An equivalent version of the Caccetta-Häggkvist conjecture in an online load balancing problem...

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