an equivalent version of the caccetta-häggkvist conjecture in an online load balancing problem...
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An equivalent version of the Caccetta-Häggkvist conjecture in an online load balancing problem
Angelo Monti1, Paolo Penna2, Riccardo Silvestri1
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 worst8 tasks
Online load balancing
How good is greedy?
Example: linear topologies [Bar-Noy et al’99]
best worst4 tasks
Online load balancing
How good is greedy?
Example: linear topologies [Bar-Noy et al’99]
worst2 tasks
Online load balancing
How good is greedy?
Example: linear topologies [Bar-Noy et al’99]
worst1 task
(log n)-competitive
Online load balancing
modified-greedy
Example: linear topologies [Bar-Noy et al’99]
worst8 tasks4 tasks2 tasks1 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-greedy2. Simple local algorithm3. 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
S1,…, Si,…, Sm
S1’,…, Si
’,…, Sm’
R1,…, Ri,…, Rm
problem “structure”
Ri = Sj : Sj’ intersects Si
’
How good is modified-greedy?
maxi |Ri|/|Si’|
[Crescenzi et al’03]
The “uniform” case
How good is modified-greedy?
comp(n,s)
Each task can be assigned to exactly s processors
ApplyCrescenzi et al’03
to uniform case
S1,…, Si,…, Sm
S1’,…, Si
’,…, Sm’
R1,…, Ri,…, Rm
Ri = Sj : Sj’ intersects Si
’
minS’ maxi |Ri|/|Si’| =
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 ConjectureEvery directed graph on n nodes and minimum outdegree d has a directed cycle of length at most n/d
A directed graph on n nodesand minimum outdegree dno directed cycle of length at most s
(n – n/s)
n/ss
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?
Thank You