Online Multi-Commodity Flow with High Demands

1. Moti Medina EE School, Tel-Aviv University Online Multi-Commodity Flow with High Demands Guy Even EE School, Tel-Aviv University WAOA 2012

2. Big vs. Small

3. Problem Definition ONMCF: Online Multi-Commodity Flow with High Demands • The Network • – set of nodes () • – set of directed edges (). • Every edge has a capacity • The Online Input • Sequence of flow requests. • - source and target nodes. • – flow demand. • – benefit. • The Output • – a multi-commodity flow. • For each , is a flow from to . • The Objective • An all-or-nothing ONMCF • That maximizes the total benefit Of the served requests. Each request is either fully served or rejected. We are credited for fully serving The requests arrive one-by-one. No information is known about a request before its arrival.

4. Competitive Analysis • We consider an online maximization problem: • Competitive analysis. • For every input sequence σ, |Alg(σ)| ≥ 1/ρ•|OPT(σ)|. • Alg– deterministic online algorithm. • OPT – offline optimum. • |Alg| - total benefit of algorithm Alg. • ρ – competitive ratio

5. Previous Work • Mostly studied in the context of single path routing. • Throughput maximization (TM) • Maximizing the total benefit gained by flow requests that are served [AAP93, BN06,EMSS12]. • Load minimization (LM) • . • Routing all requests while minimizing the maximum load of the edges [AAF+97, AAPW01, BN06,BLNZ11]. • The following variants are considered: • Permanent routing [AAF+97, BN06, EMSS12] • AAF+97 : augmentation. (LM) • BN06: using the Primal-Dual framework augmentation. Can be extended to high demands. (LM) • Unknown durations [AAPW01] • Known durations [AAP93, EMSS12, BLNZ11] • AAP93: comp. (TM) . Requires • BN06 • Primal-Dual • For the case of unit demands and caps: - comp. • EMSS12 • Primal-Dual • Embedding of traffic patterns I the context of VNETs • BLNZ11: augmentation. (LM)

6. The Main Result Assuming that caps and benefits are • An online algorithm for the ONMCF problem: • Centralized and deterministic, • There is no limitation on demands • In particular, may exceed • All-or-Nothing, • Competes with an all-or-nothing offline optimal algorithm, • – competitive, for a constant , • Violates capacities by a factor of , • Non-preemptive and monotone. Flow is never retracted

7. Approaches for ONMCF with high demands • [AAP93, BN06] • Route each request along a single path. • Requires that • Augment capacities in advance? • Might be augmentation vs. the required • Split requests into sub-requests • Demands are small • Some of the sub-requests might be rejected. Not high demand Not augmentation Not All-or-Nothing

8. Approaches for ONMCF with high , demands, cont. • Granularity of a flow • Smallest positive flow along an edge in the network. • [BN09] • Formulating ONMCF as a packing LP • Apply Primal-Dual • The caps. Augmentation of BN09 depends on , might be unbounded! Not augmentation

9. An (simple) Example • We want: • Accept ALL the requests. • Augmenting caps by a factor of at most1.25. • Granularity of 0.75. • The Network: • Caps = 1. • Two requests: • , ) • , • So…we need: • To route along multiple paths. • To reflow “small” flows, while augmenting (again) the edge caps • Might affect the competitive ratio, i.e., the chosen flow is not the “lightest” one.→ • Tri-criteria oracle (approximated, augmentation, granularity). 0.75 0.75 1 1 +0.25 0.25 1 1.25 1 1.25

10. Techniques • Main Technique • Extension of [AAP93] and [BN06] • Integrally packing paths by a centralized online algorithm. • log n– competitive. • Edge costs: exponential in the load of the edge. • Oracle: Finds a shortest path. • Alg : • If the cost of the path is higher then its benefit, then reject, • Otherwise, accept. Oracle Oracle Resource Manager Oracle

11. Techniques • Main Technique • The Reduction • Now, the requests are flow requests. • Every request increases the load of edges that it uses. • The edge cost is updated. • The Oracle finds a “min-cost flow” that fully serves the request. • The Oracle • Is an offline tri-criteria oracle. • Approximated, augmenting, granular. Oracle Oracle Resource Manager Oracle

12. Extending the Framework • Formally, -Augmentation -Granular -Approximation

13. The Tri-Criteria Oracle 2-Augmentation -Granular 2-Approximation

14. The Main Result

15. Mixed Demands • Splitting a stream of packets along multiple paths should be avoided, if possible → • One may require not to split requests with low demand, i.e., • How? We employ two oracles: • Tri-criteria oracle for high demands (as before). • An exact (shortest path) oracle for low demands. • Serves a request by a single path. • This algorithm has the same properties! • – comp., caps aug., monotone, all-or-nothing.

16. Further Extensions • Requests with known durations • Each request, upon arrival has an end time. • This talk was on requests that “stay forever”. • The same algorithm can be adapted to known durations [AAP93, BN06, EMSS12]. • Again, this algorithm has (almost) the same properties! • Cap. augmentation of • Where is the longest duration.