Online Matroid Online Matroid Intersection: Beating Half...
Transcript of Online Matroid Online Matroid Intersection: Beating Half...
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
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
1/18
Outline
Introduction
Related Work
Bipartite Matching
Extensions
Open Problems
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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?
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?
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?
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?
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?
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?
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?
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.
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.
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.
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.
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
4/18
Outline
Introduction
Related Work
Bipartite Matching
Extensions
Open Problems
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
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
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
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
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
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
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
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
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
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
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
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
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
7/18
Outline
Introduction
Related Work
Bipartite Matching
Extensions
Open Problems
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
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
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
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
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.
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.
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.
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.
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
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
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?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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
)?
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
)?
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
)?
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
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
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
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
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
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
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
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
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
14/18
Outline
Introduction
Related Work
Bipartite Matching
Extensions
Open Problems
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
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
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
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
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
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
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
OnlineMatroid
Intersection:Beating Halffor Random
Arrival
Sahil, Guru
Introduction
RelatedWork
BipartiteMatching
Extensions
OpenProblems
16/18
Outline
Introduction
Related Work
Bipartite Matching
Extensions
Open Problems
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?
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?
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?
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?
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?
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?
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?