Scheduling and Flow-Related Problems in Networks

Transcription

1 DISS. ETH NO Scheduling and Flow-Related Problems in Networks A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH for the degree of Doctor of Sciences presented by ALEXANDER HALL Dipl. Inf., Technische Universität München born citizen of Germany accepted on the recommendation of Prof. Dr. Thomas Erlebach, examiner Dr. habil. Martin Skutella, co-examiner Prof. Dr. Peter Widmayer, co-examiner 2003

2 ABSTRACT Modern communication networks give rise to many intricate combinatorial optimization problems. We investigate three different areas pertaining to scheduling and flow-related problems in the context of such networks. Recently the number of applications in the Internet which use data broadcasts, such as pay-per-view services, has increased greatly. This is motivation for the investigation of broadcast scheduling in the first part of this thesis. The basic setting here is that a server can transmit documents from a given set on a high bandwidth channel, answering previously made requests for these documents. The goal is to minimize the average response time (the average time until a request is answered) if the requests are known in advance. We prove that this problem is N P-hard, thereby answering a previously stated open problem. We propose an approximation algorithm that is allowed to use six times the given bandwidth. It achieves an average response time that is at least as good as the optimal solution for the original bandwidth. Furthermore, we present results obtained in experiments which were conducted in order to empirically compare two different variations of broadcast scheduling. Finally, from the N P-hardness of broadcast scheduling we derive a new inapproximability result for a certain unsplittable min-cost flow problem. In the second part of this thesis we consider so-called flows over time. Flow variation over time is an important feature in network flow problems arising in connection with various applications in communication and transportation networks. Such flows can be modeled as flows over time in networks with capacities and edge transit times which specify the amount of time it takes to traverse the individual edges. In the basic model these transit times are constant and their values are given in the input. While the maximum s-t-flow over time problem can be solved efficiently and min-cost s-t-flows over time are known to be N P-hard, the complexity of (fractional) multicommodity flows over time has been open for many years. We prove that this problem is N P-hard, even for series-parallel networks or instances with only two commodities. For the special case where flow traversing the network cannot be stored at intermediate nodes, we can show strong N P-hardness. Furthermore, we consider the often more realistic model where the transit time of an edge is not necessarily constant but may depend on the current inflow rate into the edge; the transit time is given as a non-decreasing function of the inflow rate. For this setting we can already prove N P-hardness for s-t-flows over time. Moreover, we propose a fully polynomial time approximation scheme for the multicommodity case, with respect to the goal of minimizing the time horizon in which given demands can be satisfied.

3 2 ABSTRACT In the third and final part we consider Internet graphs with customer-provider relationships. The Internet can be subdivided into so-called autonomous systems which are interconnected, inducing the autonomous systems graph. For each edge of this graph the two connected nodes typically take on roles as customer and provider respectively, reflecting their economic relationship. The types of the individual economic relationships impose severe restrictions on the paths along which data is allowed to move through the network. This gives rise to interesting optimization problems. The goal of the so-called type of relationship problem is to infer the respective roles of the nodes, given a set of paths which are actually used in practice. We present a strong inapproximability result for the general problem (resolving its complexity was previously posed as an open problem) and an optimal polynomial-time algorithm for a special case. For instances in which path lengths are bounded by constants we propose a constant-factor approximation algorithm and prove that the problem remains APX -hard in this setting. We report experimental results for the approximation algorithm which demonstrate that it yields excellent solutions on real data sets. If the types of the economic relationships are given or inferred e.g. by our approximation algorithm, connectivity issues can be investigated. We propose a 2-approximation for the problem of computing a maximum number of edge- or vertex-disjoint paths, adhering to the restrictions posed by the relationships, between two given nodes, and we show that no better approximation ratio is possible unless P = N P. For an important special case we give a separate N P-hardness proof. Furthermore, we present an optimal polynomialtime algorithm for the problem of computing a minimum edge cut that separates two nodes with respect to the restricted paths.

4 RESUMAZIUN Reits da communicaziun modernas porschan biars problems complex d optimaziun combinatorica. Nus lein perscrutar en treis differents secturs il scheduling e problems pertucond il flux en connex da talas reits. Dapi dacuort san ins observar in augment d applicaziuns egl Internet, sco per exempel survetschs da pagar per mirar, che fan diever d emisiuns. Quella contemplaziun motivescha nossa retscherca sur dil scheduling dad emisiuns en l emprema part dalla thesa. La disposiziun da basa ei in survient che transmetta documents ord ina quantitad avon maun sin in canal da gronda banda, e ch el rispunda cunquei sin damondas tschentadas ordavon per quels documents. La finamira ei da minimar la valita media dil temps per la risposta (la valita media dil temps ch ei drova entochen che la damonda ei vegnida satisfatga), sche tuttas rispostas ein gia conuschidas ordavon. Nus cumprovein, ch ei setracta d in problem ch ei N P-dirs, e rispundin aschia ad ina damonda muentada pli baul. Nus proponin in algoritm d approximaziun al qual nus permetin da duvrar sis gadas il total da la banda dada. El realisescha ina valita media dil temps ch ei silmeins aschi buna sco la soluziun optimala per la banda originala. Ultra da quei presentein nus resultats obteni en experiments ch ein vegni enterpreni per cumpragliar duas differentas variaziuns dal scheudling d emisiuns. Finalmein derivein nus dil fatg ch il scheduling d emisiuns ei N P-dirs, in niev resultat d inapproximabilitad per certs problems da min-cuost per flux ch ein buca divisibels. Ella secunda part da la thesa considerein nus schinumnai flux sur il temps. La variaziun dal flux sur il temps ei in impurtonta caracteristica per problems resultond ord il flux da reits, per exempel ord differentas applicaziuns da communicaziun ni ord da traffic en reits. Tals flux san vegni modelai sco flux dil temps en reits cun capazitats e cun temps da transit per ils urs, che specificescha la quantitad da temps ch ei drova per traversar ils urs. En il model da basa ein ils temps da transit constants e las valurs ein dadas sco input. Duront ch il problem maximal s-t-flux per temps sa vegir schligiaus effectivamein ed il min-cuost s-t-flux per temps ei conuschius dad esser N P-dirs, la cumplexitad da flux multicommods per temps (fractinals) ei buca vegnida tschaffada duront biars ons. Nus cumprovin che quei problem ei N P-dirs, schizun per reits ch ein serials-parallelas ni per instanzas cun mo duas commoditads. Per il cas special nua ch il flux traversescha la reit senza saver esser deponius en elements intermediats, savein nus demonstrar ferma N P- direzia. Plinavon considerein nus in model ch ei savens pli realistics, cunquei ch il

5 4 RESUMAZIUN temps da transmisiun per mintg element drova buca esser necessariamein constants, mobein ch el dapenda da la quota da flux entrond igl element; quei resultescha en ina funcziun da la quota nundigrenta. Per quella disposiziun cumprovein nus gia la N P-direzia per flux s-t per temps. Plinavon proponin nus in schema per in approximaziun dil temps pleinamein polynomica dau il cas da mulitcommoditad, respectond la minimaziun digl horizon dil temps el qual las damondas pon vegnir satisfatgas. Ella tiarza e finala part considerein nus graphs egl Internet che muossan las relaziuns denter in client ed in purschider. Igl Internet sa vegnius partius si en schinumnai systems autonomics colligiai, induzond il graph da systems autonoms. Per mintga ur da quei graph surprendan ils elements tipicamein las rollas d in client respectivamein d in purschider. Las differenetas rollas implicheschan certas restriziuns per ils viadis che l informaziun pren transcurend la reit. Quei porscha interessants problems d optimaziun. La finamira resultond ord problem dils tips da relaziuns ei d implicar las rollas, dau ina quantitad da viadis ch ein actualmein occupai ella practica. Nus presentein in ferm resultat d inapproximabilitad per il problem general (schligiar sia cumplexitad era considerau in problem nunschligiau tocchen dacheu) ed in algoritm da temps optimal polynomial per il cas special. Per instanzas nua che la lunghezia dil viadi ei restrictada entras constantas, proponin nus in algoritm che fa diever d ina approximaziun da factur-constanta e cumprovin aschia ch il problem resta APX -dirs en quella quantitad. Nus rapportein resultats experimentals per igl algoritm d approximaziun e demonstrein ch el furnescha excellentas schligiaziuns per quantitads cun datas realas. Sche las rollas ein dadas ni implicadas, sco per exempel entras nies algoritm d approximaziun, sa il pensum da connectividad vegnir perscrutaus. Nus proponin in approximaziun da 2 per il problem da quintar il diember maximal dad ur- ni vertex-disjuncts viadis (dau las restricziuns imposadas entras las rollas.) enter dus elements, e nus mussein aschia ch ei detti negina ratio d approximaziun pusseivla auter che cun P = N P. Per in cas special dein nus in separau mussament da la N P-direzia. Plinavon presentein nus in optimal algoritm da temps polynomial per il problem da quintar ora il mimimal tagl enter chantuns che separeschen dus elments respectond viadis restrictivs.