1. Online Ranking for Tournament Graphs
Claire Mathieu and Adrian Vladu
Brown University
WAOA ’10

2. Online Rank Aggregation
A → B
means
A beats B
Online Rank Aggregation
1 upset 1 1 2 3 2 3 4
What strategy to use for ranking?
Try greedy: place each vertex where it produces the smallest number of upsets
Write the legend A->B means A beats B
Add numbers s.t. it’s clear that this is a ranking

3. Performance of greedy Theorem: Greedy is (n-2)-competitive n-2
Transition slide before this!!!
The full table of results should come before the technical part.

4. Transition slide before this!!!
The full table of results should come before the technical part.

5. Results Open problem: find a tight bound algorithm, vertex arrival
Feedback Arc Set
Maximum Acyclic Subgraph
upper bounds
lower bounds
deterministic, arbitrary order
n-2
1/2 [folklore]
1/2
randomized, arbitrary order
1-1/77
random order 3
[Ailon, Charikar, Newman ‘08]
1.25
PTAS
[Mathieu, Schudy ‘07]
Open problem:
find a tight bound
Reveal the results all at a time!!!

6. K-entangled Definition:
Examples of sets of vertices that are 3-entangled : l1 l2 l3 r1 r2 r3 l1 l2 l3 r1 r2 r3 l2 l1 l3 r1 r3 r2
OPT
Greedy ordering r1 l1 r2 l2 r3 l3 r1 r2 r3 l1 l2 l3 r1 r2 l1 r3 l2 l3
Definition:
Two sets L and R are k-entangled if
there are k vertices L = {l1, …,lk} and k vertices R= {r1, …,rk} with:
in OPT:
all of L is to the left of all of R
in greedy:
the ordering of L is l1, l2, …, lk
the ordering of R is r1, r2, …, rk
entanglement:
r1, r2,…, ri are to the left of li, li+1, …, lk
Draw left and right!!
Draw example where u1…uk are not in the same order in opt as in greedy

7. K-entangled OPT Greedy ordering {1,2,7} and {3,5,8} are 3-entangled9 4 8 3 5 6
OPT
Greedy ordering 3 1 8 7 5 4 2 6
Do it for k = 3
Revise statement of Lemma 1!!!
{1,2,7} and {3,5,8} are 3-entangled
Lemma 1. Converse holds: if vertex 9 produces at least 3 transpositions no matter where it’s inserted, then there are two 3-entangled sets L and R such that all of L is to the left of 9 in OPT and all of R is to the right of 9 in OPT
if L={1,2,7} and R={3,5,8} are 3-entangled then inserting 9, which in OPT has all of L to its left and all of R to its right, produces at least 3 transpositions

8. K-entangled Why is there a subset of vertices that is k-entangled?
Examples of sets of vertices that are 3-entangled: l1 l2 l3 r1 r2 r3 l1 l2 l3 r1 r2 r3 l2 l1 l3 r1 r3 r2
OPT
Greedy ordering r1 l1 r2 l2 r3 l3 r1 r2 r3 l1 l2 l3 r1 r2 l1 r3 l2 l3
Why is there a subset of vertices that is k-entangled?
Because OPT has upsets.
Draw left and right!!
Draw example where u1…uk are not in the same order in opt as in greedy
What’s the smallest number of upsets needed to get this?
We will show that this is at least k !!!
Lemma 2

9. Putting it all together
Lemma 1: if vertex v produces at least k transpositions no matter where it’s inserted, then there are two k-entangled sets
L and R ⊆ {1, …, v-1} s.t. all of L is to the left of v, all of R is to the right of v
Lemma 2: if there are two sets
L and R ⊆ {1, …, v-1} which are k-entangled then OPT|1..v-1 contains at least k upsets
Putting it all together
#new upsets in greedy ordering when adding v
#new upsets in OPT when adding v
min #transpositions produced by v at its insertion ≤ +
Lemma 2 2 1 4 3 5 6 7 8 9
Lemma 1 =
max k for which L to the left of v and R to the right of v are k-entangled ≤ ≤
#upsets in OPT before adding v
Theorem follows since new upsets are introduced at most n-2 times
Before the proof of lemma 2!!!
max k for which there are k-entangled subsets anywhere
#new upsets in greedy ordering when adding v
#new upsets in OPT when adding v
#upsets in OPT before adding v ≤ + ≤
#upsets in OPT

10. Lemma 2: if there are two sets
L and R ⊆ {1, …, v-1} which are k-entangled then OPT|1..v-1 contains at least k upsets
1. reduction step
OPT
Reduces the number of upsets in OPT needed to obtain two k-entangled sets
Greedy ordering
OPT
Greedy ordering
Add OPT and Greedy labels
OPT
Greedy ordering

11. Lemma 2: if there are two sets
L and R ⊆ {1, …, v-1} which are k-entangled then OPT|1..v-1 contains at least k upsets
2. matching scheme k k
Any of these greedy orderings is achieved only if OPT has at least k upsets
OPT
Greedy ordering
idea:
OPT
Greedy ordering
Write the statement of Lemma 2 somewhere in a corner or instead of the title.
OPT
Greedy ordering

12. Results algorithm, vertex arrival Feedback Arc Set
Maximum Acyclic Subgraph
upper bounds
lower bounds
deterministic, arbitrary order
n-2
1/2 [folklore]
1/2
randomized, arbitrary order
1-1/77
random order 3
[Ailon, Charikar, Newman ‘08]
1.25
PTAS
[Mathieu, Schudy ‘07]
Reveal the results all at a time!!!

13. Conclusions
K-entangledness: useful property for describing how the cost of the algorithm changes with each insertion
Randomness gives power:
crucial improvement from the arbitrary to the random order model
can the use of randomization help when the vertices arrive in arbitrary order?
Can this be generalized? Does this hierarchy apply for other problems?

14. Conclusion slide!!

15. Feedback Arc Set Deterministic lower bound Randomized lower bound
OPT = 1
ALG = n-2
OPT = 0
ALG = n-2
Randomized lower bound
same idea, use Yao’s MinMax Theorem

16. Feedback Arc Set Σ Bπ(v) = ALG Bσ(1) = 0 Upper bound Bσ(2) = 1
OPT 2 4 3 1
Bπ(1) = 0
Theorem follows since
Σ Bπ(v) = ALG
and
Σ Bσ(u) = OPT
Bπ(2) = 0
Bπ(3) = 1
Greedy ordering 3 4 1 2
Bπ(4) = 2
Lemma. Bπ(v) ≤ Σ Bσ(u)
u inserted before v