An equivalent version of the Caccetta-Häggkvist conjecture in an online load balancing problem Angelo Monti 1, Paolo Penna 2, Riccardo Silvestri 1 1 Università di Roma “La Sapienza” 2 Università di Salerno

Outline Online load balancing Caccetta-Häggkvist conjecture Connection between them

Online load balancing processors task (weight, subset, duration)

Online load balancing Example: linear topologies [Bar-Noy et al’99] best worst

Online load balancing How good is greedy? Example: linear topologies [Bar-Noy et al’99] best worst 8 tasks

Online load balancing How good is greedy? Example: linear topologies [Bar-Noy et al’99] best worst 4 tasks

Online load balancing How good is greedy? Example: linear topologies [Bar-Noy et al’99] worst 2 tasks

Online load balancing How good is greedy? Example: linear topologies [Bar-Noy et al’99] worst 1 task  (log n)-competitive

Online load balancing modified-greedy Example: linear topologies [Bar-Noy et al’99] worst 8 tasks4 tasks 2 tasks 1 task 4-competitive More general approach [Crescenzi et al’03]

Online load balancing More general approach [Crescenzi et al’03]: “structure”  comp(“structure”) 1. Competitive ratio of modified-greedy 2. Simple local algorithm 3. Combinatorial approach

Online load balancing More general approach [Crescenzi et al’03]: “structure”  comp(“structure”) Optimal for “nice structures” identical, linerar, hierarchical

Online load balancing More general approach [Crescenzi et al’03]: “structure”  comp(“structure”) Optimal for “nice structures” identical, linerar, hierarchical How good on the “uniform” case? “Equivalent” to a fundamental question in graph theory

Caccetta-Häggkvist Conjecture Every directed graph on n nodes and minimum outdegree d has a directed cycle of length at most n/d

Caccetta-Häggkvist Conjecture Every directed graph on n nodes and minimum outdegree d has a directed cycle of length at most n/d ?

Modified-greedy algorithms S 1,…, S i,…, S m S 1 ’,…, S i ’,…, S m ’ R 1,…, R i,…, R m problem “structure” R i =  S j : S j ’ intersects S i ’ How good is modified-greedy? max i |R i |/|S i ’ | [Crescenzi et al’03]

The “uniform” case How good is modified-greedy? comp(n,s) Each task can be assigned to exactly s processors Apply Crescenzi et al’03 to uniform case S 1,…, S i,…, S m S 1 ’,…, S i ’,…, S m ’ R 1,…, R i,…, R m R i =  S j : S j ’ intersects S i ’ min S’ max i |R i |/|S i ’ | = complete hypergraph “best” 1.Limitations of this method 2.Local vs global

The “uniform” case Each task can be assigned to exactly s processors Trivial upper bound comp(n,s)  n/s greedy Cannot be improved unless CH-Conjecture fails

The “uniform” case Each task can be assigned to exactly s processors Cannot be improved unless CH-Conjecture fails all large

The “uniform” case Each task can be assigned to exactly s processors Cannot be improved unless CH-Conjecture fails

The “uniform” case Each task can be assigned to exactly s processors Cannot be improved unless CH-Conjecture fails high cost equivalent!

High cost d n-d Caccetta-Häggkvist Conjecture Every directed graph on n nodes and minimum outdegree d has a directed cycle of length at most n/d A directed graph on n nodes and minimum outdegree d no directed cycle of length at most s (n – n/s) n/sn/s s

High cost What are these algorithms?“Blind” algorithms “fixed” allocation

Conclusions Analyze “blind” algorithms –Diffult, interesting question Modified-greedy algos are “useless” for uniform instances Maybe a different view of the CH-Conjecture –Procedure ot check the conjecture?

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