online matroid online matroid intersection: beating half ...ssingla/presentations/omi_30mins.pdf ·...

OnlineMatroid

Intersection:Beating Halffor Random

Arrival

Sahil, Guru

Introduction

Randomarrival

Conclusion

Online Matroid Intersection:Beating Half for Random Arrival

Sahil Singla (ssingla@cmu.edu)Guru Prashanth Guruganesh (ggurugan@cs.cmu.edu)

Carnegie Mellon University

9th October, 2015

OnlineMatching:

Beating Halffor RandomEdge Arrival

Sahil, Guru

Introduction

Randomarrival

Conclusion

Online Matching: Beating Half forRandom Edge Arrival

Sahil Singla (ssingla@cmu.edu)Guru Prashanth Guruganesh (ggurugan@cs.cmu.edu)

Carnegie Mellon University

9th October, 2015

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Outline

Introduction

Random arrival

Conclusion

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Vertex arrival

I Bipartite graph

I Immediately & Irrevocably

: Adversarial/ Random arrival

I Any maximal matching 12 approx.

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Vertex arrival

I Bipartite graph

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Vertex arrival

I Bipartite graph

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Vertex arrival

I Bipartite graph

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Vertex arrival

I Bipartite graph

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Vertex arrival

I Bipartite graph

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Vertex arrival

I Bipartite graph

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Vertex arrival

I Bipartite graph

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Vertex arrival

I Bipartite graph

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Vertex arrival

I Bipartite graph

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Vertex arrival

I Bipartite graph

I Immediately & Irrevocably: Adversarial/ Random arrival

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

The Z graph

Ques. What edge should we pick?

Ans. Randomly select one! Can get 0.75 in expectation

I Randomization adds power

I Better than 12 possible?

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

The Z graph

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

The Z graph

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

The Z graph

Ques. What edge should we pick?Ans. Randomly select one! Can get 0.75 in expectation

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

The Z graph

Ques. What edge should we pick?Ans. Randomly select one! Can get 0.75 in expectation

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

I Adversarial arrival (KVV algo.1): 1− 1e ≈ 0.63

(a) Give a random rank to {u1, u2, . . . , un}(b) Match vi to lowest available uj

I Random arrival (MY algo.2): > 0.69

1Karp-Vazirani-Vazirani STOC ’902Mahdian-Yan STOC ’11

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

I Adversarial arrival (KVV algo.1): 1− 1e ≈ 0.63

(a) Give a random rank to {u1, u2, . . . , un}(b) Match vi to lowest available uj

I Random arrival (MY algo.2): > 0.69

1Karp-Vazirani-Vazirani STOC ’902Mahdian-Yan STOC ’11

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Edge arrival

I Bipartite graph

u1 v1u1 v1

I Better algo possible?

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Edge arrival

I Bipartite graph

u1 v1u1 v1

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Edge arrival

I Bipartite graph

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Edge arrival

I Bipartite graph

u1 v1u1 v1

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Edge arrival

I Bipartite graph

u1 v1u1 v1

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Edge arrival

I Bipartite graph

u1 v1u1 v1

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Edge arrival

I Bipartite graph

u1 v1u1 v1

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Edge arrival

I Bipartite graph

u1 v1u1 v1

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Edge arrival

I Bipartite graph

u1 v1u1 v1

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Edge arrival

I Bipartite graph

u1 v1u1 v1

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Edge arrival

I Bipartite graph

u1 v1u1 v1

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Edge arrival

I Bipartite graph

u1 v1u1 v1

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Edge arrival

I Bipartite graph

u1 v1u1 v1

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Edge arrival

I Bipartite graph

u1 v1u1 v1

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Edge arrival

I Bipartite graph

u1 v1u1 v1

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Comparison of vertex and edge arrival

Vertex arriv Edge arriv

Random > 0.69

> 12 + ε′

Adversarial ≈ 0.63

I Remark: Adversarial edge arrival is more general thanadversarial vertex arrival

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Random > 0.69 > 12 + ε′

Adversarial ≈ 0.63

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Random > 0.69 > 12 + ε′

Adversarial ≈ 0.63 ??

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Outline

Introduction

Random arrival

Conclusion

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

greedy algorithm – random edge arrival

I greedy algorithm: Pick the edge if you can

I Thick-Z graph:

I Only 12 + o(1) approx – bad graph

I Regular graphs > 0.63 approx

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

I Thick-Z graph:

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

I Thick-Z graph:

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

I Thick-Z graph:

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Can assume greedy is bad

I Design ALG that gives 12 + ε for ‘bad’ graphs

Good graphs Bad Graphs

greedy ≥ 12 + ε (= 50.1%)

≥ 12

ALG ≥ 0 ≥ 12 + ε (= 50.1%)

I Run greedy w.p. 1− ε (= 99.9%)and ALG w.p. ε (= 0.1%)

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

greedy ≥ 12 + ε (= 50.1%) ≥ 1

ALG ≥ 0 ≥ 12 + ε (= 50.1%)

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

greedy ≥ 12 + ε (= 50.1%) ≥ 1

ALG ≥ 0 ≥ 12 + ε (= 50.1%)

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Prior work

I Konrad-Magniez-Mathieu3:If greedy is bad then whatever it picks, it picks quickly

If E[greedy (100%)] <1

2+ ε (50.1%)

then E[greedy (10%)] ≥ 1

2− 10ε (49%)

3Maximum matching in semi-streaming with few passes., APPROX ’12

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Prior work

I Konrad-Magniez-Mathieu3:If greedy is bad then whatever it picks, it picks quickly

If E[greedy (100%)] <1

2+ ε (50.1%)

then E[greedy (10%)] ≥ 1

2− 10ε (49%)

3Maximum matching in semi-streaming with few passes., APPROX ’12

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Proof idea

Assume we know greedy is bad

I Suppose greedy for first 10% edges

– close to half

I Would like to ‘mark’ some edges and ‘augment’ them later

I What edges are augmentable?

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Proof idea

I Suppose greedy for first 10% edges

– close to half

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Proof idea

I Suppose greedy for first 10% edges – close to half

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Proof idea

I Would like to ‘mark’ some edges

and ‘augment’ them later

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Proof idea

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Proof idea

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Algorithm ALG

(a) greedy for 10% edges

– but randomly mark 20%

(b) Try augmenting marked – For next 90% edgesRun greedy (U1,V1) and greedy (U2,V2)

I Augmentations kill each other?

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Algorithm ALG

(a) greedy for 10% edges – but randomly mark 20%

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Algorithm ALG

(b) Try augmenting marked

– For next 90% edgesRun greedy (U1,V1) and greedy (U2,V2)

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Algorithm ALG

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Algorithm ALG

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Random sampling

I Bip. graph (T ,S) with S-perfect matching

I S ′ ⊆ S with sampling prob 0.2

I E[greedy (T ,S ′)]: Better than E[|S ′|](12

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Random sampling

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Random sampling

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Random sampling works

Q. E[greedy (T ,S ′)]: Better than E[|S ′|](12

A. Yes, ≥ E[|S ′|](

11+0.2

I Note s2 marked w.p. only 0.2

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

A. Yes, ≥ E[|S ′|](

11+0.2

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

A. Yes, ≥ E[|S ′|](

11+0.2

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

A. Yes, ≥ E[|S ′|](

11+0.2

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

A. Yes, ≥ E[|S ′|](

11+0.2

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

A. Yes, ≥ E[|S ′|](

11+0.2

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

A. Yes, ≥ E[|S ′|](

11+0.2

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

A. Yes, ≥ E[|S ′|](

11+0.2

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Outline

Introduction

Random arrival

Conclusion

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

I Random edge arrivalI Showed 1

2 + ε′ approx. – bipartite graphs

I Extends to general graphsI Extends to online matroid intersection

I Adversarial edge arrivalI General than adversarial vertex arrivalI 0.591 upper bound4 on approx.

Open Problem

Can we beat half for adversarial edge arrival?

QUESTIONS?

4Epstein et al., STACS ’13

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

2 + ε′ approx. – bipartite graphsI Extends to general graphs

I Extends to online matroid intersection

Open Problem

QUESTIONS?

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

2 + ε′ approx. – bipartite graphsI Extends to general graphsI Extends to online matroid intersection

Open Problem

QUESTIONS?

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

I Adversarial edge arrivalI General than adversarial vertex arrival

I 0.591 upper bound4 on approx.

Open Problem

QUESTIONS?

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Open Problem

QUESTIONS?

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Open Problem

QUESTIONS?

OnlineMatching:

Sahil, Guru

Introduction

Randomarrival

Conclusion

Open Problem

QUESTIONS?

online matroid online matroid intersection: beating half ...ssingla/presentations/omi_30mins.pdf ·...

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