Contributions on Secretary Problems, Independent Sets of Rectangles - PowerPoint PPT Presentation

Contributions on Secretary Problems, Independent Sets of Rectangles and Related Problems Jos´ e A. Soto Doctoral Thesis Defense. Department of Mathematics. M.I.T. April 15th, 2011 Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 1

Outline Matroid Secretary Problem 1 Jump Number Problem and Independent Sets of Rectangles. 2 (joint work with C. Telha) Symmetric Submodular Function Minimization under Hereditary 3 Constraints.

Outline Matroid Secretary Problem 1 Jump Number Problem and Independent Sets of Rectangles. 2 (joint work with C. Telha) Symmetric Submodular Function Minimization under Hereditary 3 Constraints.

MSP: Introduction Given a matroid. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

MSP: Introduction 3 Given a matroid. Elements’ weights are revealed in certain (random) order. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

MSP: Introduction 3 1 Given a matroid. Elements’ weights are revealed in certain (random) order. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

MSP: Introduction 4 3 1 Given a matroid. Elements’ weights are revealed in certain (random) order. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

MSP: Introduction 4 3 15 1 Given a matroid. Elements’ weights are revealed in certain (random) order. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

MSP: Introduction 4 3 15 1 10 Given a matroid. Elements’ weights are revealed in certain (random) order. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

MSP: Introduction 4 3 2 15 1 10 Given a matroid. Elements’ weights are revealed in certain (random) order. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

MSP: Introduction 4 3 2 15 1 10 20 Given a matroid. Elements’ weights are revealed in certain (random) order. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

MSP: Introduction 4 3 2 15 1 36 10 20 Given a matroid. Elements’ weights are revealed in certain (random) order. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

MSP: Introduction 4 3 2 15 1 9 36 10 20 Given a matroid. Elements’ weights are revealed in certain (random) order. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

MSP: Introduction 5 4 3 2 15 1 9 36 10 20 Given a matroid. Elements’ weights are revealed in certain (random) order. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

MSP: Introduction 5 4 3 2 15 1 9 36 10 20 Given a matroid. Elements’ weights are revealed in certain (random) order. Want to select independent set of high weight. (In online way / secretary problem setting) Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 3

MSP: Introduction (II) Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

MSP: Introduction (II) 3 Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

MSP: Introduction (II) 3 1 Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

MSP: Introduction (II) 4 3 1 Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

MSP: Introduction (II) 4 3 15 1 Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

MSP: Introduction (II) 4 3 15 1 10 Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

MSP: Introduction (II) 4 3 2 15 1 10 Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

MSP: Introduction (II) 4 3 2 15 1 10 20 Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

MSP: Introduction (II) 4 3 2 15 1 36 10 20 Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

MSP: Introduction (II) 4 3 2 15 1 9 36 10 20 Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

MSP: Introduction (II) 5 4 3 2 15 1 9 36 10 20 Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

MSP: Introduction (II) 5 5 4 3 2 15 4 3 2 15 1 9 1 9 36 10 36 10 20 20 w (OPT) = 80 w (ALG) = 42 Rules We accept or reject an element when its weight is revealed. Accepted elements must form an independent set. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 4

Special Cases Classical / Multiple choice Hire one person (or at most r ). Sell one item to best bidder (or sell r identical items). Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 5

Models Opponent selects n weights . w 1 ≥ w 2 ≥ · · · ≥ w n ≥ 0 then The weights are assigned either: adversarially or at random . and independently The presentation order is chosen: adversarially or at random . Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 6

Models Adv.-Assign. Opponent selects n weights . Adv.-Order w 1 ≥ w 2 ≥ · · · ≥ w n ≥ 0 then Random-Assign. Adv.-Assign. Adv.-Order Random-Order The weights are assigned either: adversarially or at random . Random-Assign. and independently Random-Order The presentation order is chosen: I.I.D. Weights adversarially or at random . Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 6

Models Adv.-Assign. (Adv.-Assign. Adv.-Order) Adv.-Order Hard: n -competitive ratio [Babaioff, Immorlica, Kleinberg 07] Random-Assign. Adv.-Assign. Conjecture: O ( 1 ) -competitive Adv.-Order Random-Order algorithm for all other models. Random-Assign. Random-Order I.I.D. Weights Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 6

Models Adv.-Assign. (Adv.-Assign. Adv.-Order) Adv.-Order Hard: n -competitive ratio [Babaioff, Immorlica, Kleinberg 07] Random-Assign. Adv.-Assign. Conjecture: O ( 1 ) -competitive Adv.-Order Random-Order algorithm for all other models. (Adv.-Assign. Random-Order) Random-Assign. O ( 1 ) for partition, graphic, transversal, laminar. Random-Order [L61,D63,K05,BIK07,DP08,KP09,BDGIT09,IW11] O ( log rk ( M )) for general I.I.D. Weights matroids [BIK07] . Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 6

Models Adv.-Assign. (Adv.-Assign. Adv.-Order) Adv.-Order Hard: n -competitive ratio [Babaioff, Immorlica, Kleinberg 07] Random-Assign. Adv.-Assign. Conjecture: O ( 1 ) -competitive Adv.-Order Random-Order algorithm for all other models. (Adv.-Assign. Random-Order) Random-Assign. O ( 1 ) for partition, graphic, transversal, laminar. Random-Order [L61,D63,K05,BIK07,DP08,KP09,BDGIT09,IW11] O ( log rk ( M )) for general I.I.D. Weights matroids [BIK07] . (Random-Assign. Random-Order) [S11] O ( 1 ) for general matroids. Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 6

Random-Assignment Random-Order. Data σ ← − − − − W : w 1 ≥ w 2 ≥ · · · ≥ w n ≥ 0 . r . a . Hidden weight list Random assignment. σ : [ n ] → E . Known Matroid Random order. π : E → { 1 , . . . , n } . Objective Return an independent set ALG ∈ I such that: E π,σ [ w ( ALG )] ≥ Ω( 1 ) · E σ [ w ( OPT )] , where OPT is the optimum base of M under assignment σ . (Greedy) Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 7

Divide and Conquer to get O ( 1 ) -competitive algorithm. For a general matroid M = ( E , I ) : M 1 , E 1 Find matroids M i = ( E i , I i ) with E = � k i = 1 E i . M i admits O ( 1 ) -competitive algorithm 1 M 2 , E 2 (Easy parts). M , E Union of independent sets in each M i is 2 independent in M . I ( � k i = 1 M i ) ⊆ I ( M ) . (Combine nicely). . . . Optimum in � k i = 1 M i is comparable with 3 Optimum in M . (Don’t lose much). M k , E k Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 8

(Easiest matroids): Uniform. [Independent sets = Sets of size ≤ r .] For r = 1: Dynkin’s Algorithm � �� � n / e Observe n / e objects. Accept the first record after that. Top weight is selected w.p. ≥ 1 / e . Jos´ e A. Soto - M.I.T. Thesis Defense April 15th, 2011 9