online matroid online matroid intersection: beating half...

OnlineMatroid

Intersection:Beating Halffor Random

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

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Online Matroid Intersection:Beating Half for Random Arrival

Sahil Singla ([email protected])Guru Prashanth Guruganesh ([email protected])

Carnegie Mellon University

26th June, 2017

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Outline

Introduction

Related Work

Bipartite Matching

Extensions

Open Problems

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

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OpenProblems

Edge arrival

I Bipartite graph: Intersection of partition matroids

u1 v1u1 v1

I Immediately & Irrevocably: Maximize size of matching

I greedy (pick an edge if possible): maximal matching12 ≤

ALGOPT : Competitive Ratio

I Better algo possible? Adversarial/Random arrival

OnlineMatroid

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Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Edge arrival

u1 v1u1 v1

OnlineMatroid

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Sahil, Guru

Introduction

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BipartiteMatching

Extensions

OpenProblems

Edge arrival

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Edge arrival

u1 v1u1 v1

OnlineMatroid

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Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Edge arrival

u1 v1u1 v1

OnlineMatroid

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Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Edge arrival

u1 v1u1 v1

OnlineMatroid

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Sahil, Guru

Introduction

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BipartiteMatching

Extensions

OpenProblems

Edge arrival

u1 v1u1 v1

OnlineMatroid

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Sahil, Guru

Introduction

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BipartiteMatching

Extensions

OpenProblems

Edge arrival

u1 v1u1 v1

OnlineMatroid

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Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Edge arrival

u1 v1u1 v1

OnlineMatroid

Arrival

Sahil, Guru

Introduction

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BipartiteMatching

Extensions

OpenProblems

Edge arrival

u1 v1u1 v1

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Edge arrival

u1 v1u1 v1

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Edge arrival

u1 v1u1 v1

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Edge arrival

u1 v1u1 v1

I greedy (pick an edge if possible): maximal matching

12 ≤

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Edge arrival

u1 v1u1 v1

OnlineMatroid

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Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Edge arrival

u1 v1u1 v1

I Better algo possible?

Adversarial/Random arrival

OnlineMatroid

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Sahil, Guru

Introduction

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BipartiteMatching

Extensions

OpenProblems

Edge arrival

u1 v1u1 v1

OnlineMatroid

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Sahil, Guru

Introduction

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BipartiteMatching

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OpenProblems

The Z graph

Q. Should we pick the first edge?

I Best deterministic is 12 -competitive (adversarial arrival)

I Select w.p. 23 . Gets 4

3 edges in expectation!

I Randomization adds power: E[ALG ]OPT Competitive Ratio

I Now, is better than 12 possible?

OnlineMatroid

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Sahil, Guru

Introduction

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BipartiteMatching

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OpenProblems

The Z graph

OnlineMatroid

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Sahil, Guru

Introduction

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OpenProblems

The Z graph

OnlineMatroid

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Sahil, Guru

Introduction

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OpenProblems

The Z graph

OnlineMatroid

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Sahil, Guru

Introduction

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OpenProblems

The Z graph

OnlineMatroid

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Sahil, Guru

Introduction

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OpenProblems

The Z graph

OnlineMatroid

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Sahil, Guru

Introduction

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The Z graph

OnlineMatroid

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Sahil, Guru

Introduction

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Online Matroid Intersection

I Two unknown matroids M1 = (E , I1) and M2 = (E , I2)

I Elements revealed one-by-one: Adversarial/Random arrival

I Matroids oracles only on the revealed elements

I Immediately & Irrevocably decide

I greedy (pick an element if possible) is 12 competitive

Theorem

There exists a (12

+ ε)-competitive algorithm when theelements are revealed in a random order, where ε > 10−5.

OnlineMatroid

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Introduction

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OpenProblems

Theorem

There exists a (12

OnlineMatroid

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Sahil, Guru

Introduction

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Extensions

OpenProblems

Theorem

There exists a (12

OnlineMatroid

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Sahil, Guru

Introduction

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Extensions

OpenProblems

Theorem

There exists a (12

OnlineMatroid

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Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Outline

Introduction

Related Work

Bipartite Matching

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Open Problems

OnlineMatroid

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Sahil, Guru

Introduction

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Comparison to Vertex Arrival

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

Vertex arriv Edge arriv

Random > 0.69

> 12 + ε & < 0.822

Adversarial ≈ 0.63

≥ 12 & < 0.572

1Karp-Vazirani-Vazirani STOC ’902Mahdian-Yan STOC ’113Wajc, Unpublished

OnlineMatroid

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Sahil, Guru

Introduction

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Extensions

OpenProblems

Random > 0.69

> 12 + ε & < 0.822

≥ 12 & < 0.572

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Random > 0.69

> 12 + ε & < 0.822

≥ 12 & < 0.572

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Random > 0.69

> 12 + ε & < 0.822

Adversarial ≈ 0.63 ≥ 12 & < 0.5723

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Random > 0.69 > 12 + ε & < 0.822

Adversarial ≈ 0.63 ≥ 12 & < 0.5723

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

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OpenProblems

Faster Algorithms

Offline Algorithms

I Linear time (1− ε)-approx max cardinality matching4

I Recent works give quadratic time (1− ε)-approx algos formax-weight matroid intersection5

I Our algorithm gives first linear time (1/2 + ε)-approx algofor max-cardinality matroid intersection

I Even for exact matroid intersection, only linear time lowerbounds known6

4Hopcroft-Karp SICOMP’735Chekuri-Quanrud, SODA’16 and Huang et al., SODA’166Harvey, SODA’08

OnlineMatroid

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Sahil, Guru

Introduction

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BipartiteMatching

Extensions

OpenProblems

Faster Algorithms

Offline Algorithms

OnlineMatroid

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Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Faster Algorithms

Offline Algorithms

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Faster Algorithms

Offline Algorithms

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Faster Algorithms

Offline Algorithms

OnlineMatroid

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Sahil, Guru

Introduction

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BipartiteMatching

Extensions

OpenProblems

Other Edge Arrival Models

I Edge Weighted Bipartite Matching(a) Maximize weight of matching(b) No constant approx possible for adversarial arrival(c) For random arrival, constant approx possible7

I Semi-Streaming Models(a) Decisions for O(n) edges can be postponed(b) For edge-weighted, 1/2− ε recently shown8

(c) For unweighted, 1/2 + ε known when edges arrive randomly9

7Korula-Pal, ICALP’09 and Kesselheim et al., ESA’138Paz-Schwartzman, SODA’179Konrad et al., APPROX’12

OnlineMatroid

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Sahil, Guru

Introduction

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BipartiteMatching

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Other Edge Arrival Models

I Edge Weighted Bipartite Matching(a) Maximize weight of matching(b) No constant approx possible for adversarial arrival(c) For random arrival, constant approx possible7

I Semi-Streaming Models(a) Decisions for O(n) edges can be postponed(b) For edge-weighted, 1/2− ε recently shown8

(c) For unweighted, 1/2 + ε known when edges arrive randomly9

7Korula-Pal, ICALP’09 and Kesselheim et al., ESA’138Paz-Schwartzman, SODA’179Konrad et al., APPROX’12

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Outline

Introduction

Related Work

Bipartite Matching

Extensions

Open Problems

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

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

OnlineMatroid

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Sahil, Guru

Introduction

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BipartiteMatching

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OpenProblems

I Thick-Z graph:

OnlineMatroid

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Sahil, Guru

Introduction

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I Thick-Z graph:

OnlineMatroid

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Sahil, Guru

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I Thick-Z graph:

OnlineMatroid

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Sahil, Guru

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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%)

I Now, E[Good ] ≥ (1/2 + ε)(1− ε) + 0 = 1/2 + ε/2− ε2and E[Bad ] ≥ 1/2(1− ε) + ε(1/2 + ε) = 1/2 + ε2.

OnlineMatroid

Arrival

Sahil, Guru

Introduction

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greedy ≥ 12 + ε (= 50.1%) ≥ 1

ALG ≥ 0 ≥ 12 + ε (= 50.1%)

OnlineMatroid

Arrival

Sahil, Guru

Introduction

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BipartiteMatching

Extensions

OpenProblems

greedy ≥ 12 + ε (= 50.1%) ≥ 1

ALG ≥ 0 ≥ 12 + ε (= 50.1%)

OnlineMatroid

Arrival

Sahil, Guru

Introduction

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BipartiteMatching

Extensions

OpenProblems

greedy ≥ 12 + ε (= 50.1%) ≥ 1

ALG ≥ 0 ≥ 12 + ε (= 50.1%)

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Prior work

I Hastiness Lemma [Konrad-Magniez-Mathieu10]: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%)

10Maximum matching in semi-streaming with few passes., APPROX ’12

OnlineMatroid

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Sahil, Guru

Introduction

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BipartiteMatching

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Prior work

I Hastiness Lemma [Konrad-Magniez-Mathieu10]: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%)

10Maximum matching in semi-streaming with few passes., APPROX ’12

OnlineMatroid

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Sahil, Guru

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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?

OnlineMatroid

Arrival

Sahil, Guru

Introduction

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Proof idea

I Suppose greedy for first 10% edges

– close to half

OnlineMatroid

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Sahil, Guru

Introduction

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BipartiteMatching

Extensions

OpenProblems

Proof idea

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

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Proof idea

I Would like to ‘mark’ some edges

and ‘augment’ them later

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Proof idea

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Proof idea

OnlineMatroid

Arrival

Sahil, Guru

Introduction

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OpenProblems

Two Phase 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?

OnlineMatroid

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

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(b) Try augmenting marked

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

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

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OpenProblems

OnlineMatroid

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Sahil, Guru

Introduction

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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

OnlineMatroid

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Sahil, Guru

Introduction

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BipartiteMatching

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OpenProblems

Random sampling

OnlineMatroid

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Sahil, Guru

Introduction

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BipartiteMatching

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OpenProblems

Random sampling

OnlineMatroid

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Sampling Lemma

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

OnlineMatroid

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Sampling Lemma

A. Yes, ≥ E[|S ′|](

11+0.2

OnlineMatroid

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OpenProblems

Sampling Lemma

A. Yes, ≥ E[|S ′|](

11+0.2

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Sampling Lemma

A. Yes, ≥ E[|S ′|](

11+0.2

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Sampling Lemma

A. Yes, ≥ E[|S ′|](

11+0.2

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Sampling Lemma

A. Yes, ≥ E[|S ′|](

11+0.2

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Sampling Lemma

A. Yes, ≥ E[|S ′|](

11+0.2

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Sampling Lemma

A. Yes, ≥ E[|S ′|](

11+0.2

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Outline

Introduction

Related Work

Bipartite Matching

Extensions

Open Problems

OnlineMatroid

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Sahil, Guru

Introduction

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BipartiteMatching

Extensions

OpenProblems

General Matching

Assume greedy is bad

I U denotes vertices matched by greedy (in Phase (a))

I Reduces to bipartite matching problem

OnlineMatroid

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Introduction

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BipartiteMatching

Extensions

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General Matching

Assume greedy is bad

I U denotes vertices matched by greedy (in Phase (a))

I Reduces to bipartite matching problem

OnlineMatroid

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Matroid Intersection

I Assume greedy is bad

I Extend Hastiness Lemma

I Run greedy with Marking in Phase (a):let Tf be the greedy and S be the picked elements

I In Phase (b):I Consider e only if in span of exactly one matroid, say

span1(Tf )I Pick only if e independent w.r.t. S in M1 and w.r.t. Tf

in M2, along with the newly picked elements.

I Extend Sampling Lemma

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

OnlineMatroid

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Sahil, Guru

Introduction

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BipartiteMatching

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OnlineMatroid

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Sahil, Guru

Introduction

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BipartiteMatching

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OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

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OpenProblems

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Outline

Introduction

Related Work

Bipartite Matching

Extensions

Open Problems

OnlineMatroid

Arrival

Sahil, Guru

Introduction

RelatedWork

BipartiteMatching

Extensions

OpenProblems

Open Problems

Question 1

Is there a linear time (1− ε)-approximation algorithm foroffline matroid intersection?

Question 2

Can we beat half for adversarial edge arrival?

Question 3

For OMI, can we “significantly” improve the (12

+ ε)-competitive ratio?

OnlineMatroid

Arrival

Sahil, Guru

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Open Problems

Question 1

Question 2

Question 3

OnlineMatroid

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Sahil, Guru

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Open Problems

Question 1

Question 2

Question 3

OnlineMatroid

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Conclusion

I Random edge arrivalI Showed ( 1

2 + ε)-approx for bipartite graphsI Use Hastiness Lemma and Sampling LemmaI Cannot do better than 0.822

I ExtensionsI General GraphsI Online Matroid Intersection

I Open problemsI Linear time (1− ε)-approx matroid intersection?I Can we beat half for adversarial edge arrival?

QUESTIONS?

OnlineMatroid

Arrival

Sahil, Guru

Introduction

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BipartiteMatching

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Conclusion

QUESTIONS?

OnlineMatroid

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Sahil, Guru

Introduction

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Conclusion

QUESTIONS?

OnlineMatroid

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Sahil, Guru

Introduction

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Conclusion

QUESTIONS?

online matroid online matroid intersection: beating half...

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