* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download The Simulated Greedy Algorithm for Several Submodular Matroid Secretary Problems Princeton University
Birthday problem wikipedia , lookup
Computational electromagnetics wikipedia , lookup
Lateral computing wikipedia , lookup
Mathematical optimization wikipedia , lookup
Hardware random number generator wikipedia , lookup
Genetic algorithm wikipedia , lookup
Knapsack problem wikipedia , lookup
Computational complexity theory wikipedia , lookup
Exact cover wikipedia , lookup
Travelling salesman problem wikipedia , lookup
Smith–Waterman algorithm wikipedia , lookup
Algorithm characterizations wikipedia , lookup
Simplex algorithm wikipedia , lookup
Expectation–maximization algorithm wikipedia , lookup
Simulated annealing wikipedia , lookup
Dijkstra's algorithm wikipedia , lookup
Factorization of polynomials over finite fields wikipedia , lookup
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!