Matroid Secretary Problem in the Random Assignment Model Jos e - PowerPoint PPT Presentation

Matroid Secretary Problem in the Random Assignment Model Jos´ e Soto Department of Mathematics M.I.T. SODA 2011 Jan 25, 2011. Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 1

Classical / multiple-choice Secretary Problem Rules Given a set E of elements with hidden nonnegative weights . 1 Each element reveals its weight in uniform random order . 2 We accept or reject before the next weight is revealed . 3 Maintain a feasible set: Set of size at most r . 4 Goal: Maximize the sum of weights of selected set. 5 Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 2

Matroid secretary problem Babaioff, Immorlica, Kleinberg [SODA07] Rules Given a set E of elements with hidden nonnegative weights . 1 E is the ground set of a known matroid M = ( E , I ) . Each element reveals its weight in uniform random order . 2 We accept or reject before the next weight is revealed . 3 Maintain a feasible set: Set of size at most r . 4 Feasible set = Independent Set in I . Goal: Maximize the sum of weights of selected set. 5 Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 3

Algorithms for classical problem (uniform matroid). 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 Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 4

Algorithms for classical problem (uniform matroid). 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 . General r · · · � � �� � � �� � � �� � �� � � �� � n / r n / r n / r n / r n / r Divide in r classes and apply Dynkin’s algorithm in each class. Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 4

Algorithms for classical problem (uniform matroid). 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 . General r · · · � � �� � � �� � � �� � �� � � �� � n / r n / r n / r n / r n / r Divide in r classes and apply Dynkin’s algorithm in each class. Each of the r top weights is the best of its class with prob. ≥ ( 1 − 1 / r ) r − 1 ≥ C > 0 . Thus it is selected with prob. ≥ C / e . Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 4

Algorithms for classical problem (uniform matroid). 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 . General r · · · � � �� � � �� � � �� � �� � � �� � n / r n / r n / r n / r n / r Divide in r classes and apply Dynkin’s algorithm in each class. Each of the r top weights is the best of its class with prob. ≥ ( 1 − 1 / r ) r − 1 ≥ C > 0 . Thus it is selected with prob. ≥ C / e . e / C (constant) competitive algorithm. Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 4

Harder example: Gammoid Independent Sets: Servers Clients that can be simultaneously Clients ← Elements. connected to Servers using edge-disjoint paths. Connections Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 5

Models of Weight Assignment: Adversarial Assignment: 1 Hidden weights are arbitrary. Random Assignment: 2 A hidden (adversarial) list of weights is assigned uniformly. Unknown distribution: 3 Weights selected i.i.d. from unknown distribution. Known Distribution: 4 Weights selected i.i.d. from known distribution. Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 6

Models of Weight Assignment: Conjecture [BIK07]: O ( 1 ) -competitive algorithm for all these models Adversarial Assignment: 1 Hidden weights are arbitrary. Random Assignment: 2 A hidden (adversarial) list of weights is assigned uniformly. Unknown distribution: 3 Weights selected i.i.d. from unknown distribution. Known Distribution: 4 Weights selected i.i.d. from known distribution. Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 6

Models of Weight Assignment: Conjecture [BIK07]: O ( 1 ) -competitive algorithm for all these models Adversarial Assignment: 1 Hidden weights are arbitrary. O ( 1 ) -competitive alg. for partition, graphic, transversal, laminar. [L61,D63,K05,BIK07,DP08,KP09,BDGIT09,IW11] O ( log rk ( M )) -competitive algorithms for general matroids. [BIK07] Random Assignment: 2 A hidden (adversarial) list of weights is assigned uniformly. Unknown distribution: 3 Weights selected i.i.d. from unknown distribution. Known Distribution: 4 Weights selected i.i.d. from known distribution. Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 6

Models of Weight Assignment: Conjecture [BIK07]: O ( 1 ) -competitive algorithm for all these models Adversarial Assignment: 1 Hidden weights are arbitrary. O ( 1 ) -competitive alg. for partition, graphic, transversal, laminar. [L61,D63,K05,BIK07,DP08,KP09,BDGIT09,IW11] O ( log rk ( M )) -competitive algorithms for general matroids. [BIK07] Random Assignment: [S11] O ( 1 ) -competitive algorithm. 2 A hidden (adversarial) list of weights is assigned uniformly. Unknown distribution: 3 Weights selected i.i.d. from unknown distribution. Known Distribution: 4 Weights selected i.i.d. from known distribution. Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 6

Random Assignment. Data: Known matroid M = ( E , I ) on n elements. Hidden list of weights: W : w 1 ≥ w 2 ≥ w 3 ≥ · · · ≥ w n ≥ 0 . Random assignment. σ : W → E . Random order. π : E → { 1 , . . . , n } . Objective Return an independent set ALG ∈ I such that: E π,σ [ w ( ALG )] ≥ α · E σ [ w ( OPT )] , where w ( S ) = � e ∈ S σ − 1 ( e ) . OPT is the optimum base of M under assignment σ . (Greedy) α : Competitive Factor. Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 7

Divide and Conquer to get O ( 1 ) -competitive algorithm. For a general matroid M = ( E , I ) : Find matroids M i = ( E i , I i ) with E = � k i = 1 E i . M 1 , E 1 M i admits O ( 1 ) -competitive algorithm 1 (Easy parts). M 2 , E 2 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). M k , E k Optimum in � k i = 1 M i is comparable with 3 Optimum in M . (Don’t lose much). Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 8

(Easy matroids) : Uniformly dense matroids are like Uniform Definition (Uniformly dense) A loopless matroid M = ( E , I ) is uniformly dense if | F | | E | rk ( F ) ≤ rk ( E ) , for all F � = ∅ . e.g. Uniform ( rk ( F ) = min ( | F | , r ) ). Graphic K n . Projective Spaces. Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 9

(Easy matroids) : Uniformly dense matroids are like Uniform Definition (Uniformly dense) A loopless matroid M = ( E , I ) is uniformly dense if | F | | E | rk ( F ) ≤ rk ( E ) , for all F � = ∅ . e.g. Uniform ( rk ( F ) = min ( | F | , r ) ). Graphic K n . Projective Spaces. Property: Sets of rk ( E ) elements have almost full rank. E ( X : | X | = rk ( E )) [ rk ( X )] ≥ rk ( E )( 1 − 1 / e ) . Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 9

Uniformly dense matroid: Simple algorithm · · · Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 10

Uniformly dense matroid: Simple algorithm · · · � � �� � � �� � � �� � �� � � �� � n / r n / r n / r n / r n / r Simulate e / C -comp. alg. for Uniform Matroids with r = rk M ( E ) . Try to add each selected weight to the independent set. Selected elements have expected rank ≥ r ( 1 − 1 / e ) . Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 10

Uniformly dense matroid: Simple algorithm · · · � � �� � � �� � � �� � �� � � �� � n / r n / r n / r n / r n / r Simulate e / C -comp. alg. for Uniform Matroids with r = rk M ( E ) . Try to add each selected weight to the independent set. Selected elements have expected rank ≥ r ( 1 − 1 / e ) . Lemma: Constant competitive algorithm for Uniformly Dense. � � r E π,σ [ w ( ALG )] ≥ C 1 − 1 � w i ≥ K · E π [ w ( OPT M )] . e e i = 1 � �� � K In fact: E π,σ [ w ( ALG )] ≥ K · E σ [ w ( OPT P )] , where P is the uniform matroid in E with bound r = rk M ( E ) . Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 10

Uniformly Dense (sub)matroids That combine nicely Densest Submatroid Let M = ( E , I ) be a loopless matroid. M , E Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 11

Uniformly Dense (sub)matroids That combine nicely Densest Submatroid Let M = ( E , I ) be a loopless matroid. M 1 , E 1 Let E 1 be the densest set of M of maximum cardinality. | F | | E 1 | γ ( M ) := max rk M ( F ) = rk M ( E 1 ) . M , E F ⊆ E M 1 = M| E 1 is uniformly dense. Jos´ e Soto - M.I.T. Matroid Secretary - Random Assignment SODA 2011 11