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The Simulated Greedy Algorithm for Several
Submodular Matroid Secretary Problems
Tengyu Ma
Princeton University
Kiel, Germany 2011
Joint work with
Bo Tang,
University of Liverpool
Yajun Wang,
Microsoft Research Asia
Classical Secretary Problem

𝑛 candidates arrive in random order for interview

Hire or reject immediately and irrevocably after interview

To hire the best secretary
Classical Secretary Problem

𝑛 candidates arrive in random order for interview

Hire or reject immediately and irrevocably after interview

To hire the best secretary
Algorithm [Dynkin ‘63]

Not hire the first 𝑛/𝑒 candidates (denoted by 𝐻)

Hire the first one that is better than all ones in 𝐻
Classical Secretary Problem

𝑛 candidates arrive in random order for interview

Hire or reject immediately and irrevocably after interview

To hire the best secretary
Algorithm [Dynkin ‘63]

Not hire the first 𝑛/𝑒 candidates (denoted by 𝐻)

Hire the first one that is better than all ones in 𝐻

Algorithm succeeds with const. prob. when:
• The best ∉ 𝐻 and the second best ∈ 𝐻
Classical Secretary Problem

𝑛 candidates arrive in random order for interview

Hire or reject immediately and irrevocably after interview

To hire the best secretary
Algorithm [Dynkin ‘63]

Not hire the first 𝑛/𝑒 candidates (denoted by 𝐻)

Hire the first one that is better than all ones in 𝐻

Algorithm succeeds with const. prob. when:
• The best ∉ 𝐻 and the second best ∈ 𝐻
• Careful analysis gives successful probability 1/𝑒 (optimal)
Hiring more secretaries?
Potential constraints

To hire at most 𝑘 secretaries

To hire secretaries with different specialties

etc…
Hiring more secretaries?
Potential constraints

To hire at most 𝑘 secretaries

To hire secretaries with different specialties

etc…
Matroid ℳ = 𝐸, ℐ

𝐸 a ground set of 𝑛 elements

ℐ a (nonempty) family of (independent) subsets of 𝐸
Hiring more secretaries?
Potential constraints

To hire at most 𝑘 secretaries

To hire secretaries with different specialties

etc…
Matroid ℳ = 𝐸, ℐ

𝐸 a ground set of 𝑛 elements

ℐ a (nonempty) family of (independent) subsets of 𝐸
with following properties

If 𝐵 ∈ ℐ, and 𝐴 ⊂ 𝐵, then 𝐴 ∈ ℐ

If 𝐴, 𝐵 ∈ ℐ, 𝐴 < |𝐵|, then ∃𝑒 ∈ 𝐵\A, s.t. 𝐴 ∪ 𝑒 ∈ ℐ
Hiring more secretaries?
Potential constraints
→ uniform matroids

To hire at most 𝑘 secretaries

To hire secretaries with different specialties

etc…
Matroid ℳ = 𝐸, ℐ

𝐸 a ground set of 𝑛 elements

ℐ a (nonempty) family of (independent) subsets of 𝐸
with following properties

If 𝐵 ∈ ℐ, and 𝐴 ⊂ 𝐵, then 𝐴 ∈ ℐ

If 𝐴, 𝐵 ∈ ℐ, 𝐴 < |𝐵|, then ∃𝑒 ∈ 𝐵\A, s.t. 𝐴 ∪ 𝑒 ∈ ℐ
Hiring more secretaries?
Potential constraints



→ uniform matroids
→ partition
To hire secretaries with different specialties
matroids
To hire at most 𝑘 secretaries
etc…
Matroid ℳ = 𝐸, ℐ

𝐸 a ground set of 𝑛 elements

ℐ a (nonempty) family of (independent) subsets of 𝐸
with following properties

If 𝐵 ∈ ℐ, and 𝐴 ⊂ 𝐵, then 𝐴 ∈ ℐ

If 𝐴, 𝐵 ∈ ℐ, 𝐴 < |𝐵|, then ∃𝑒 ∈ 𝐵\A, s.t. 𝐴 ∪ 𝑒 ∈ ℐ
Matroid Secretary Problem
Given:
Constraints:
Goal:
Matroid Secretary Problem
Given:

Matroid ℳ = 𝐸, ℐ

Weight function 𝑤: 𝐸 → ℝ≥0

Elements of 𝐸 arrive in random order
Constraints:
Goal:
Matroid Secretary Problem
Given:

Matroid ℳ = 𝐸, ℐ

Weight function 𝑤: 𝐸 → ℝ≥0

Elements of 𝐸 arrive in random order
Constraints:

Online Accept or reject an element immediately and irrevocably

Independence 𝑆, the set of accepted elements, is independent
Goal:
Matroid Secretary Problem
Given:

Matroid ℳ = 𝐸, ℐ

Weight function 𝑤: 𝐸 → ℝ≥0

Elements of 𝐸 arrive in random order
Constraints:

Online Accept or reject an element immediately and irrevocably

Independence 𝑆, the set of accepted elements, is independent
Goal:

To maximize total weight of accepted elements
max 𝑤 𝑆 =
𝑤(𝑒)
𝑒∈𝑆
Submodular Matroid Secretary Problem
Given:

Matroid ℳ = 𝐸, ℐ

Weight function 𝑤: 𝐸 → ℝ≥0

Elements of 𝐸 arrive in random order
Constraints:

Online Accept an element or not immediately and irrevocably

Independence 𝑆, the set of accepted elements, is independent
Goal

To maximize total weight of accepted elements
max 𝑤 𝑆 =
𝑤(𝑒)
𝑒∈𝑆
Submodular Matroid Secretary Problem
Given:

Matroid ℳ = 𝐸, ℐ

Elements of 𝐸 arrive in random order
Constraints:

Online Accept an element or not immediately and irrevocably

Independence 𝑆, the set of accepted elements, is independent
Goal

To maximize total weight of accepted elements
max 𝑤 𝑆 =
𝑤(𝑒)
𝑒∈𝑆
Submodular Matroid Secretary Problem
Given:

Matroid ℳ = 𝐸, ℐ

Submodular valuation function 𝑓: 2𝐸 ⟶ ℝ≥0

Elements of 𝐸 arrive in random order
Constraints:

Online Accept an element or not immediately and irrevocably

Independence 𝑆, the set of accepted elements, is independent
Goal

To maximize total weight of accepted elements
max 𝑤 𝑆 =
𝑤(𝑒)
𝑒∈𝑆
Submodular Matroid Secretary Problem
Given:

Matroid ℳ = 𝐸, ℐ

Submodular valuation function 𝑓: 2𝐸 ⟶ ℝ≥0

Elements of 𝐸 arrive in random order
Constraints:

Online Accept an element or not immediately and irrevocably

Independence 𝑆, the set of accepted elements, is independent
Goal
Submodular Matroid Secretary Problem
Given:

Matroid ℳ = 𝐸, ℐ

Submodular valuation function 𝑓: 2𝐸 ⟶ ℝ≥0

Elements of 𝐸 arrive in random order
Constraints:

Online Accept an element or not immediately and irrevocably

Independence 𝑆, the set of accepted elements, is independent
Goal

To maximize the valuation of the accepted elements
max 𝑓(𝑆)
Offline optimization problem

Given ℳ = 𝐸, ℐ , submodular function 𝑓: 2𝐸 → ℝ≥0

Find 𝑆 ∈ ℐ that maximizes 𝑓(𝑆)
GREEDY

𝑆 ← ∅, 𝑇 ← 𝐸. Greedy in marginal values w.r.t 𝑆

For 𝑖 = 1 to 𝑛
• 𝑒𝑖 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑒 ∈ 𝑇: 𝑓𝑆 𝑒 = 𝑓 𝑆 ∪ 𝑒 − 𝑓(𝑆) , 𝑇 ⟵ 𝑇\{𝑒}
• Accept 𝑒𝑖 as long as 𝑆 ∪ {𝑒𝑖 } is independent

Obtain approximate solution 𝑓 𝑆 ≥ 𝑂𝑃𝑇/2
• If 𝑓 is linear, then exact optimal solution
Competitive ratio

Online algorithm is 𝛼-competitive if
• 𝛼⋅𝔼 𝑓 𝑆
≥ 𝑂𝑃𝑇
Related work
Matroid secretary problem

log 𝑟-competitive algorithm for general matroids
[Chakraborty/Lachish’12]
• 𝑟 = 𝑟𝑎𝑛𝑘 ℳ = max A : A ∈ ℐ

Constant competitive algorithm for specific matroids
• uniform/paritition/graphical [Babaioff/Immorlica/Kleinberg ‘07] ,
Laminar [Im/Wang ‘11], tranversal[Dimitrov/Plaxton’08, Korula/Pál’09 ],
regular and decomposable [Dinitz/Kortsarz’13]
Related work
Matroid secretary problem

log 𝑟-competitive algorithm for general matroids
[Chakraborty/Lachish’12]
• 𝑟 = 𝑟𝑎𝑛𝑘 ℳ = max A : A ∈ ℐ

Constant competitive algorithm for specific matroids
• uniform/paritition/graphical [Babaioff/Immorlica/Kleinberg ‘07] ,
laminar [Im/Wang ‘11], tranversal[Dimitrov/Plaxton’08, Korula/Pál’09 ],
regular and decomposable [Dinitz/Kortsarz’13]
Submodular matroid secretary problem

𝑂(log 𝑟)-competitive for general
matroids[Gupta/Roth/Schoenebeck/Talwar’10]

Constant competitive for uniform and partition matroids
[Bateni/Hajiaghayi/Zadimoghaddam’10], [FeldmanNaor,/Schwartz’11]
Main Contribution
First constant competitive algorithm for submodular matoid
secretary problem with

Laminar matroids

Transversal matroids

Intersection of laminar matroids
Main Contribution
First constant competitive algorithm for submodular matoid
secretary problem with

Laminar matroids

Transversal matroids

Intersection of laminar matroids
o
Corollary: this algorithm improves the competitive ratio for
linear matroid secretary problem with laminar matroid from
16000/3 [IW’11] to 9.6
Main Algorithm
 𝑀, 𝑁, 𝑆 ← ∅, 𝑚 ← 𝑏𝑖𝑛𝑜𝑚 𝑛, 𝑝
 Reject the first 𝑚 elements
(denoted by 𝐻, 𝐻 = 𝑚)
𝑒1 𝑒2
. . . 𝑒𝑚 𝑒𝑚+1
. . . 𝑒𝑛
Main Algorithm
 𝑀, 𝑁, 𝑆 ← ∅, 𝑚 ← 𝑏𝑖𝑛𝑜𝑚 𝑛, 𝑝
𝑒1 𝑒2
. . . 𝑒𝑚 𝑒𝑚+1
 Reject the first 𝑚 elements
(denoted by 𝐻, 𝐻 = 𝑚)
GREEDY
 𝑀 ← 𝐺𝑅𝐸𝐸𝐷𝑌 𝐻
𝑀
. . . 𝑒𝑛
Main Algorithm
 𝑀, 𝑁, 𝑆 ← ∅, 𝑚 ← 𝑏𝑖𝑛𝑜𝑚 𝑛, 𝑝
𝑒1 𝑒2
. . . 𝑒𝑚 𝑒𝑚+1
 Reject the first 𝑚 elements
(denoted by 𝐻, 𝐻 = 𝑚)
GREEDY
 𝑀 ← 𝐺𝑅𝐸𝐸𝐷𝑌 𝐻
 For subsequent element 𝑒
 If 𝐺𝑅𝐸𝐸𝐷𝑌 𝐻 ∪ {𝑒} ≠ 𝑀 then
 𝑁 ← 𝑁 ∪ {𝑒}
𝑀
. . . 𝑒𝑛
Main Algorithm
 𝑀, 𝑁, 𝑆 ← ∅, 𝑚 ← 𝑏𝑖𝑛𝑜𝑚 𝑛, 𝑝
𝑒1 𝑒2
. . . 𝑒𝑚 𝑒𝑚+1
 Reject the first 𝑚 elements
(denoted by 𝐻, 𝐻 = 𝑚)
GREEDY
. . . 𝑒𝑛
large w.r.t 𝑀 ?
 𝑀 ← 𝐺𝑅𝐸𝐸𝐷𝑌 𝐻
 For subsequent element 𝑒
𝑒𝑖 𝑒𝑗 . . .
𝑒𝑚+1
 If 𝐺𝑅𝐸𝐸𝐷𝑌 𝐻 ∪ {𝑒} ≠ 𝑀 then
 𝑁 ← 𝑁 ∪ {𝑒}
𝑀
𝑁
Main Algorithm
 𝑀, 𝑁, 𝑆 ← ∅, 𝑚 ← 𝑏𝑖𝑛𝑜𝑚 𝑛, 𝑝
𝑒1 𝑒2
. . . 𝑒𝑚 𝑒𝑚+1
 Reject the first 𝑚 elements
(denoted by 𝐻, 𝐻 = 𝑚)
GREEDY
. . . 𝑒𝑛
large w.r.t 𝑀 ?
 𝑀 ← 𝐺𝑅𝐸𝐸𝐷𝑌 𝐻
 For subsequent element 𝑒
𝑒𝑖 𝑒𝑗 . . .
𝑒𝑚+1
 If 𝐺𝑅𝐸𝐸𝐷𝑌 𝐻 ∪ {𝑒} ≠ 𝑀 then
 𝑁 ← 𝑁 ∪ {𝑒}
 IF 𝑆 ∪ 𝑒 is independent
 Accept 𝑒
 𝑆 ← 𝑆 ∪ {𝑒}
𝑀
𝑁
Main Algorithm
 𝑀, 𝑁, 𝑆 ← ∅, 𝑚 ← 𝑏𝑖𝑛𝑜𝑚 𝑛, 𝑝
𝑒1 𝑒2
. . . 𝑒𝑚 𝑒𝑚+1
 Reject the first 𝑚 elements
(denoted by 𝐻, 𝐻 = 𝑚)
GREEDY
. . . 𝑒𝑛
large w.r.t 𝑀 ?
 𝑀 ← 𝐺𝑅𝐸𝐸𝐷𝑌 𝐻
 For subsequent element 𝑒
𝑒𝑖 𝑒𝑗 . . .
𝑒𝑚+1
 If 𝐺𝑅𝐸𝐸𝐷𝑌 𝐻 ∪ {𝑒} ≠ 𝑀 then
 𝑁 ← 𝑁 ∪ {𝑒}
 IF 𝑆 ∪ 𝑒 is independent
 Accept 𝑒
 𝑆 ← 𝑆 ∪ {𝑒}
𝑀
𝑁
independent?
𝑒𝑚+1 𝑒𝑗 . . .
𝑆
Simulated Greedy Algorithm
 𝑀, 𝑁, 𝑆, 𝐻 ← ∅,
ℎ
ℎ
ℎ
ℎ
...
 Add each element 𝑒 to 𝐻 with
probability 𝑝
 While
∃𝑒 = 𝑎𝑟𝑔𝑚𝑎𝑥 {𝑓𝑀 𝑒 = 𝑓 𝑀 ∪ 𝑒 −
with prob. 𝑝
𝑓 𝑀 | 𝑒 ∉ 𝑀 ∪ 𝑁, 𝑀 ∪ 𝑒 ∈ ℐ}
largest w.r.t 𝑀
with prob. 1-𝑝
 If 𝑒 ∈ 𝐻, then 𝑀 ← 𝑀 ∪ {𝑒}
 Otherwise 𝑁 ← 𝑁 ∪ 𝑒
 End while
𝑀
𝑁
Simulated Greedy Algorithm
 𝑀, 𝑁, 𝑆, 𝐻 ← ∅,
ℎ
ℎ
ℎ
...
 Add each element 𝑒 to 𝐻 with
probability 𝑝
 While
∃𝑒 = 𝑎𝑟𝑔𝑚𝑎𝑥 {𝑓𝑀 𝑒 = 𝑓 𝑀 ∪ 𝑒 −
with prob. 𝑝
𝑓 𝑀 | 𝑒 ∉ 𝑀 ∪ 𝑁, 𝑀 ∪ 𝑒 ∈ ℐ}
 If 𝑒 ∈ 𝐻, then 𝑀 ← 𝑀 ∪ {𝑒}
with prob. 1-𝑝
ℎ
 Otherwise 𝑁 ← 𝑁 ∪ 𝑒
 End while
largest w.r.t 𝑀
𝑀
𝑁
Simulated Greedy Algorithm
 𝑀, 𝑁, 𝑆, 𝐻 ← ∅,
ℎ
ℎ
ℎ
...
 Add each element 𝑒 to 𝐻 with
probability 𝑝
 While
∃𝑒 = 𝑎𝑟𝑔𝑚𝑎𝑥 {𝑓𝑀 𝑒 = 𝑓 𝑀 ∪ 𝑒 −
with prob. 𝑝
𝑓 𝑀 | 𝑒 ∉ 𝑀 ∪ 𝑁, 𝑀 ∪ 𝑒 ∈ ℐ}
 If 𝑒 ∈ 𝐻, then 𝑀 ← 𝑀 ∪ {𝑒}
with prob. 1-𝑝
ℎ
 Otherwise 𝑁 ← 𝑁 ∪ 𝑒
 End while
largest w.r.t 𝑀
𝑀
𝑁
Simulated Greedy Algorithm
 𝑀, 𝑁, 𝑆, 𝐻 ← ∅,
ℎ
ℎ
ℎ
...
 Add each element 𝑒 to 𝐻 with
probability 𝑝
 While
∃𝑒 = 𝑎𝑟𝑔𝑚𝑎𝑥 {𝑓𝑀 𝑒 = 𝑓 𝑀 ∪ 𝑒 −
with prob. 𝑝
𝑓 𝑀 | 𝑒 ∉ 𝑀 ∪ 𝑁, 𝑀 ∪ 𝑒 ∈ ℐ}
 If 𝑒 ∈ 𝐻, then 𝑀 ← 𝑀 ∪ {𝑒}
with prob. 1-𝑝
ℎ
 Otherwise 𝑁 ← 𝑁 ∪ 𝑒
 End while
largest w.r.t 𝑀
𝑀
𝑁
Simulated Greedy Algorithm
ℎ
ℎ
...
 𝑀, 𝑁, 𝑆, 𝐻 ← ∅,
 Add each element 𝑒 to 𝐻 with
probability 𝑝
 While
∃𝑒 = 𝑎𝑟𝑔𝑚𝑎𝑥 {𝑓𝑀 𝑒 = 𝑓 𝑀 ∪ 𝑒 −
with prob. 𝑝
𝑓 𝑀 | 𝑒 ∉ 𝑀 ∪ 𝑁, 𝑀 ∪ 𝑒 ∈ ℐ}
 If 𝑒 ∈ 𝐻, then 𝑀 ← 𝑀 ∪ {𝑒}
ℎ
with prob. 1-𝑝
ℎ
 Otherwise 𝑁 ← 𝑁 ∪ 𝑒
 End while
largest w.r.t 𝑀
𝑀
𝑁
Simulated Greedy Algorithm
ℎ
ℎ
...
 𝑀, 𝑁, 𝑆, 𝐻 ← ∅,
 Add each element 𝑒 to 𝐻 with
probability 𝑝
 While
∃𝑒 = 𝑎𝑟𝑔𝑚𝑎𝑥 {𝑓𝑀 𝑒 = 𝑓 𝑀 ∪ 𝑒 −
with prob. 𝑝
𝑓 𝑀 | 𝑒 ∉ 𝑀 ∪ 𝑁, 𝑀 ∪ 𝑒 ∈ ℐ}
 If 𝑒 ∈ 𝐻, then 𝑀 ← 𝑀 ∪ {𝑒}
ℎ
with prob. 1-𝑝
ℎ
 Otherwise 𝑁 ← 𝑁 ∪ 𝑒
 End while
largest w.r.t 𝑀
𝑀
𝑁
Main Algorithm
 𝑀, 𝑁, 𝑆 ← ∅, 𝑚 ← 𝑏𝑖𝑛𝑜𝑚 𝑛, 𝑝
𝑒1 𝑒2
. . . 𝑒𝑚 𝑒𝑚+1
 Reject the first 𝑚 elements
(denoted by 𝐻, 𝐻 = 𝑚)
GREEDY
. . . 𝑒𝑛
large w.r.t 𝑀 ?
 𝑀 ← 𝐺𝑅𝐸𝐸𝐷𝑌 𝐻
 For subsequent element 𝑒
𝑒𝑖 𝑒𝑗 . . .
𝑒𝑚+1
 If 𝐺𝑅𝐸𝐸𝐷𝑌 𝐻 ∪ {𝑒} ≠ 𝑀 then
 𝑁 ← 𝑁 ∪ {𝑒}
 IF 𝑆 ∪ 𝑒 is independent
 Accept 𝑒
 𝑆 ← 𝑆 ∪ {𝑒}
𝑀
𝑁
independent?
𝑒𝑚+1 𝑒𝑗 . . .
𝑆
Simulated Greedy Algorithm
ℎ
ℎ
...
 𝑀, 𝑁, 𝑆, 𝐻 ← ∅,
 Add each element 𝑒 to 𝐻 with
probability 𝑝
 While
∃𝑒 = 𝑎𝑟𝑔𝑚𝑎𝑥 {𝑓𝑀 𝑒 = 𝑓 𝑀 ∪ 𝑒 −
with prob. 𝑝
𝑓 𝑀 | 𝑒 ∉ 𝑀 ∪ 𝑁, 𝑀 ∪ 𝑒 ∈ ℐ}
 If 𝑒 ∈ 𝐻, then 𝑀 ← 𝑀 ∪ {𝑒}
ℎ
with prob. 1-𝑝
ℎ
 Otherwise 𝑁 ← 𝑁 ∪ 𝑒
 End while
largest w.r.t 𝑀
𝑀
𝑁
Simulated Greedy Algorithm
ℎ
ℎ
...
 𝑀, 𝑁, 𝑆, 𝐻 ← ∅,
 Add each element 𝑒 to 𝐻 with
probability 𝑝
 While
∃𝑒 = 𝑎𝑟𝑔𝑚𝑎𝑥 {𝑓𝑀 𝑒 = 𝑓 𝑀 ∪ 𝑒 −
with prob. 𝑝
𝑓 𝑀 | 𝑒 ∉ 𝑀 ∪ 𝑁, 𝑀 ∪ 𝑒 ∈ ℐ}
 If 𝑒 ∈ 𝐻, then 𝑀 ← 𝑀 ∪ {𝑒}
ℎ
 Random shuffle 𝑁
 For each element 𝑒 ∈ 𝑁
 If 𝑆 ∪ 𝑒 is independent
 𝑆 ← 𝑆 ∪ {𝑒}
with prob. 1-𝑝
ℎ
 Otherwise 𝑁 ← 𝑁 ∪ 𝑒
 End while
largest w.r.t 𝑀
𝑀
𝑁
Simulated Greedy Algorithm
ℎ
ℎ
...
 𝑀, 𝑁, 𝑆, 𝐻 ← ∅,
 Add each element 𝑒 to 𝐻 with
probability 𝑝
 While
∃𝑒 = 𝑎𝑟𝑔𝑚𝑎𝑥 {𝑓𝑀 𝑒 = 𝑓 𝑀 ∪ 𝑒 −
with prob. 𝑝
𝑓 𝑀 | 𝑒 ∉ 𝑀 ∪ 𝑁, 𝑀 ∪ 𝑒 ∈ ℐ}
 If 𝑒 ∈ 𝐻, then 𝑀 ← 𝑀 ∪ {𝑒}
ℎ
 Random shuffle 𝑁
with prob. 1-𝑝
ℎ
 Otherwise 𝑁 ← 𝑁 ∪ 𝑒
 End while
largest w.r.t 𝑀
𝑀
𝑁
independent?
 For each element 𝑒 ∈ 𝑁
 If 𝑆 ∪ 𝑒 is independent
 𝑆 ← 𝑆 ∪ {𝑒}
𝑆
Proof outlines: Linear case

𝔼𝑤 𝑀
=
𝑝
𝔼
1−𝑝
𝑤 𝑁
ℎ
≥
ℎ
ℎ
...
𝑝 ⋅ 𝑂𝑃𝑇

𝔼𝑤 𝑆
≥𝛼𝔼 𝑤 𝑁

Goal: lower bound 𝔼[𝑤 𝑆 ]
largest w.r.t 𝑀
with prob. 𝑝
ℎ
with prob. 1 − 𝑝
ℎ
𝑀
𝑁
independent
𝑆
When 𝑤 𝑆 ≈ 𝑤(𝑁) ?
Uniform matroids (integer 𝑘)

ℐ = {𝐴 ⊂ 𝐸: 𝐴 ≤ 𝑘}
Partition matroids (partition of 𝐸, 𝐸 = 𝐵1 ∪ ⋯ ∪ 𝐵𝑠 )

ℐ = {𝐴 ⊂ 𝐸: ∀𝑖, |𝐴 ∩ 𝐵𝑖 | ≤ 1}
Laminar matroids (Laminar family ℱ = {ℬ𝑗 }, 𝜇(𝐵𝑗 ))

ℐ = {𝐴 ⊂ 𝐸: ∀𝑗, 𝐴 ∩ 𝐵𝑗 ≤ 𝜇 𝐵𝑗 }
𝐵2
𝐵1
𝐵4
𝐵3
𝐵5
When 𝑤 𝑆 ≈ 𝑤(𝑁) ? – Laminar matroids


ℎ
𝑀 ∩ 𝐵1 ≤ 𝜇(𝐵1 ) implies
𝔼 𝑁 ∩ 𝐵1 =
1−𝑝
𝔼[|𝑀
𝑝
ℎ
∩ 𝐵1 |] ≤
𝑝
𝜇(𝐵1 )
1−𝑝

Pr 𝑁 ∩ 𝐵1 ≤ 𝜇 𝐵1
≥ 1 − exp( −𝜇(𝐵1 ))
with prob. 𝑝
ℎ
ℎ
...
largest w.r.t 𝑀
with prob. 1 − 𝑝
ℎ
𝐵1
𝑀
𝑁
independent?
When 𝑤 𝑆 ≈ 𝑤(𝑁) ? – Laminar matroids

Pr 𝑁 ∩ 𝐵𝑗 ≤ 𝜇 𝐵𝑗

For any 𝑒 ∈ 𝑁,
ℎ
Pr ∀𝐵𝑗 ∋ 𝑒, 𝑁 ∩ 𝐵𝑗 ≤ 𝜇 𝐵𝑗
1−
ℎ
> 1 − exp( −𝜇(𝐵𝑗 ))
Pr 𝑒 ∈ 𝑆 𝑒 ∈ 𝑁] ≥ 𝑐𝑜𝑛𝑠𝑡

𝑤 𝑆 ≈ 𝑤(𝑁)
largest w.r.t 𝑀
with prob. 𝑝
ℎ
𝐵𝑘
𝑒
with prob. 1 − 𝑝
ℎ
𝐵1
𝐵𝑗
...
>
exp(−𝜇 𝐵𝑗 ) > const

ℎ
𝑀
𝑁
independent?
Proof outlines: Submodular case

𝑀𝑒 ≜ the set 𝑀 when 𝑒 appears

𝒘 𝑒 ≜ 𝑓𝑀𝑒 𝑒 = 𝑓 𝑀𝑒 ∪ 𝑒 −
ℎ
ℎ
𝑓(𝑀𝑒 ),


𝑓 𝑀 =
𝔼𝑓 𝑀
𝑒∈𝑀 𝒘(𝑒)
=𝔼𝒘 𝑀
=
𝑝
𝔼
1−𝑝
≥ ⋅ 𝑂𝑃𝑇

𝔼𝒘 𝑀

𝔼𝒘 𝑆
≥ 𝛼 𝔼 𝒘 𝑁 ,𝛼 ≈ 1

𝔼𝑓 𝑆
≥𝛽⋅𝔼 𝒘 𝑆
𝛾𝔼 𝒘 𝑁 ,

𝒘 𝑁
−
...
largest w.r.t 𝑀
≜ 𝒘(𝑀)
𝑝
2
ℎ
with prob. 𝑝
ℎ
with prob. 1 − 𝑝
ℎ
𝑀
𝑁
independent
𝛽≫𝛾
Goal : lower bound 𝑓(𝑆) by 𝑂𝑃𝑇
𝑆
Open Problems

Constant competitive ratio for general matroid secretary
problem?

Competitive ratio for general matroid under submodular
valuation function better than 𝑂(log 𝑟)? (say, 𝑂

log 𝑟 ?)
Any other assumption of inputs on which our algorithm gives
constant competitive ratio for more types of matroids? (say,
random assignment of weights, known independent
distribution)
Thank you for your attention!