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OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
1/15
Online Matroid Intersection:Beating Half for Random Arrival
Sahil Singla ([email protected])Guru Prashanth Guruganesh ([email protected])
Carnegie Mellon University
9th October, 2015
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
2/15
Online Matching: Beating Half forRandom Edge Arrival
Sahil Singla ([email protected])Guru Prashanth Guruganesh ([email protected])
Carnegie Mellon University
9th October, 2015
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
2/15
Outline
Introduction
Random arrival
Conclusion
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
3/15
Vertex arrival
I Bipartite graph
u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v2
u1
u2
u3
u4
v2
u1
u2
u3
u4
v3
v2
u1
u2
u3
u4 v4
v2
u1
u2
u3
u4 v4
u1
u2
u3
u4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
3/15
Vertex arrival
I Bipartite graph
u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v2
u1
u2
u3
u4
v2
u1
u2
u3
u4
v3
v2
u1
u2
u3
u4 v4
v2
u1
u2
u3
u4 v4
u1
u2
u3
u4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
3/15
Vertex arrival
I Bipartite graph
u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v2
u1
u2
u3
u4
v2
u1
u2
u3
u4
v3
v2
u1
u2
u3
u4 v4
v2
u1
u2
u3
u4 v4
u1
u2
u3
u4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
3/15
Vertex arrival
I Bipartite graph
u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v2
u1
u2
u3
u4
v2
u1
u2
u3
u4
v3
v2
u1
u2
u3
u4 v4
v2
u1
u2
u3
u4 v4
u1
u2
u3
u4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
3/15
Vertex arrival
I Bipartite graph
u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v2
u1
u2
u3
u4
v2
u1
u2
u3
u4
v3
v2
u1
u2
u3
u4 v4
v2
u1
u2
u3
u4 v4
u1
u2
u3
u4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
3/15
Vertex arrival
I Bipartite graph
u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v2
u1
u2
u3
u4
v2
u1
u2
u3
u4
v3
v2
u1
u2
u3
u4
v4
v2
u1
u2
u3
u4 v4
u1
u2
u3
u4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
3/15
Vertex arrival
I Bipartite graph
u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v2
u1
u2
u3
u4
v2
u1
u2
u3
u4
v3
v2
u1
u2
u3
u4 v4
v2
u1
u2
u3
u4
v4
u1
u2
u3
u4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
3/15
Vertex arrival
I Bipartite graph
u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v2
u1
u2
u3
u4
v2
u1
u2
u3
u4
v3
v2
u1
u2
u3
u4 v4
v2
u1
u2
u3
u4 v4
u1
u2
u3
u4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
3/15
Vertex arrival
I Bipartite graph
u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v2
u1
u2
u3
u4
v2
u1
u2
u3
u4
v3
v2
u1
u2
u3
u4 v4
v2
u1
u2
u3
u4 v4
u1
u2
u3
u4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
3/15
Vertex arrival
I Bipartite graph
u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v2
u1
u2
u3
u4
v2
u1
u2
u3
u4
v3
v2
u1
u2
u3
u4 v4
v2
u1
u2
u3
u4 v4
u1
u2
u3
u4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
3/15
Vertex arrival
I Bipartite graph
u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v1u1
u2
u3
u4
v2
u1
u2
u3
u4
v2
u1
u2
u3
u4
v3
v2
u1
u2
u3
u4 v4
v2
u1
u2
u3
u4 v4
u1
u2
u3
u4
I Immediately & Irrevocably: Adversarial/ Random arrival
I Any maximal matching 12 approx.
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
4/15
The Z graph
u1
u2
v1
u2
v1
u2 v2
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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
4/15
The Z graph
u1
u2
v1
u2
v1
u2 v2
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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
4/15
The Z graph
u1
u2
v1
u2
v1
u2 v2
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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
4/15
The Z graph
u1
u2
v1
u2
v1
u2 v2
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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
4/15
The Z graph
u1
u2
v1
u2
v1
u2 v2
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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
5/15
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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
5/15
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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
6/15
Edge arrival
I Bipartite graph
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1u1 v1
u2
v1
u4
v2
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
I Immediately & Irrevocably: Adversarial/ Random arrival
I Any maximal matching 12 approx.
I Better algo possible?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
6/15
Edge arrival
I Bipartite graph
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1u1 v1
u2
v1
u4
v2
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
I Immediately & Irrevocably: Adversarial/ Random arrival
I Any maximal matching 12 approx.
I Better algo possible?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
6/15
Edge arrival
I Bipartite graph
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u1 v1
u2
v1
u4
v2
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
I Better algo possible?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
6/15
Edge arrival
I Bipartite graph
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1u1 v1
u2
v1
u4
v2
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
I Better algo possible?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
6/15
Edge arrival
I Bipartite graph
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1u1 v1
u2
v1
u4
v2
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
I Better algo possible?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
6/15
Edge arrival
I Bipartite graph
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1u1 v1
u2
v1
u4
v2
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
I Better algo possible?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
6/15
Edge arrival
I Bipartite graph
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1u1 v1
u2
v1
u4
v2
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
I Better algo possible?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
6/15
Edge arrival
I Bipartite graph
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1u1 v1
u2
v1
u4
v2
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
I Better algo possible?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
6/15
Edge arrival
I Bipartite graph
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1u1 v1
u2
v1
u4
v2
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
I Better algo possible?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
6/15
Edge arrival
I Bipartite graph
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1u1 v1
u2
v1
u4
v2
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
I Better algo possible?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
6/15
Edge arrival
I Bipartite graph
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1u1 v1
u2
v1
u4
v2
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
I Better algo possible?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
6/15
Edge arrival
I Bipartite graph
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1u1 v1
u2
v1
u4
v2
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
I Better algo possible?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
6/15
Edge arrival
I Bipartite graph
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1u1 v1
u2
v1
u4
v2
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
I Immediately & Irrevocably
: Adversarial/ Random arrival
I Any maximal matching 12 approx.
I Better algo possible?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
6/15
Edge arrival
I Bipartite graph
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1u1 v1
u2
v1
u4
v2
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
I Immediately & Irrevocably: Adversarial/ Random arrival
I Any maximal matching 12 approx.
I Better algo possible?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
6/15
Edge arrival
I Bipartite graph
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1u1 v1
u2
v1
u4
v2
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
u1 v1
u2 v2
u3 v3
u4 v4
I Immediately & Irrevocably: Adversarial/ Random arrival
I Any maximal matching 12 approx.
I Better algo possible?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
7/15
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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
7/15
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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
7/15
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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
7/15
Outline
Introduction
Random arrival
Conclusion
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
8/15
greedy algorithm – random edge arrival
I greedy algorithm: Pick the edge if you can
I Thick-Z graph:
U1
U2
V1
V2
I Only 12 + o(1) approx – bad graph
I Regular graphs > 0.63 approx
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
8/15
greedy algorithm – random edge arrival
I greedy algorithm: Pick the edge if you can
I Thick-Z graph:
U1
U2
V1
V2
I Only 12 + o(1) approx – bad graph
I Regular graphs > 0.63 approx
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
8/15
greedy algorithm – random edge arrival
I greedy algorithm: Pick the edge if you can
I Thick-Z graph:
U1
U2
V1
V2
I Only 12 + o(1) approx – bad graph
I Regular graphs > 0.63 approx
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
8/15
greedy algorithm – random edge arrival
I greedy algorithm: Pick the edge if you can
I Thick-Z graph:
U1
U2
V1
V2
I Only 12 + o(1) approx – bad graph
I Regular graphs > 0.63 approx
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
9/15
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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
9/15
Can assume greedy is bad
I Design ALG that gives 12 + ε for ‘bad’ graphs
Good graphs Bad Graphs
greedy ≥ 12 + ε (= 50.1%) ≥ 1
2
ALG ≥ 0 ≥ 12 + ε (= 50.1%)
I Run greedy w.p. 1− ε (= 99.9%)and ALG w.p. ε (= 0.1%)
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
9/15
Can assume greedy is bad
I Design ALG that gives 12 + ε for ‘bad’ graphs
Good graphs Bad Graphs
greedy ≥ 12 + ε (= 50.1%) ≥ 1
2
ALG ≥ 0 ≥ 12 + ε (= 50.1%)
I Run greedy w.p. 1− ε (= 99.9%)and ALG w.p. ε (= 0.1%)
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
10/15
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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
10/15
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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
11/15
Proof idea
Assume we know greedy is bad
I Suppose greedy for first 10% edges
– close to half
U1
U2
V1
V2
I Would like to ‘mark’ some edges and ‘augment’ them later
I What edges are augmentable?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
11/15
Proof idea
Assume we know greedy is bad
I Suppose greedy for first 10% edges
– close to half
U1
U2
V1
V2
I Would like to ‘mark’ some edges and ‘augment’ them later
I What edges are augmentable?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
11/15
Proof idea
Assume we know greedy is bad
I Suppose greedy for first 10% edges – close to half
U1
U2
V1
V2
I Would like to ‘mark’ some edges and ‘augment’ them later
I What edges are augmentable?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
11/15
Proof idea
Assume we know greedy is bad
I Suppose greedy for first 10% edges – close to half
U1
U2
V1
V2
I Would like to ‘mark’ some edges
and ‘augment’ them later
I What edges are augmentable?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
11/15
Proof idea
Assume we know greedy is bad
I Suppose greedy for first 10% edges – close to half
U1
U2
V1
V2
I Would like to ‘mark’ some edges and ‘augment’ them later
I What edges are augmentable?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
11/15
Proof idea
Assume we know greedy is bad
I Suppose greedy for first 10% edges – close to half
U1
U2
V1
V2
I Would like to ‘mark’ some edges and ‘augment’ them later
I What edges are augmentable?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
12/15
Algorithm ALG
(a) greedy for 10% edges
– but randomly mark 20%
U1
U2
V1
V2
(b) Try augmenting marked – For next 90% edgesRun greedy (U1,V1) and greedy (U2,V2)
I Augmentations kill each other?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
12/15
Algorithm ALG
(a) greedy for 10% edges – but randomly mark 20%
U1
U2
V1
V2
(b) Try augmenting marked – For next 90% edgesRun greedy (U1,V1) and greedy (U2,V2)
I Augmentations kill each other?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
12/15
Algorithm ALG
(a) greedy for 10% edges – but randomly mark 20%
U1
U2
V1
V2
(b) Try augmenting marked
– For next 90% edgesRun greedy (U1,V1) and greedy (U2,V2)
I Augmentations kill each other?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
12/15
Algorithm ALG
(a) greedy for 10% edges – but randomly mark 20%
U1
U2
V1
V2
(b) Try augmenting marked – For next 90% edgesRun greedy (U1,V1) and greedy (U2,V2)
I Augmentations kill each other?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
12/15
Algorithm ALG
(a) greedy for 10% edges – but randomly mark 20%
U1
U2
V1
V2
(b) Try augmenting marked – For next 90% edgesRun greedy (U1,V1) and greedy (U2,V2)
I Augmentations kill each other?
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
13/15
Random sampling
S ′
T S
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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
13/15
Random sampling
S ′
T S
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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
13/15
Random sampling
S ′
T S
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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
14/15
Random sampling works
Q. E[greedy (T ,S ′)]: Better than E[|S ′|](12
)?
A. Yes, ≥ E[|S ′|](
11+0.2
)
t1 s1
t2 s2
t3 s3
t4
s4
T S
s1
I Note s2 marked w.p. only 0.2
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
14/15
Random sampling works
Q. E[greedy (T ,S ′)]: Better than E[|S ′|](12
)?
A. Yes, ≥ E[|S ′|](
11+0.2
)
t1 s1
t2 s2
t3 s3
t4
s4
T S
s1
I Note s2 marked w.p. only 0.2
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
14/15
Random sampling works
Q. E[greedy (T ,S ′)]: Better than E[|S ′|](12
)?
A. Yes, ≥ E[|S ′|](
11+0.2
)
t1 s1
t2 s2
t3 s3
t4 s4
T S
s1
I Note s2 marked w.p. only 0.2
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
14/15
Random sampling works
Q. E[greedy (T ,S ′)]: Better than E[|S ′|](12
)?
A. Yes, ≥ E[|S ′|](
11+0.2
)
t1 s1
t2 s2
t3 s3
t4 s4
T S
s1
I Note s2 marked w.p. only 0.2
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
14/15
Random sampling works
Q. E[greedy (T ,S ′)]: Better than E[|S ′|](12
)?
A. Yes, ≥ E[|S ′|](
11+0.2
)
t1 s1
t2 s2
t3 s3
t4
s4
T S
s1
I Note s2 marked w.p. only 0.2
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
14/15
Random sampling works
Q. E[greedy (T ,S ′)]: Better than E[|S ′|](12
)?
A. Yes, ≥ E[|S ′|](
11+0.2
)
t1 s1
t2 s2
t3 s3
t4
s4
T S
s1
I Note s2 marked w.p. only 0.2
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
14/15
Random sampling works
Q. E[greedy (T ,S ′)]: Better than E[|S ′|](12
)?
A. Yes, ≥ E[|S ′|](
11+0.2
)
t1 s1
t2 s2
t3 s3
t4
s4
T S
s1
I Note s2 marked w.p. only 0.2
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
14/15
Random sampling works
Q. E[greedy (T ,S ′)]: Better than E[|S ′|](12
)?
A. Yes, ≥ E[|S ′|](
11+0.2
)
t1 s1
t2 s2
t3 s3
t4
s4
T S
s1
I Note s2 marked w.p. only 0.2
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
14/15
Outline
Introduction
Random arrival
Conclusion
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
15/15
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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
15/15
Conclusion
I Random edge arrivalI Showed 1
2 + ε′ approx. – bipartite graphsI Extends to general graphs
I 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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
15/15
Conclusion
I Random edge arrivalI Showed 1
2 + ε′ approx. – bipartite graphsI 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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
15/15
Conclusion
I Random edge arrivalI Showed 1
2 + ε′ approx. – bipartite graphsI Extends to general graphsI Extends to online matroid intersection
I Adversarial edge arrivalI General than adversarial vertex arrival
I 0.591 upper bound4 on approx.
Open Problem
Can we beat half for adversarial edge arrival?
QUESTIONS?
4Epstein et al., STACS ’13
OnlineMatching:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
15/15
Conclusion
I Random edge arrivalI Showed 1
2 + ε′ approx. – bipartite graphsI 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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
15/15
Conclusion
I Random edge arrivalI Showed 1
2 + ε′ approx. – bipartite graphsI 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:
Beating Halffor RandomEdge Arrival
Sahil, Guru
Introduction
Randomarrival
Conclusion
15/15
Conclusion
I Random edge arrivalI Showed 1
2 + ε′ approx. – bipartite graphsI 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
