Download - How to Rank with Fewer Errors A PTAS for Minimum Feedback Arc Set in Tournaments Warren Schudy Claire Mathieu, Warren Schudy Brown University Thanks to:

How to Rank with Fewer Errors

A PTAS for Minimum Feedback Arc Set in Tournaments

Claire Mathieu, Warren Warren SchudySchudy

Brown University

Thanks to:Nir Ailon, Marek Karpinski, and Eli Upfal (for useful discussions)

Cora Borradaile, Aparna Das, Micha Elsner, Dave McClosky, Fabio Vandin, and Matt Wronka (for useful comments on practice talks)

Feedback arc set in Feedback arc set in tournamentstournaments

•FAS problem: minimize number of upsets

•NP-hard [Ailon Charikar Newman ’05, Alon ’06, Charbit Thomassè Yeo ‘07]

•Applications– Ranking by pairwise comparisons– Kemeny rank aggregation

A

B

C DA B C D

1 upset

(Animating…)

Best … Worst

Algorithms for FAS-TAlgorithms for FAS-T

• Slater and Alway 1961– Heuristic algorithm

• Fernandez de la Vega ’96, Arora Frieze Kaplan ’96 and many others– small additive error

• Ailon Charikar Newman ‘05– Quicksort 3 approx– LP + quicksort 2.5 approx

• Coppersmith Fleischer Rudra ‘06– Sort by Wins 5-approx

• Our Main Result: a PTAS for FAS-T:

OPTCOST 1

OPTnCOST 2

OPTCOST 3

Aside: FAS in general Aside: FAS in general graphsgraphs

•Log n log log n approx [Seymour ‘95, Even, Noar, Rao, Schieber ‘95]

•At least as hard as vertex cover, which cannot be approximated better than 1.36 [Karp ’72, Dinur ‘02]

A B C D

OutlineOutline

• Intro

•Algorithm

•Analysis

•Application (Kemeny Rank Aggregation)

Towards our algorithm (1/6)Towards our algorithm (1/6)• An easy instance: when there is an

ordering with zero upsets

B D A C

1 win

Algorithm 1: Sort by Wins5-approx [CFR]

A

B

C

D

3 wins

0 wins

2 wins

This means David beat

Charlie

23 1 0Wins:

Towards our algorithm (2/6)Towards our algorithm (2/6)

• A difficult instance for sort by wins

• Cost Θ(n²), so additive approx Θ(εn²) works!• Algorithm 2: return better of additive

approximation and sort by wins

Sort by Wins

A B C D2 2 2 2

E2

E D C B2 2 2 2

A2

OPT


• A hard instance for Algorithm 2:

• Algorithm 3: Sort by wins. Recursively divide in two until Cost=Ω(n²), then run additive approximation

E F G HA B C D I J7 4 3 17 7 7 7 1 1AE D C B F G J I H

Sorted by wins:

Additive Approx

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• Algorithm 4: like Algorithm 3 but divide into two parts of random sizes

E F G HA B C D I J5 4 3 28 8 7 6 1 1

Uh-oh, we just “committed” ourselves to paying 3 but

OPT=2

• A hard instance for algorithm 3:

Sorted by wins:

(Animating…)


E F G HA B C D I J

OPTn 5/

K L M N OSorted by wins

Uh-oh, with constant probability we commit to paying Θ(n)

(Animating…)

E F GA B C D I J

5/n

K L M N OHH

• Final ingredient: after sorting by wins do single vertex move local optimization


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Our algorithmOur algorithmAlgorithm:• Sort by Wins• Single vertex moves• Result ordering: • ImproveRec(all)

ImproveRec(vertex set S):If (Cost(S, )≤Θ(ε²|S|²))

Choose k at random between 1 and |S|Recurse on the first k vertices of SRecurse on the other vertices

elseRun additive approx. Θ(ε³|S|²) on S

E F G HA B C D I J6 3 3 37 7 7 7 2 0AE D C B

Etc…

loc

loc

(Animating…)

OutlineOutline

• Intro

•Algorithm

•Analysis


Proof planProof plan• After first divide step “commit” to putting some

vertices before others. For talk, will show there is a good ordering “consistent” with the first divide step

• Therefore

verticesof pairsorder -second #)()( :1 Lemma * OCC good

2

)()(pairsorder -second # :2 Lemma

n

CCO

locloc E

)()(

)()(

)()(

*

2*

COCO

Cn

COCC

loc

locloc

good

E

good

Defined later Lemmas

Non-Stopping condition Constant-factor approx

Example 1Example 1

BB CC FF DD EE

loc :ImproveRec Input to

* :Optimal

AA BB CC

good

locA

locA AA ))(( *

kk

DD EE FF

BB CC FF DD AA EE

AA

BB CC DD AA EE FF

1 so

F, extrabut S.V.M. Same* CC good

For this section of talk, “align” permutations.

Ignore this detail.

otherwise)(

is if)()(

* u

splituuu

locgood

)( and )(between is if is where

*loc uuksplitu

It is cheaper to put A to the left of B,C,D than to

the right

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Example 2Example 2

loc

*

good

kk

AA BB CC DD EE

EE AA CC BB DD

AA BB CC DD EE

'good EE AA BB CC DD)'C()C( and

)C()'C( implies

of optimality Local

good*

goodgood

loc

Example 3Example 3

loc

*

good

kk

AA BB CC DD EE FF

BB EE CC AA DD FF

AA BB EE CC DD FF 1)C()C( implies of optimality Local

*good loc

Second order pairsSecond order pairs

Ex. 1

BB CC FF DD AA EE

AA BB CC DD EE FF

AA BB CC DD EE

EE AA CC BB DD

AA BB CC DD EE FF

BB EE CC AA DD FF

Ex. 2

Ex. 3

loc

*

loc*

loc

*

1 Recall * CC good

* Recall CC good

1 Recall * CC good

AFAF alternating, A split-> {A, F} 2nd order pair

EBBE not alternating-> {B, E} not 2nd order pair

AEAE alternating, A split-> {A, E} 2nd order pair

AA BB CC DD EE FF

BB EE CC AA DD FF

Ex. 3’ loc* * good

AEAE alternating, neither split-> {A, E} not 2nd order pair

Definition: {u,v} are second order pair if u and v are alternating and at least one of them is split

are the root of all evilare the root of all evil errorerror

Bound this

Bounding (1/2)Bounding (1/2)

TCC

πCπC

ππ ππππ

locE

locB

good

loclocE

loclocB

*good

locloc

*

C C

suggestswhich

notation Abusing

loc

*

good

AA BB CC DD EE

EE AA CC BB DD

AA BB CC DD EE

AA CC BB DD EElocB

EE AA BB CC DDlocE

0Csplit

Csplit Csplit :generalIn

loc

,

locloc

u vuv

locuuv

u vwwv

locuwv

u

locu

Cu

CuCuT

1

11

uvlocvuv

loc

vuuv

locuuv

locuv

*uv

good

*good*good

πCπCv

πCπCuπCπC

TπCπCπCπC

split

split

)optimality local(by

,

1

1

(Example 2 again))()( * CC good

BB CC EE AA DD

BB AA EE CC DD

AA BB CC DD EE

AA CC BB DD EE

BB CC AA DD EE

uv

locvuv

loc

vuuv

locuuv

locuv

*uv

good

*good

πCπCvπCπCuπCπC

πCπC

split split

slide previous From

,

11

• If {u,v} second-order pair, bound by 3

• Otherwise, show zero by cases:– e.g. {E,D} (Neither split)

• Proof: by def’ns and

– e.g. {C,D} (u<u<v<v pattern)• Proof: edge {D,C} oriented same

way in all orderings so terms cancel

– e.g. {C,A} (u<v<v<u pattern)• Proof:

and

loc

*

EDED ,*

,good )C()C(

0split split DE 11

good

CACACACA ,locA,

loc,

*,

good )C()C()C()C(

locC

locA

AClocCAC

loc πCπC

Bounding (2/2)Bounding (2/2))()( * CC good

(Animating…)

OutlineOutline

• Intro

•Algorithm

•Analysis–

–


pairsorder -2 #)()( :1 Lemma nd* OCC good

2


n

CCO

locloc E

Bounding (1/2)Bounding (1/2)

•Def’n: u

loc uuF *

loc AA BB CC DD EE

BB CC EE AA DD*

21113 F

galternatin : #split Prpairsorder second # u,vvuuE

n

uuu

loc |)()(|split Pr

*

paper) (see galternatin : # FOu,vv

n

FFOFO

n

uu

u

loc |)()(|pairsorder second #

*E

pairs s.o. #E

•Claim: •Proof:

– [Diaconis and Graham ’77]:

– If the pair are in different orders, at least one of the orders has to pay for the pair(This is where the restriction to tournament graphs helps)

)(4)(2)(2 * locloc CCCF

} and in ordersdifferent in and :},{{2* locyxyxF

loc AA BB CC DD EE

BB CC EE AA DD*Bounding (2/2)Bounding (2/2) pairs s.o. #E

Reminder: overall proofReminder: overall proof

verticesof pairsorder -second #)()( :1 Lemma * CC good

2


n

CCO

locloc E

)()(

)()(

)()(

*

2*

COCO

Cn

COCC

loc

locloc

good

E

Lemmas

Non-Stopping condition Constant-factor approx

Questions on proof?

OutlineOutline

• Intro

•Algorithm

•Analysis


An application to votingAn application to voting• Kemeny Rank Aggregation:

(1959) – Voters submit rankings of candidates

– Translate rankings into tournaments

– Add those tournaments together

– Find feedback arc set of resulting weighted tourney

– Nice properties, e.g. ranks Condorcet winner first [Young & Levenglick ’78, Young ‘95]

• Our PTAS generalizes to this!

A>B>C

A

B

C

C>A>B

A

B

C

A>C>B

A

B

C

A

B

C2

1

2103

A BC2

1

2

10

3

Open QuestionOpen Question•Real rankings often have ties,

e.g. restaurant guides with ratings 1-5

•Exists 1.5-approx [Ailon 07]; is there a PTAS?

A

BC

A: 5 C: 4B: 5 D: 3

D