Matroid Secretary Problem in the Random Assignment Model

Transcription

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

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

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

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. José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

5 Algorithms for classical problem (uniform matroid). For r = 1: Dynkin s Algorithm n/e Observe n/e objects. Accept the first record after that. General r Top weight is selected w.p. 1/e. Divide in r classes and apply Dynkin s algorithm in each class. José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

6 Algorithms for classical problem (uniform matroid). For r = 1: Dynkin s Algorithm n/e Observe n/e objects. Accept the first record after that. General r Top weight is selected w.p. 1/e. 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. José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

7 Algorithms for classical problem (uniform matroid). For r = 1: Dynkin s Algorithm n/e Observe n/e objects. Accept the first record after that. General r Top weight is selected w.p. 1/e. 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. José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

8 Harder example: Gammoid Servers Clients Elements. Connections Independent Sets: Clients that can be simultaneously connected to Servers using edge-disjoint paths. José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

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

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

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

12 Models of Weight Assignment: Conjecture [BIK07]: O(1)-competitive algorithm for all these models 1 Adversarial Assignment: 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] 2 Random Assignment: [S11] O(1)-competitive algorithm. A hidden (adversarial) list of weights is assigned uniformly. 3 Unknown distribution: Weights selected i.i.d. from unknown distribution. 4 Known Distribution: Weights selected i.i.d. from known distribution. José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

13 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. José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

14 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. 1 M i admits O(1)-competitive algorithm (Easy parts). 2 Union of independent sets in each M i is independent in M. I( k i=1 M i) I(M). (Combine nicely). M 1, E 1 M 2, E 2. M, E 3 Optimum in k i=1 M i is comparable with Optimum in M. (Don t lose much). M k, E k José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

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

16 (Easy matroids): Uniformly dense matroids are like Uniform Definition (Uniformly dense) A loopless matroid M = (E, I) is uniformly dense if F rk(f) E, for all F. rk(e) 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). José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

17 Uniformly dense matroid: Simple algorithm José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

18 Uniformly dense matroid: Simple algorithm 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). José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

19 Uniformly dense matroid: Simple algorithm 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. E π,σ [w(alg)] C ( 1 1 ) r 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). José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

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

21 Uniformly Dense (sub)matroids That combine nicely Densest Submatroid Let M = (E, I) be a loopless matroid. Let E 1 be the densest set of M of maximum cardinality. M 1, E 1 γ(m) := max F E F rk M (F) = E 1 rk M (E 1 ). M, E M 1 = M E1 is uniformly dense. José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

22 Uniformly Dense (sub)matroids That combine nicely Densest Submatroid Let M = (E, I) be a loopless matroid. Let E 1 be the densest set of M of maximum cardinality. M 1, E 1 γ(m) := max F E F rk M (F) = E 1 rk M (E 1 ). M, E M 1 = M E1 is uniformly dense. M = M/E 1 is loopless and M, E \ E 1 γ(m ) := max F E\E 1 F rk M (F) < γ(m). I 1 I 1, I I implies I 1 I I. José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

23 Uniformly Dense (sub)matroids That combine nicely Densest Submatroid Let E 2 be the densest set of M of maximum cardinality. M 2 = M E2 is uniformly dense. M = M/(E 1 E 2 ) is loopless and γ(m ) < γ(m 2 ) < γ(m 1 ) = γ(m). M 1, E 1 M 2, E 2 M, E \ (E 1 E 2 ) M, E José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

24 Uniformly Dense (sub)matroids That combine nicely Densest Submatroid Let E 2 be the densest set of M of maximum cardinality. M 2 = M E2 is uniformly dense. M = M/(E 1 E 2 ) is loopless and γ(m ) < γ(m 2 ) < γ(m 1 ) = γ(m). Iterate... M 1, E 1 M 2, E 2. M, E M k, E k José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

25 Principal Partition of a matroid Theorem (Principal Partition) Given M = (E, I) loopless, there exists a partition E = k i=1 E i such that 1 The principal minor M i = (M/E i 1 ) Ei is a uniformly dense matroid with density 2 λ 1 > λ 2 > > λ k 1. Note: λ i = γ(m i ) = E i r i. If I i I(M i ), then I 1 I 2 I k I(M). Can compute the partition in polynomial time. M 1, E 1 M 2, E 2. M k, E k M, E José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

26 Algorithm for a General Matroid M Algorithm 1 Remove the loops from M. 2 Let M 1, M 2,..., M k be the principal minors. 3 In each M i use the K -competitive algorithm for uniformly dense matroids to obtain an independent set I i. 4 Return ALG = I 1 I 2 I k. José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

27 Algorithm for a General Matroid M Algorithm 1 Remove the loops from M. 2 Let M 1, M 2,..., M k be the principal minors. 3 In each M i use the K -competitive algorithm for uniformly dense matroids to obtain an independent set I i. 4 Return ALG = I 1 I 2 I k. José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

28 Analysis. From Uniformly Dense Matroids to Uniform Matroids Each M i is uniformly dense. Let P i be the uniform matroid on E i with bounds r i = rk Mi (E i ). Let P = k i=1 P i be the corresponding partition matroid. By Lemma: E π,σ [w(alg E i )] K E σ [w(opt Pi )]. Hence: E π,σ [w(alg)] K E σ [w(opt P )]. José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

29 Analysis. From Uniformly Dense Matroids to Uniform Matroids Each M i is uniformly dense. Let P i be the uniform matroid on E i with bounds r i = rk Mi (E i ). Let P = k i=1 P i be the corresponding partition matroid. By Lemma: E π,σ [w(alg E i )] K E σ [w(opt Pi )]. To conclude we show: Hence: E π,σ [w(alg)] K E σ [w(opt P )]. ( ): E σ [w(opt P )] (1 1/e) E σ [w(opt M )]. ( ): E[rk P (X j )] (1 1/e) E[rk M (X j )], for all j, where X j is a uniform random set of j elements in E. José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

30 Analysis II: Proof of E[rk P (X j )] (1 1/e) E[rk M (X j )]. E[rk P (X j )] = = k k E[rk P (X j E i )] = E[min( X j E i, r i )] i=1 i=1 k (1 1/e) min(e[ X j E i ], r i ) i=1 k (1 1/e) min( E i j n, r i). i=1 Since λ i = E i /r i is a decreasing sequence, there is an index i = i (j) such that: i k E[rk P (X j )] (1 1/e) r i + E i j. n i=1 i=i +1 José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

31 Analysis III: Proof of E[rk P (X j )] (1 1/e) E[rk M (X j )]. i k E[rk P (X j )] (1 1/e) r i + E i j n i=1 i=i +1 ( ) j (1 1/e) rk M (E 1 E i ) + (E }{{} i +1 E k ) n E = (1 1/e) (rk M (E ) + E[ X j (E \ E ) ] ) (1 1/e) E[rk M (X j E ) + rk M (X j (E \ E )] (1 1/e) E[rk M (X j )]. José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

32 Analysis III: Proof of E[rk P (X j )] (1 1/e) E[rk M (X j )]. i k E[rk P (X j )] (1 1/e) r i + E i j n i=1 i=i +1 ( ) j (1 1/e) rk M (E 1 E i ) + (E }{{} i +1 E k ) n E = (1 1/e) (rk M (E ) + E[ X j (E \ E ) ] ) (1 1/e) E[rk M (X j E ) + rk M (X j (E \ E )] (1 1/e) E[rk M (X j )]. Therefore: E π,σ [w(alg)] K (1 1/e) E σ [w(opt M )]. José Soto - M.I.T. Matroid Secretary - Random Assignment SODA

33 Conclusions and Open Problems. Summary Open First constant competitive algorithm for Matroid Secretary Problem in Random Assignment Model. Corollary: Also holds for i.i.d. weights from known or unknown distributions. Algorithm does not use hidden weights (only relative ranks). Find constant competitive algorithm for General Matroids under Adversarial Assignment. Extend to other independent systems: Note[BIK07]: Ω(log(n)/ log log(n)) lower bound. José Soto - M.I.T. Matroid Secretary - Random Assignment SODA