Online Matroid Online Matroid Intersection: Beating Half...
Transcript of Online Matroid Online Matroid Intersection: Beating Half...
![Page 1: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/1.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
1/18
Online Matroid Intersection:Beating Half for Random Arrival
Sahil Singla ([email protected])Guru Prashanth Guruganesh ([email protected])
Carnegie Mellon University
26th June, 2017
![Page 2: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/2.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
1/18
Outline
Introduction
Related Work
Bipartite Matching
Extensions
Open Problems
![Page 3: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/3.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
2/18
Edge arrival
I Bipartite graph: Intersection of partition matroids
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: Maximize size of matching
I greedy (pick an edge if possible): maximal matching12 ≤
ALGOPT : Competitive Ratio
I Better algo possible? Adversarial/Random arrival
![Page 4: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/4.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
2/18
Edge arrival
I Bipartite graph: Intersection of partition matroids
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: Maximize size of matching
I greedy (pick an edge if possible): maximal matching12 ≤
ALGOPT : Competitive Ratio
I Better algo possible? Adversarial/Random arrival
![Page 5: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/5.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
2/18
Edge arrival
I Bipartite graph: Intersection of partition matroids
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: Maximize size of matching
I greedy (pick an edge if possible): maximal matching12 ≤
ALGOPT : Competitive Ratio
I Better algo possible? Adversarial/Random arrival
![Page 6: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/6.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
2/18
Edge arrival
I Bipartite graph: Intersection of partition matroids
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: Maximize size of matching
I greedy (pick an edge if possible): maximal matching12 ≤
ALGOPT : Competitive Ratio
I Better algo possible? Adversarial/Random arrival
![Page 7: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/7.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
2/18
Edge arrival
I Bipartite graph: Intersection of partition matroids
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: Maximize size of matching
I greedy (pick an edge if possible): maximal matching12 ≤
ALGOPT : Competitive Ratio
I Better algo possible? Adversarial/Random arrival
![Page 8: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/8.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
2/18
Edge arrival
I Bipartite graph: Intersection of partition matroids
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: Maximize size of matching
I greedy (pick an edge if possible): maximal matching12 ≤
ALGOPT : Competitive Ratio
I Better algo possible? Adversarial/Random arrival
![Page 9: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/9.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
2/18
Edge arrival
I Bipartite graph: Intersection of partition matroids
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: Maximize size of matching
I greedy (pick an edge if possible): maximal matching12 ≤
ALGOPT : Competitive Ratio
I Better algo possible? Adversarial/Random arrival
![Page 10: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/10.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
2/18
Edge arrival
I Bipartite graph: Intersection of partition matroids
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: Maximize size of matching
I greedy (pick an edge if possible): maximal matching12 ≤
ALGOPT : Competitive Ratio
I Better algo possible? Adversarial/Random arrival
![Page 11: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/11.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
2/18
Edge arrival
I Bipartite graph: Intersection of partition matroids
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: Maximize size of matching
I greedy (pick an edge if possible): maximal matching12 ≤
ALGOPT : Competitive Ratio
I Better algo possible? Adversarial/Random arrival
![Page 12: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/12.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
2/18
Edge arrival
I Bipartite graph: Intersection of partition matroids
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: Maximize size of matching
I greedy (pick an edge if possible): maximal matching12 ≤
ALGOPT : Competitive Ratio
I Better algo possible? Adversarial/Random arrival
![Page 13: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/13.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
2/18
Edge arrival
I Bipartite graph: Intersection of partition matroids
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: Maximize size of matching
I greedy (pick an edge if possible): maximal matching12 ≤
ALGOPT : Competitive Ratio
I Better algo possible? Adversarial/Random arrival
![Page 14: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/14.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
2/18
Edge arrival
I Bipartite graph: Intersection of partition matroids
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: Maximize size of matching
I greedy (pick an edge if possible): maximal matching12 ≤
ALGOPT : Competitive Ratio
I Better algo possible? Adversarial/Random arrival
![Page 15: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/15.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
2/18
Edge arrival
I Bipartite graph: Intersection of partition matroids
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: Maximize size of matching
I greedy (pick an edge if possible): maximal matching
12 ≤
ALGOPT : Competitive Ratio
I Better algo possible? Adversarial/Random arrival
![Page 16: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/16.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
2/18
Edge arrival
I Bipartite graph: Intersection of partition matroids
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: Maximize size of matching
I greedy (pick an edge if possible): maximal matching12 ≤
ALGOPT : Competitive Ratio
I Better algo possible? Adversarial/Random arrival
![Page 17: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/17.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
2/18
Edge arrival
I Bipartite graph: Intersection of partition matroids
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: Maximize size of matching
I greedy (pick an edge if possible): maximal matching12 ≤
ALGOPT : Competitive Ratio
I Better algo possible?
Adversarial/Random arrival
![Page 18: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/18.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
2/18
Edge arrival
I Bipartite graph: Intersection of partition matroids
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: Maximize size of matching
I greedy (pick an edge if possible): maximal matching12 ≤
ALGOPT : Competitive Ratio
I Better algo possible? Adversarial/Random arrival
![Page 19: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/19.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
3/18
The Z graph
u1
u2
v1
v2
u2
v1u1
u2
v1
v2
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?
![Page 20: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/20.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
3/18
The Z graph
u1
u2
v1
v2u2
v1
u1
u2
v1
v2
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?
![Page 21: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/21.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
3/18
The Z graph
u1
u2
v1
v2u2
v1u1
u2
v1
v2
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?
![Page 22: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/22.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
3/18
The Z graph
u1
u2
v1
v2u2
v1u1
u2
v1
v2
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?
![Page 23: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/23.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
3/18
The Z graph
u1
u2
v1
v2u2
v1u1
u2
v1
v2
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?
![Page 24: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/24.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
3/18
The Z graph
u1
u2
v1
v2u2
v1u1
u2
v1
v2
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?
![Page 25: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/25.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
3/18
The Z graph
u1
u2
v1
v2u2
v1u1
u2
v1
v2
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?
![Page 26: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/26.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
4/18
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
I Better algo possible?
Theorem
There exists a (12
+ ε)-competitive algorithm when theelements are revealed in a random order, where ε > 10−5.
![Page 27: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/27.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
4/18
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
I Better algo possible?
Theorem
There exists a (12
+ ε)-competitive algorithm when theelements are revealed in a random order, where ε > 10−5.
![Page 28: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/28.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
4/18
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
I Better algo possible?
Theorem
There exists a (12
+ ε)-competitive algorithm when theelements are revealed in a random order, where ε > 10−5.
![Page 29: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/29.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
4/18
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
I Better algo possible?
Theorem
There exists a (12
+ ε)-competitive algorithm when theelements are revealed in a random order, where ε > 10−5.
![Page 30: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/30.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
4/18
Outline
Introduction
Related Work
Bipartite Matching
Extensions
Open Problems
![Page 31: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/31.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
5/18
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
3
1Karp-Vazirani-Vazirani STOC ’902Mahdian-Yan STOC ’113Wajc, Unpublished
![Page 32: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/32.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
5/18
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
3
1Karp-Vazirani-Vazirani STOC ’902Mahdian-Yan STOC ’113Wajc, Unpublished
![Page 33: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/33.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
5/18
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
3
1Karp-Vazirani-Vazirani STOC ’902Mahdian-Yan STOC ’113Wajc, Unpublished
![Page 34: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/34.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
5/18
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 & < 0.5723
1Karp-Vazirani-Vazirani STOC ’902Mahdian-Yan STOC ’113Wajc, Unpublished
![Page 35: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/35.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
5/18
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 & < 0.5723
1Karp-Vazirani-Vazirani STOC ’902Mahdian-Yan STOC ’113Wajc, Unpublished
![Page 36: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/36.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
6/18
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
![Page 37: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/37.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
6/18
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
![Page 38: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/38.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
6/18
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
![Page 39: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/39.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
6/18
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
![Page 40: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/40.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
6/18
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
![Page 41: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/41.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
7/18
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
![Page 42: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/42.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
7/18
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
![Page 43: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/43.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
7/18
Outline
Introduction
Related Work
Bipartite Matching
Extensions
Open Problems
![Page 44: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/44.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
8/18
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
![Page 45: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/45.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
8/18
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
![Page 46: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/46.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
8/18
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
![Page 47: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/47.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
8/18
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
![Page 48: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/48.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
9/18
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.
![Page 49: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/49.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
9/18
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%)
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.
![Page 50: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/50.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
9/18
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%)
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.
![Page 51: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/51.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
9/18
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%)
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.
![Page 52: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/52.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
10/18
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
![Page 53: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/53.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
10/18
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
![Page 54: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/54.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
11/18
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?
![Page 55: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/55.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
11/18
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?
![Page 56: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/56.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
11/18
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?
![Page 57: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/57.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
11/18
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?
![Page 58: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/58.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
11/18
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?
![Page 59: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/59.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
11/18
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?
![Page 60: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/60.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
12/18
Two Phase 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?
![Page 61: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/61.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
12/18
Two Phase 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?
![Page 62: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/62.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
12/18
Two Phase 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?
![Page 63: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/63.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
12/18
Two Phase 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?
![Page 64: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/64.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
12/18
Two Phase 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?
![Page 65: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/65.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
13/18
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
)?
![Page 66: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/66.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
13/18
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
)?
![Page 67: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/67.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
13/18
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
)?
![Page 68: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/68.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
14/18
Sampling Lemma
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
![Page 69: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/69.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
14/18
Sampling Lemma
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
![Page 70: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/70.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
14/18
Sampling Lemma
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
![Page 71: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/71.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
14/18
Sampling Lemma
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
![Page 72: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/72.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
14/18
Sampling Lemma
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
![Page 73: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/73.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
14/18
Sampling Lemma
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
![Page 74: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/74.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
14/18
Sampling Lemma
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
![Page 75: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/75.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
14/18
Sampling Lemma
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
![Page 76: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/76.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
14/18
Outline
Introduction
Related Work
Bipartite Matching
Extensions
Open Problems
![Page 77: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/77.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
15/18
General Matching
Assume greedy is bad
I U denotes vertices matched by greedy (in Phase (a))
I Reduces to bipartite matching problem
![Page 78: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/78.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
15/18
General Matching
Assume greedy is bad
I U denotes vertices matched by greedy (in Phase (a))
I Reduces to bipartite matching problem
![Page 79: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/79.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
16/18
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
![Page 80: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/80.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
16/18
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
![Page 81: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/81.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
16/18
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
![Page 82: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/82.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
16/18
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
![Page 83: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/83.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
16/18
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
![Page 84: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/84.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
16/18
Outline
Introduction
Related Work
Bipartite Matching
Extensions
Open Problems
![Page 85: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/85.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
17/18
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?
![Page 86: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/86.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
17/18
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?
![Page 87: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/87.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
17/18
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?
![Page 88: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/88.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
18/18
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?
![Page 89: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/89.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
18/18
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?
![Page 90: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/90.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
18/18
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?
![Page 91: Online Matroid Online Matroid Intersection: Beating Half ...singla/presentations/OMI30minIPCO.… · Sahil, Guru Introduction Related Work Bipartite Matching Extensions Open Problems](https://reader036.fdocuments.net/reader036/viewer/2022071017/5fd0950c134505363a5db813/html5/thumbnails/91.jpg)
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
18/18
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?