth unform fle szes has a smple schedulng n the concurrent fle transfer (see Secton 4), n the Non-concurrent model t s equvalent to properly colorng the edges of a graph, whch s an NP-Complete problem (see for example [8]). We can also gve a smple example for the MnSum crtera. Let G be a path of length wth edges e and e. Let l(e) M, l(e ) 1, r(e) 0 and r(e ) M/. If we start transferrng fles rght on ther release tmes, the sum of completon tmes would be f(e) + f(e ) (M + 1) + (M/ + ) 1.5M + 3. But f we are not able to transfer the fles ncdent to a host concurrently we must transfer them n some order. If we start transferrng e at tme zero, then the total cost would be M + (M + 1) M + 1. If we want to transfer e before e, then the total cost would be (M/ + 1) + (M/ M) M + 3. Therefore the cost of the optmum soluton s almost M n the Non-concurrent model but the cost of the optmum n concurrent model s almost 1.5M. In ths paper, we desgn our algorthms by mostly relyng on nonconcurrent schedulng algorthms, however, the approxmaton rato s defned by comparng the cost of an algorthm wth that of an optmal concurrent soluton. Note that an algorthm wth an approxmaton rato α n our model has at most the same approxmaton rato compared aganst an optmal non-concurrent soluton. However, the reverse s not true: a soluton whch approxmates the optmal non-concurrent soluton wthn a factor β, s not necessary a β-approxmaton soluton n our model. 1.4 Our Contrbutons THEOREM 1. The problem of computng a schedule wth mnmum MakeSpan s NP-complete. The problem of computng that of mnmum average completon tme s NP-complete even on trees, and even wth fle szes 1 or. In Sectons 3. and 3.1, we then present constant factor approxmatons for our problems. THEOREM. There exst polynomal-tme algorthms that acheve the followng approxmaton factors for varous versons of the problems we study 1. Here e.718 stands for the base of the natural logarthms. 1 The approxmaton rato for the Avg. Resp. Tme on General Non-Unform verson s clamed to be 6e n the conference verson. However, due to a subtle flaw t s corrected to 9e.

3 Fle szes Release Tmes MakeSpan Avg. Resp. Tme Non-Unform General e 3.3 Zero-Release 3. 4e 3.3 Unform General e 3.3 Zero-Release Bpartte: 4 Our technques. We generalze the technque gven by Queyranne and Svrdenko [18] and Kortsarz et al. [3] for our framework n secton 3.1. They use a method to reduce the MnSum crtera to MakeSpan crtera. Ther basc dea s to frst partton the vertces nto dsjont subsets accordng to a certan node-weght functon. Next, they reduce the MnSum problem on every subset to a MakeSpan problem nduced by that subset. However, ther analyss strongly reles on specfc propertes of both the parttonng functon and the partcular MakeSpan algorthm used n ther soluton. We generalze ths technque to partton the edges (nstead of vertces) usng an arbtrary parttonng functon and then usng a schedulng algorthm wth very smple restrctons. Ths Metaalgorthm provdes a general tool for schedulng conflctng jobs under the MnSum crtera. Usng ths approach, we present constant approxmaton algorthms for the general fle transfer problem wth the MnSum crtera. 1.5 Related Work A closely related problem to the fle transfer problem s the Data Mgraton problem. The data mgraton problem arses n large storage systems, such as Storage allocaton or Schedulng on dedcated processors [3], where a network of hosts s used to store multmeda data. As the data access pattern changes over tme, the load across the hosts needs to be re-balanced. Ths s done by computng a new data layout and then mgratng data to convert the ntal data layout to the target data layout. The mgraton s done by transferrng fles between the hosts. A host completes when all the fles concernng that host s transferred. Clearly t s mportant to compute a data mgraton schedule that converts the ntal layout to the target layout quckly. Ths problem can be modeled as a transfer graph, n whch the vertces represent the hosts and an edge between two vertces u and v corresponds to a data object that must be transferred from u to v, or vce-versa. Each edge has a length that represents the transfer tme of a data object between the hosts correspondng to the endponts of the edge. In data mgraton problem we assume that any host can be nvolved n at most one transfer at any tme. In contrast to the fle transfer problem, n the data mgraton problem the goal s to mnmze the completon tme of the hosts (vertces) nstead of that of fles (edges). Several varatons of the data mgraton problem have been studed, arsng ether due to dfferent objectve functons or due to addtonal constrants. There are usually three dfferent objectve functons to optmze. One common objectve functon s to mnmze the MakeSpan of the mgraton schedule,.e., the tme by whch all mgratons complete. Coffman et al. [9] ntroduced ths problem. They showed that when edges may have arbtrary lengths, a class of greedy algorthms yelds a -approxmaton to the mnmum MakeSpan. In the specal case where the lengths are unform,.e., edges have equal (unt) lengths, the problem reduces to edge colorng of the transfer (mult)graph of the system for whch an asymptotc approxmaton scheme s now known [13]. Another objectve functon s to mnmze the average completon tme over all hosts (vertces). In ths verson we usually consder the weghted sum of fnshng tmes of vertces,.e., each vertex v s assocated wth a weght w(v) and the completon tme of each vertex C(v) s the last fnshng tme of edges adjacent to v and we want to mnmze v w(v)c(v). Km [6] proved that the problem s NP-hard and showed that Graham s lst schedulng algorthm [14], when guded by an optmal soluton to a lnear programmng relaxaton, gves an approxmaton rato of 3. She also gave a 9-approxmaton algorthm for the case where edges have arbtrary lengths. Gandh et al. [11] showed that the analyss of the 3-approxmaton algorthm s tght. They also gave a approxmaton algorthm for a more general case when edges have release tmes and arbtrary lengths. Bar-Noy et al. [10] studed the data mgraton problem wth the objectve to mnmze the average completon tme over all data mgratons (edges). They showed that the problem s NPhard and gave a smple -approxmaton algorthm for the unform case (whch s also known as Mn Sum Edge Colorng Problem). Halldórsson et al. [] mproved ths rato to For arbtrary edge lengths, the best known rato s 7.68 by [3]. A problem related to the data mgraton problem s open shop schedulng. In ths problem, we have a set of machnes and a set of jobs wth postve weghts. Each job conssts of a set of operatons whch can be performed n any order. Each operaton has a processng tme and must be processed on a specfc machne. Each machne can process a sngle operaton at any tme, and two operatons that belong to the same job cannot be processed smultaneously. The objectve s to mnmze the sum of weghted completon tmes of all jobs. Ths problem s a specal case of the data mgraton problem [11]. Open shop schedulng problem has been studed n [11, 15, 16, 17, 18]. For dfferent models of data mgraton, see [1, 0, 19].. NP-COMPLETENESS RESULTS NP-Completeness for mn. avg. completon tme. THEOREM 3. The problem of computng a schedule wth mnmum average completon tme s NP-hard even n trees and even f the jobs have length 1 or. The reducton uses the deas of Marx []. The problem Marx consdered s preemptve sum multcolorng of the edges of the a tree (MEPS). In ths problem we are gven a tree and every edge has an ntegral length l(e). We have to color the edges of the tree wth postve ntegers 1,, 3,. If e and e share a vertex, then ther color sets must be dsjont. Thus a soluton must choose a matchng at every round (and the edges n the matchng get the color of the round number). Every edge e must belong to l(e) matchngs. Let Ψ be a proper colorng and f Ψ(e) be the largest color assgned to e by Ψ. The goal s to mnmze e E fψ(e). In the nonpreemptve case, every e must receve l(e) consecutve ntegers. Consder MEPS. The soluton of [] for a YES nstance happens to be non-preemptve as we shall see. Note that ths mples a hardness for the non preemptve case as well. For us ths fact s mportant as t fts our model whch s non-preemptve. We now state some observatons used by [] (albet, not made explctly n []). Defnton Gven an undrected graph G(V, E) a vertex cover (of the edges) s a subset A V so that for every edge e uv E ether u A or v A. An exact vertex cover A, s a vertex cover A, so that for every e uv exactly one of u or v belongs to A. Consder an exact vertex cover A of the edges of the graph. Say that v, u A. Let E v, E u be the edges of v and u n G. CLAIM 1. E v E u.

4 Proof. If E v E u then E v E u uv as uv s the only edge that can appear both n E v and n E u. But our assumpton that u A and v A mples that A s not an exact cover (uv s covered twce). Ths s a contradcton. Hence the collecton of sets of edges {E v v A} s a collecton of edge-dsjont stars, wth v the center of E v. Note that every edge appears n exactly one of these stars. Defnton A perfect colorng of a star E v of A s a colorng that takes the star, deletes the rest of the edges from the graph, and assgns ths star ts optmum colorng (dsregardng conflcts that may occur wth other stars). A perfect colorng, s a perfect colorng of all stars {E v, v A}. It s not clear a-pror that a perfect colorng exsts. However, we can prove the followng. CLAIM. If we can fnd an exact cover A and t s possble to fnd perfect and proper colorng of all stars, then the colorng s optmal. Proof. Snce the stars are edge-dsjont we consder every star separately. Gven a star collecton of A, the perfect sum colorng correspondng to each star lower bounds the contrbuton of ths star to the sum. Ths s because ths colorng s locally optmal (dsregardng all other stars). Thus the sum of the contrbuton of a perfect colorng over all stars of A s a lower bound on the optmum sum. Snce we assume that a perfect colorng can be obtaned n a consstent way, the colorng s optmal. Consder a general star wth center v and let E v e 1, e,..., e k. Assume wthout loss of generalty that l(e 1) l(e )... l(e k ). CLAIM 3. In the MnSum problem, a perfect colorng of the star frst schedules e 1 for l(e 1) tme unts, and then schedules e for l(e ) tme unts, and so on. Proof. Let < j. The edges e and e j share a vertex. Ths means that one of the edges wll add to the delay of the other or vce versa. Snce l(e ) l(e j), the contrbuton to the delay of ths par s at least mn{l(e ), l(e j)} l(e ). For ths e has to be transferred fully before the transfer of e j starts. The opton of schedulng edges together, whch holds n our fle transfer model, does not produce a perfect colorng. It works n half speed. More precsely, f we schedule e, e j for ɛ > 0 tme unts together, then e and e j both get ɛ unts of delay and ɛ/ unts of ther jobs were fnshed by now. Now, even f the ɛ/ tme unts of the mutual schedule were the last tme unts e needed, the rest of the job e delayed e j by l(e ) ɛ/. Thus the total delay correspondng to these two edges s ɛ + (l(e ) ɛ/) l(e ) + ɛ/ whch s a non-perfect colorng. It s easy to see that after e and e j were scheduled for ɛ > 0 tme unts together, no colorng completon can derve a perfect colorng. Now, f we schedule e 1 fully frst and then e and so own, for every < j the delay caused by the par e to e j s the mnmum possble l(e ). Hence all delays for all edge pars s the mnmum possble and the colorng s perfect. The followng theorem s proved n []. We start wth a 3 SAT nstance so that every lteral appears twce non-negated and twce negated. Ths problem s stll NP-hard (see [4]). We denote ths problem by 3 SAT 4. In [] a reducton from 3 SAT 4 to MEPS s gven. The nstance of MEPS s called a YES nstance f t corresponds to a satsfable 3 SAT 4 formula. Else t s called a NO nstance. A colorng s proper f no two edges of the same color share an endpont. THEOREM 4. [] In the YES nstance of MEPS, there exsts an exact vertex cover A and a perfect proper colorng of ts stars. In addton, the colorng s non-preemptve. A NO nstance, does not admt a perfect proper colorng of A and so the sum s larger than the one gven by a perfect sum. In [] t s shown that there exst an optmum soluton for a MEPS nstance such that the maxmum color s at most p ( 1) wth p the maxmum demand and the maxmum degree n the graph. If p s exponental n n then a soluton for MEPS s exponental (n the NO nstance, as the preemptve colorng s concerned, there may be no short descrpton of the colorng). Therefore we assume that p s bounded by a polynomal n n. COROLLARY 1. If there exsts a polynomal tme algorthm for the MnSum problem, then P NP. Proof. By Clam 3, one can compute the cost of a perfect colorng n polynomal tme. Thus t s suffcent to show an nstance s a YES nstance f and only f OPT ms s equal to the cost of a perfect colorng. Snce the soluton of [] for a YES nstance s non-preemptve, the solutons of MEPS and the MnSum problem are dentcal for a YES nstance. Hence by Theorem 4, n a YES nstance the optmum soluton corresponds to a perfect colorng. On the other hand, n a NO nstance the optmum (preemptve) soluton for MEPS and the optmum soluton for the MnSum problem have nothng n common. However, by Clam the value of the NO nstance for both models s larger than the value of a perfect colorng. Therefore gven a polynomal algorthm for the MnSum problem, we can dstngush between a YES nstance and a NO nstance of the 3 SAT 4 problem. NP-Completeness for MakeSpan. We reduce the strongly NP-hard problem called 3-partton to the problem of computng a schedule wth mnmum MakeSpan. An nstance of the 3-partton problem s gven by an nteger B > 0 and 3n ntegers s 1,..., s 3n such that B/4 < s < B/ for all [3n] and 3n 1 s nb. Here [k] stands for {1,..., k}. An nstance s called a YES nstance f the 3n ntegers can be parttoned nto subsets G 1,..., G n such that each G j has 3 elements addng up to exactly B. An nstance that s not a YES nstance s called a NO nstance. It s well known that t s strongly NP-hard to dstngush between YES and NO nstances [4]. We now gve a polynomal-tme procedure that, gven an nstance of the 3-partton problem, creates an nstance of the MakeSpan mnmzaton problem. The graph G of the MakeSpan mnmzaton nstance created s gven n Fgure 1. The release tmes and lengths of the varous edges are also gven n the adjacent table. LEMMA 1. An nstance obtaned from a YES nstance of the 3- partton problem has optmum MakeSpan (B + 1)n 1, whle an nstance obtaned from a NO nstance of the 3-partton problem has optmum MakeSpan strctly more than (B + 1)n 1. Proof. Consder an nstance obtaned from a YES nstance of the 3-partton problem. We create a non-concurrent schedule (.e., a schedule whch does not schedule multple edges ncdent to a host at any pont n tme) wth MakeSpan (B + 1)n 1. Frst schedule all the edges of the form {v, b j}, {b j, a j}, {b j, c j} for j [n 1] startng at ther respectve release tmes. Note that no two adjacent edges among them wll be actve at any pont n tme. Now the only tme wndows that are open for schedulng edges of the form

5 (a) Edge e Release tme r(e) Length l(e) {v, u } for [3n] 0 s {v, b j } for j [n 1] (B + 1)j 1 1 {b j, a j } for j [n 1] 0 (B + 1)j 1 {b j, c j } for j [n 1] (B + 1)j (B + 1)(n j) 1 (b) Fgure 1: MakeSpan mnmzaton nstance n the reducton {v, u } for [3n] are W j [(B+1)(j 1), (B+1)(j 1)+B) for j [n]. Note that each of these wndows s of length exactly B. Let G 1,..., G n be the partton of the 3n ntegers nto subsets of sze 3 each addng up to B. Fx j [n] and let G j {s j1, s j, s j3 } so that s j1 + s j + s j3 B. Now schedule the edges {v, u j1 }, {v, u j }, {v, u j3 } n wndow W j one after the other. Ths n fact gves a non-concurrent schedule wth MakeSpan (B + 1)n 1. Now t s enough to show that f there s a schedule (ether concurrent or non-concurrent) wth MakeSpan at most (B + 1)n 1, the nstance must have been created from a YES nstance of the 3-partton problem. Note that the edges ncdent to each b j have total length exactly (B + 1)n 1. Consderng ther release tmes, t s clear that for the MakeSpan to be at most (B + 1)n 1, these edges must be schedules at ther respectve release tmes n a nonconcurrent manner. Ths agan gves that the only tme wndows that are open for schedulng edges of the form {v, u } for [3n] are W j for j [n] defned above. Furthermore no edge {v, u } can be actve n more than one wndow. Thus each edge {v, u } maps to a unque wndow W j. The edges mappng to any sngle wndow must have total length exactly B. Ths naturally nduces a feasble soluton to the 3-partton problem and hence gves that the startng nstance of the 3-partton problem must have been a YES nstance. 3. NON-UNIFORM TRANSFER MODEL In ths secton we present dfferent algorthms for fndng schedules under the non-unform transfer model. In Secton 3.1, we generalze the approach n [4] and [3] to gve a meta-algorthm for solvng the MnSum problem by parttonng the jobs (fles) nto dfferent blocks and then usng an algorthm wth good MakeSpan tme for each block. In Secton 3., we gve constant compettve algorthms for the MakeSpan problem. Fnally n Secton 3.3, by provdng dfferent bucketng functons and usng algorthms n Secton 3., we gve constant compettve algorthms for the MnSum problem. 3.1 Overvew of the Approach for Solvng the MnSum Problem We use a meta-algorthm whch provdes a very general tool for schedulng conflctng jobs under the mnmum sum crtera. The meta-algorthm uses a bucketng functon f to dvde the jobs nto dsjont blocks such that each block has a "unformty property" (e.g., n a near optmum schedulng these blocks end up roughly havng the same fnshng tme). Then we smply schedule each block usng a MakeSpan algorthm A. The trck s to fnd a bucketng functon such that the sum of values assgned to jobs s n a small constant approxmaton of OPT sum (ths mposes an upper bound on bucket values), and the maxmum bucket value should also be a constant approxmaton of OPT max (ths mposes a lower bound on bucket values). Now we elaborate how to partton the job set E nto dfferent blocks E 0, E 1,..., E k. For an nstance (G, l, r), assume a value f (e) (whch we call the bucket value of e) s assocated wth each job e. Let a > 1 be a constant real number and α be a value chosen unformly at random from [0, 1). Let l a α+, for 1, 0, 1,..., k. Defne the block E {e E l 1 < f (e) l }, for 0,..., k. So the edge set E, s dvded nto dsjont blocks of E 0,..., E k. Denote by b e the block nto whch edge e belongs (whch of course, s a functon of α). The meta-algorthm ALG(A, f ) gven n the fgure, apples A (whch wll gve us a near optmum MakeSpan tme) on all the blocks separately. We may use the same notaton ALG(A, f ) as the schedule gven by ths algorthm on an nstance of the MnSum problem. Algorthm 1 ALG(A, f ) 1: Choose α unformly at random from [0, 1). : Usng the bucket functon f, partton the edges nto blocks E 0,..., E k. 3: Schedule the blocks n sequence usng algorthm A. We gve suffcent propertes for the bucket functon (regardng the MakeSpan algorthm A and the optmum answer OPT sum) n order for ALG to have a constant approxmaton rato. Let f be a bucketng functon and A be a schedulng algorthm such that for any nstance of the transfer problem G, l, r : (P1) For any subgraph H (V, E ) G we have e E e E fa(e) βf (e ) where f A(e) s the fnshng tme of e f we run A only on H. (P) e E f (e) γopt sum where β and γ are constants. Lemma gves an upper-bound on the expected value of fnshng tmes of edges n ALG. LEMMA. Let f and A have the property (P1) and let ALG(A, f ) be the correspondng schedule. For each edge e E, E[f ALG (e)] β a f (e) βef (e). Here e.718 ln a stands for the base of the natural logarthms. Proof. In ALG before schedulng the block, we wat for blocks 0 j < to fnsh transferrng all ther fles, and then we schedule the edges n E, usng A. To bound the fnshng tme of edges we consder an arbtrary block separately and then we consder the watng tme requred for transferrng prevous blocks. Let f A(e) be the fnshng tme of edge e when the block E be has been scheduled separately by A. Accordng to (P1) for G V, E be we have: f A(e) βf (e ) βl be βl be n whch the last equalty follows from the fact that e and e are n the same block. Ths shows that f we consder each block

6 separately, an edge e fnshes at most on βl be. Recall that l a α+. Consderng the watng tme for blocks before b e, we have: f ALG (e) b e 0 βl β b e 0 a α+ βaα+be+1 a 1 β a a 1 t b e where t be a α+be. Snce α s a random varable, then b e and t be are also random varables. We use the same method as n [3] to compute the expected value of t be. Let z log a f (e), for e E. Defne y e α + b e z. Snce b e s the smallest nteger such that α + b e z, then y e s unformly dstrbuted on [0, 1). Therefore E[f ALG (e)] β a a 1 E[aye+z ] β a a 1 f (e) β a a 1 f (e) a 1 ln a a β a ln a f (e). 1 0 a t dt The functon f(a) s maxmzed when a e.718, ln a therefore E[f ALG (e)] βef (e). Consderng Lemma we would have the followng. THEOREM 5. The sum cost of ALG(A, f ) s less than βγeopt sum for any nstance of transfer problem (G, l, r) and bucketng functon f and schedulng algorthm A wth both propertes (P1) and (P). Proof. By Lemma, SUM(ALG) e E f ALG(e) βe e E f (e). Thus by (P) we have SUM(ALG) βγeopt. 3. MakeSpan Problem We call a schedule S, non-concurrent schedule f by applyng that schedule no processor runs more than one transfer at any tme,.e., for any two adjacent edges e and e, we have ether s(e) f(e ) or s(e ) f(e). Let deg v for any v V denote the degree of v n G. Next theorem shows that a greedy algorthm can guarantee a 3-factor of the optmum soluton n the MakeSpan verson. THEOREM 6. There s an algorthm Greedy MakeSpan (or GMS) whch for any nstance of transfer problem (G, l, r) gves a non-concurrent schedule wth a MakeSpan cost at most 3OPT ms. In addton for every edge e {u, v} E the fnshng tme s at most r(e) + e E u l(e ) + e E v l(e ) l(e). Proof. We use a sweep lne method to make a non-concurrent schedule (thus f(e) would be equal to s(e) + l(e) for any edge e E). In each step of the algorthm gven n the fgure, we schedule the edge whch can start sooner than all other edges whch are not yet scheduled, breakng tes arbtrarly. Formally, consder U as the subset of edges whch are scheduled by our algorthm before step. Intally U 1 s empty and at each step of the algorthm we schedule a new edge. Recall that E v for any v V, denotes the edges adjacent to v. In step, for any edge e {u, v} / U, consder p (e) as the frst possble startng tme of e,.e., the tme after ts release tme and after transferrng all currently scheduled edges whch are adjacent to u and v. Let e be the edge wth mnmum possble startng tme n step. We set the startng tme of e equal to p (e), set U +1 U {e }, and contnue the algorthm n the next step. We observe that the resultng sequence of startng tmes s(e 1),..., s(e E ) s non-decreasng. Now we show that for each edge e {u, v}, s(e ) s at most r(e ) + e E u l(e) l(e ) + e E v l(e) l(e ). To show ths, we wll prove by contradcton that u and v are never dle n Algorthm Greedy MakeSpan (GMS) Input: An nstance of transfer problem (G (V, E), l, r). Output: A schedule S wth correspondng startng tmes for each edge n E. 1: Defne U 1 φ. : for 1 to E do 3: For every edge e {u, v} / U, defne p (e) max{r(e), max e U (E u E v) f S(e )}. 4: Let e argmn e / U p (e). 5: Set s S(e ) p (e ) and thus f S(e ) s S(e ) + l(e ). 6: Defne U +1 U {e }. the same tme between r(e ) and s(e ). Let r(e ) t < s(e ) be the moment that both u and v are dle. Let k be the frst step whch we schedule an edge after t,.e., k mn { s(e )>t}. In step k, p k (e k ) > t and s(e) t for all edges e U k. Snce u and v are both dle on tme t and no scheduled edge s started after that tme, the mnmum possble startng tme of e at step k s p k (e ) t. But the edge wth mnmum possble startng tme at step k s e k wth p k (e k ) > t, whch s a contradcton. Therefore the fnshng tme of each edge e {u, v} s at most f(e ) s(e ) + l(e ) r(e ) + l(e) + l(e) l(e ) e E u e E v 3OPT l(e ). The last nequalty follows from the fact that OPT e E v l(e) for any vertex v V and OPT r(e) + l(e) for any edge e E. We note that by Theorem 6, any edge e {u, v} wth release tme of zero fnshes at most on e E u l(e ) + e E v l(e ) l(e) OPT. Hence n a zero-release transfer problem GMS s always n -approxmaton of the optmum. COROLLARY. For any nstance of zero-release transfer problem (G, l), GMS gves a non-concurrent schedule wth a MakeSpan cost at most OPT ms. 3.3 MnSum Problem In ths secton we present a bucketng functon wth both propertes (P1) and (P) accordng to the MakeSpan algorthm GMS gven n secton 3.. For any vertex v V consder the edges adjacent to v ordered by the length. For any e E v, let S e (v) denote the sum of the length of edges whch come before e plus the length of e. For any edge e {u, v} E, let the bucket value of e be f (e) max{r(e), S e (u), S e (v)}. The followng two lemmas show the exstence of both propertes P1 and P for the bucketng functon f and algorthm GMS. LEMMA 3. Let H (V, E ) be a subgraph of G. For any edge e {u, v} E there exsts an edge e E such that f GMS (e) 3f (e ) and thus satsfyng property P1 wth β 3, where f GMS (e) s the fnshng tme of e when we run GMS restrcted to H. In addton f e s released at tme zero (.e. r(e) 0) then f GMS (e) f (e ) for some edge e E. Proof. By Theorem 6 we have f(e) r(e) + e E u l(e ) + e E v l(e ) l(e). Assume that p and q are the fles wth the longest length adjacent to u and v n H, respectvely. Thus S p (u) (S q (v)) s at least the sum of the lengths of all edges n E u (E v). By the defnton of the bucketng functon f we have

7 f (e) r(e). f (p) S p (u) e E u l(e ), f (q) S q (v) e E v l(e ), Therefore f GMS (e) r(e) + e E u l(e ) + e E v f (e) + f (p) + f (q) l(e) 3f (e ) l(e) l(e ) l(e) where e s the edge wth the maxmum bucket value among all edges n E. We note that f r(e) 0 then the above nequalty would be f GMS (e) f (e ) l(e). LEMMA 4. The sum of the bucket values of all edges n E s at most 3OPT sum and thus satsfyng property P wth γ 3. Furthermore, f all the release tmes are zero, or f all lengths are unform, P holds wth γ. Proof. Let opt be the optmum schedule. Consder the fnshng tmes of edges adjacent to an arbtrary vertex v V n opt and let x v denote the th smallest fnshng tme among them. Note that degv v V 1 xv s exactly equal to OP T. Vertex v V should have fnshed transferrng at least k fles at tme x vk for any 1 k deg v. Thus () x vk cannot be less than the sum of lengths of the smallest k edges n E v. We denote ths sum by sm k (v); () x vk cannot be less than the kth smallest release tme of edges n E v. Now we show that e E f (e) s wthn 3 factor of OP T. We have: f (e) max {r(e), S e (u), S e (v)} e E e(u,v) e(u,v) r(e) + e(u,v) (S e (u) + S e (v)) The frst term, e(u,v) r(e), cannot be more than OP T. The second term s the sum of edges shorter than e adjacent to u or v, when summed over all edges. We can rewrte the second term by summng these values over the edges adjacent to each vertex. f (e) OP T + S e (v) e E v V e E v OP T + v V OP T + v V 3OP T deg v k1 deg v k1 sm k (v) x vk We note that f all the release tmes are zero, then e E f (e) OP T. We can prove the same rato under unform length assumpton. For a vertex v V, let e v1,..., e v degv denote the edges adjacent to v sorted by the release tme, breakng tes arbtrarly. We note that snce all the lengths are unform for all k deg v, S e vk (v) sm k (v) and thus by () and () we have x vk max{r(e vk ), S e vk (v)}. Ths shows that under unform length assumpton e E f (e) cannot be more than OP T. f (e) max {r(e), S e (u), S e (v)} e E e(u,v) e(u,v) (max {r(e), S e (u)} + max {r(e), S e (v)}) max {r(e), S e (v)} v V e E v v V deg v k1 deg v x vk v V k1 OP T max{r(e vk ), S e vk (v)} Fnally usng the meta-algorthm provded n Secton 3.1, we get a constant compettve algorthm for the general transfer problem. THEOREM 7. For an nstance of transfer problem (G, l, r), ALG(GMS, f ) s a 9e-approxmaton algorthm for the Mn- Sum problem. Proof. By Lemma 3 the bucketng functon f has (P1) property wth β 3 and by Lemma 4 has (P) property wth γ 3. The clam follows drectly from Theorem 5. We note that smlar to Corollary, by Lemma 3 for a zerorelease transfer problem or a unform transfer problem we get property (P1) wth β whch mproves the approxmaton rato of the algorthm. COROLLARY 3. For an nstance of zero-release transfer problem (G, l), ALG(GMS, f ) s a 4e-approxmaton algorthm for the MnSum problem. COROLLARY 4. For an nstance of unform transfer problem (G, r), ALG(GMS, f ) s a 6e-approxmaton algorthm for the MnSum problem. 4. UNIFORM ZERO-RELEASE TRANS- FER MODEL In ths secton we consder the unform zero-release transfer model. Frst we show that by runnng all the fle transfers smultaneously, we could obtan an exact soluton for the MakeSpan verson. We call ths schedulng algorthm n whch e E, s(e) 0 by Smultaneous Start or smply SS. THEOREM 8. For any graph G (V, E) wth unform zerorelease fle transfers, MAX(SS) OPT ms. Proof. Let (G) be the maxmum degree of vertces of G and let u be one of the vertces wth degree (G). The vertex u needs to transfer (G) unts of fle and transferrng these fles takes at least (G) unts of tme. So we have OPT (G). However, f we start all the jobs smultaneously, n each unt of tme we run at least 1 (G) of each transfer. Therefore MAX(SS) (G) OPT. We note that f the fles can have arbtrary lengths, SS may not have the optmum cost. We gve an example for zero-release nonunform fle transfer model where the cost of SS can be almost two

8 THEOREM 9. For any graph G (V, E) wth unform zerorelease fle transfers, there s a non-concurrent schedule S wth SUM(S) OPT sum wth ntegral startng tmes. Fgure : An example where SS s not optmum tmes the OPT. Consder the tree shown n Fgure. Vertex v s adjacent to n vertces u 1,..., u n through the edges e 1,..., e n. The length of all edges adjacent to v s an arbtrary nteger M. For every [n] the vertex u s adjacent to (n 1)M leaves through the edges of length 1. We can show that OPT ms nm but the MakeSpan cost of SS s MAX(SS) nm (M + n 1). The optmum schedule has n stages. In stage vertex u starts transferrng e and for every j the vertex u j starts transferrng wth M of ts adjacent leaves. Therefore each stage takes exactly M unts of tme and OPT ms would be equal to nm (the sum of edges adjacent to v s nm and thus OPT cannot be smaller). The schedule SS starts all edges smultaneously at tme zero. Snce all edges are adjacent to a vertex wth degree (n 1)M + 1, the edges adjacent to the leaves fnsh at tme (n 1)M + 1. After that the remanng M 1 unts of data on edges adjacent to v wll be transferred wth speed of 1/n and SS fnshes all transfers at tme (n 1)M (M 1)n. Therefore by choosng n M the rato between the optmum cost and the cost of SS would be MAX(SS) OPT ms n n n n 1 n. Now we show there s always a non-concurrent schedule wth a cost less than twce the optmum soluton n the MnSum problem. Frst we need to gve a lower bound on the optmum cost. LEMMA 5. For any graph G (V, E) wth unform zerorelease fle transfers, OPT sum 1 4 v V (deg v + deg v). Proof. Let opt be the optmum schedule. Consder the fnshng tmes of edges adjacent to an arbtrary vertex v V n opt and let x v denote the th smallest fnshng tme among them. Snce x v for all 1 deg v, we have e E v f opt(e) deg v 1 xv deg v(deg v+1). Therefore OPT e E f opt(e) f opt(e) v V e E v v V degv + deg v. Proof. Let opt be the optmum schedule for G. Assume that the set of edges e 1, e,..., e E are sorted accordng to ther fnshng tme n opt,.e., f opt(e 1) f opt(e ) f opt(e E ). Now we make a non-concurrent schedule S based on ths sequence wth the total cost at most OPT. For any vertex v V and 1 E, let Eu be the subset of edges from e 1,..., e whch are adjacent to u. We start transferrng edge e {u, v}, whenever both ts endponts have fnshed transferrng Eu 1 e s and E 1 v s S(e ) mn{t t Z 0, e E 1 u. Formally, the startng tme for edge E 1 v [t s S(e)]}. By defnton of s S(e ), t s clear that () the resultng schedule s a non-concurrent schedule, therefore f S(e ) s S(e ) + 1; and () s S(e ) Eu 1 + Ev 1 whch means that f S(e ) Eu + Ev 1. We know n opt, the fnshng tmes of all the edges n Eu and Ev are less than or equal to f opt(e ). Snce the speed of transfer s at most 1 for any processor, vertex u cannot fnsh all the edges of Eu n less than Eu ; hence f opt(e ) max{ Eu, Ev }. Therefore for every edge e we have f S(e ) f opt(e ) and so SUM(S) OPT. COROLLARY 5. The unform fle transfer problem wthout release tmes on planar graphs admts a + ɛ approxmaton rato. Ths follows from the non-concurrent PTAS of Marx [3] for colorng the edges of a planar graph. In unform transfer model, f all the startng tmes are ntegers n a non-concurrent schedule, the schedule s ndeed a partton of the edges E, nto k matchngs M 1,..., M k for some k, where M s the set of all edges startng on tme 1 and therefore fnshng at tme. The cost of ths schedule would be 1k M. Halldórsson et. al. [] gve an approxmaton algorthm of rato for fndng the mnmum sum cost edge colorng. By Theorem 9, the same algorthm gves us a approxmaton algorthm for the zero-release unform transfer model. COROLLARY 6. There s a polynomal tme algorthm whch gves a non-concurrent schedule wth the cost less than factor of OPT sum for the zero-release unform transfer model. 5. FILE TRANSFER ON BIPARTITE GRAPHS For certan classes of graphs we may get a smaller constant approxmaton. Consder a graph G (V, E) and a non-concurrent schedule of E nto k matchngs M 1,..., M k. If for every vertex v V all edges adjacent to v are n the frst deg v matchngs, then the sum of edges adjacent to v would be deg v 1 deg v(deg v + 1)/. Therefore the sum of fnshng tmes over all edges would be 1 v V degv(degv + 1)/ whch by Lemma 5 s equal to the optmum cost. For example, n k-regular bpartte graphs we can always fnd such a schedule by smply parttonng the edges nto k perfect matchngs, thus: COROLLARY 7. For any regular bpartte graph G we can fnd a schedule wth the cost equal to OPT sum n polynomal tme. The nature of many transfer problems are transferrng fles between hosts and clents thus bpartte graphs are of separate nterest.

9 We present a approxmaton algorthm for fndng a schedule wth mnmum sum cost n bpartte graphs. We rely on the algorthm [5] n non-concurrent settngs but we need to change t somewhat. We frst argue a smple rato approxmaton MnSum fle transfer of unt jobs on bpartte graphs (wthout release tme). Then we show a approxmaton based on [5] for the sum verson. Let G (V, E) be a graph. We say vertex v s full n G when deg v (G) 3. A graph G s n class 1 ff χ (G) (G) where χ (G) s the edge-chromatc number of G. Theorem 17. of [1] shows any bpartte graph G s n class 1 and can be parttoned nto χ (G) matchngs n polynomal tme. Assume an nstance of the unform zero-release fle transfer problem wth a bpartte graph G (V, E). Snce χ (G) (G) we can partton the edge set E, nto (G) matchngs. Consder M as one of these matchngs. If we remove the edges of M from the graph, the resultng graph would have an edge chromatc number of (G) 1 and thus the maxmum degree of (G) 1 (snce the resultng graph s also n class 1). Therefore every full vertex v n G must have one edge n M. Let E (G) be the subset of the edges of M whch are adjacent to at least one full vertex n G. Let G (G) 1 (V, E\E (G) ). Snce every full vertex n G has an edge n E (G) we have (G (G) 1 ) (G) 1. Wth the same argument we can fnd a matchng E (G) 1 n G (G) 1, such that every full vertex n G (G) 1 has one adjacent edge n that matchng, and each edge n that matchng s adjacent to at least one full vertex n G (G) 1. Now the graph G (G) ( V, E\(E (G) E (G) 1 ) ) has the maxmum degree of (G). By repeatng ths procedure we partton E nto (G) matchngs E (G), E (G) 1,..., E 1. Algorthm 3 Input: An nstance of zero-release unform fle transfer problem G (V, E) where G s bpartte. Output: A schedule S wth correspondng startng tmes for each edge n E. 1: Defne G (G) G. : for (G) to 1 do 3: Partton the edges of G nto χ (G ) (G ) matchngs and let M be one of the matchngs. 4: Let Vfull V be the set full vertces n G. 5: Let E M be the subset of edges n M whch are adjacent to at least one of the vertces n Vfull. 6: For any edge e n E, set s S(e) 1. 7: Defne G 1 (V, E\ (G) j E j). It can easly be shown that Algorthm 3 s a -approxmaton algorthm. THEOREM 10. Algorthm 3 s a -approxmaton algorthm for the MnSum cost n unform zero-release fle transfer problem n bpartte graphs. Proof. Consder the bpartte graph G (V, E). Runnng Algorthm 3 gves us a schedule S. The schedule S parttons the edges nto (G) matchngs E 1,..., E (G) such that for any, 1 (G), any full vertex n G has an edge n E and any edge n E s adjacent to at least one full vertex n G (where Ej)). Let V full V be the set of full ver- G (V, E\ (G) j+1 tces n G. Snce each edge n E s adjacent to at least one vertex n V full we have E V full. 3 For a graph H, (H) s the maxmum degree n H Let n() denote the number of vertces n G wth degree at least,.e., n() {v V deg v }. Recall that a vertex s full n G ff the degree of v n G s equal to (G ). The maxmum degree (G ) s equal to snce G s the unon of matchngs and thus the degree of any full vertex n G would be at least n G. Therefore Vfull n(). Consderng the sum cost of the schedule we have SUM(S) E v V 1 (G) 1 (G) 1 (G) V full 1 (G) {v deg v } 1 v V deg(v)(deg(v) + 1) The last lne s the result of Lemma 5. n() 1deg(v) OPT sum. In [5] t s shown that the algorthm s -approxmate and there analyss s almost tght. One can change ther proof to obtan the same approxmaton rato. The proof s presented n the next subsecton. COROLLARY 8. There s a polynomal tme algorthm whch gves a non-concurrent schedule wth the cost less than -factor of OPT sum for the zero-release unform transfer problem n bpartte graphs. 5.1 Approxmaton Rato of Algorthm 3 Gandh and Mestre [5] use a class of matchngs whch are strongly mnmal. One can prove that the same property gven s suffcent for a schedulng to be -approxmate n the unform zero-release fle transfer model. Not all graphs admt strongly mnmal matchngs but n some classes of graphs such as bpartte graphs we can gve polynomal tme algorthms to fnd such matchngs. For the sake of completeness we present a slghtly modfed proof. Let G (V, E) and let S be a non-concurrent schedule of E. Recall that S can be shown as the partton ( of E nto matchngs M 1,..., M k. For, 1 k, let G V, ) j1 M (thus G k G). Let deg v for a vertex v V and 1 k denote the degree of vertex v n G. By borrowng notatons from [5], we say vertex v s full n G when deg v (G ). We call S strongly mnmal f for every, 1 k: For every full vertex v n G, there s one edge adjacent to v n M. At least one of the endponts of every edge of E s full n G. Another way to look at ths property s that G s a maxmal - matchng w.r.t G for every k. We note that by defnton of strongly mnmal schedule the number of matchngs k s ndeed equal to (G). Furthermore, snce every full vertex n E has exactly one edge n M we have (G 1 ) (G ) 1 and n general for every, 1 (G), (G ). Thus f v s full n G for some, then t s full n G j for every j. THEOREM 11. For G (V, E) n the unform zero-release fle transfer model any strongly mnmal schedule S s - approxmate.

10 Proof. The dea s to make a full vertex n G responsble for payng the fnshng tme of ts adjacent edge n E. Formally, for an edge e {u, v} assume that that e s n the matchng M. By the defnton of S at lease one of u or v s full n G. If both endponts are full then each of u and v are half-responsble for e. If only one of the endponts, say u, s full then u s fully-responsble for e. Now consder an arbtrary vertex v V. Assume that v s fullyresponsble and half-responsble for the set of edges Rv 1 and Rv f opt (e) respectvely. Defne C opt(v) e R fopt(e) + v 1 e Rv where opt s the optmum schedule for G. Smlarly defne C S(v) e R fs(e) + f S (e) v 1 e R. In other words v always pays the fnshng tmes of edges n Rv 1 and pays the half of v the fnshng tmes of edges n Rv. Thus SUM(opt) v Copt(v) and SUM(S) v CS(v). To show that S s α-approxmate t s suffcent to show that C S (v) C opt (v) α. Fx a vertex v and let n1 R1 v and n R v. Vertex v s full n G 1,..., G n 1+n snce each vertex gets at least a half responsblty when t s full n some G. Thus for an edge e R 1 v R v the fnshng tme of e n S s not greater than n 1+n. More precsely, the set of fnshng tmes of edges n R 1 v R v s exactly {1,..., n 1 + n }. Therefore C S(v) f S(e) + e R 1 v e R 1 v R v n 1 +n 1 n 1 +n 1 e R v f S(e) e R v n 1 f S(e) e R v f S(e) (n1 + n)(n1 + n + 1) n 1 + n 1 + n + n 4 f S(e) + n 1n. (n)(n + 1) 4 Snce the edges n R 1 v and R v are all adjacent opt cannot fnsh transferrng all of them sooner than n 1 + n, thus C opt(v) f opt(e) + e R 1 v e R 1 v R v n 1 +n 1 n 1 +n 1 f opt(e) + e R 1 v n1 + 1 e R v + f opt(e) e R 1 v f opt(e) (n1 + n)(n1 + n + 1) 4 n 1 + n 1 + n + n f opt(e) + + n1n. We need to determne the smallest α such that (n1)(n1 + 1) 4 α (n 1 + n 1) + (n + n )/ + (n 1n ) (n 1 + n1) + (n + n)/ + (n1n) n 1 + n + 4n 1n n 1 + n + n1n. By changng the varable to x n 1 n 1 +n we get α 1 + x x 1 + x. Fnally the rght hand sde s maxmzed for x 1, whch gves us α. Runnng Algorthm 3 on a bpartte graph G (V, E) gves us a schedule S. The schedule S parttons the edges nto (G) matchngs E 1,..., E (G) such that for any, 1 (G), any full vertex n G has an edge n E and any edge n E s adjacent to at least one full vertex n G (where G (V, E\ (G) j+1 Ej)). Ths shows that the schedule gven by Algorthm 3 s strongly mnmal and thus proves Corollary CONCLUSION AND OPEN PROBLEMS Ths paper studes a local protocol for fle transfer problems whch to the best of our knowledge, has not been studed before n theoretcal computer scence. Among the problems we consder, we hghlght one open problem of prmary concern: Is there a gap between the optmum concurrent schedule and optmum nonconcurrent schedule when mnmzng the average fnshng tme n the case of the zero-release fle transfer model? In ths paper most of our algorthms gve non-concurrent solutons payng a constant factor compared to the optmum concurrent schedule. It would be nstructve to see whether we can desgn concurrent algorthms wth better approxmaton factors. Ths shows the mportance of fndng the gap between optmum concurrent and optmum non-concurrent schedules. It would also be nterestng to consder the onlne verson of the problem where the release tmes are revealed to the algorthm n an onlne fashon. Could we get constant approxmatons n nonpreemptve model or do we need to add preemptve assumptons to the problem? 7. ACKNOWLEDGEMENTS The authors would lke to thank the anonymous revewers for ther valuable comments and suggestons to mprove the qualty of the paper. 8. REFERENCES [1] J.A. Bondy and U.S.R. Murty. Graph Theory. Graduate Texts n Mathematcs, 44. Sprnger, New York, 008. [] M.M. Halldórsson, G. Kortsarz and M. Svrdenko. Mn Sum Edge Colorng n Multgraphs Va Confguraton LP. In Proc. 13th Conf. Integer Prog. Combn. Optmz. (IPCO), 008. [3] R. Gandh, M. M. Halldórsson, G. Kortsarz and H. Shachna. Improved Bounds for Schedulng Conflctng Jobs wth Mnsum Crtera. ACM Transactons on Algorthms. Vol. 4, No. 1, 008. [4] M.M. Halldórsson and G. Kortsarz. Tools for multcolorng wth applcatons to planar graphs and partal k-trees. Journal of Algorthms 4,, , 00. [5] R. Gandh and J. Mestre. Combnatoral Algorthms for Data Mgraton to Mnmze Average Completon Tme. Algorthmca 54, 1,pp 54-71, 009. [6] Y. Km. Data Mgraton to Mnmze the Average Completon Tme. Journal of Algorthms,55:4-57, 005. [7] M.K. Goldberg, Edge-colorng of multgraphs: recolorng technque. J. Graph Theory, 8:11-137, 1984 [8] D.S. Hochbaum, T. Nshzek, and D.B. Shmoys. A better than "Best Possble" algorthm to edge color multgraphs. Journal of Algorthm 7:79-104, [9] E. G. Coffman, M. R. Garey, D. S. Johnson, and A. S. Lapaugh. Schedulng fle transfers. SIAM Journal on Computng, 14(3): , 1985.