Maximum Coverage over a Matroid Constraint

Transcription

1 Maximum Coverage over a Matroid Constraint Yuval Filmus Justin Ward University of Toronto STACS 2012, Paris

2 Max Coverage: History Location of bank accounts: Cornuejols, Fisher & Nemhauser 1977, Management Science Official definition: Hochbaum & Pathria 1998, Naval Research Quarterly Lower bound: Feige 1998 Extended to Submodular Max. over a Matroid: Calinescu, Chekuri, Pál & Vondrák 2008 (with help from Ageev & Sviridenko 2004) We consider Maximum Coverage over a Matroid.

3 Maximum Coverage... Input: Universe U with weights w 0 Sets S i U Number n Goal: Find n sets S i that maximize w(s i1 S in )

4 Maximum Coverage... Input: Universe U with weights w 0 Sets S i U Number n Goal: Find n sets S i that maximize w(s i1 S in ) Greedy algorithm gives 1 1/e approximation. Feige ( 98): optimal unless P=NP.

5 ... over a Matroid Input: Goal: Universe U with weights w 0 Sets S i U Matroid m over set of all S i Find collection of sets S m that maximizes w( S)

6 What is a matroid? Invented by Whitney (1935). Definition: Matroid A collection of independent sets s.t. 1 A independent, B A B independent. 2 A, B independent, A > B there exists some x A \ B s.t. B x is independent.

7 What is a matroid? Invented by Whitney (1935). Definition: Matroid A collection of independent sets s.t. 1 A independent, B A B independent. 2 A, B independent, A > B there exists some x A \ B s.t. B x is independent. Partition Matroid F 1,..., F n disjoint sets. Independent set: 1 set from each F i.

8 Max Coverage over a Partition Matroid Input: Goal: Universe U with weights w 0 n families F i 2 U Find collection of sets S i F i that maximizes w(s 1 S n )

9 Some algorithms Greedy 1 Pick set S 1 of maximal weight. 2 Pick set S 2 of maximal additional weight. 3 And so on.

10 Some algorithms Greedy 1 Pick set S 1 of maximal weight. 2 Pick set S 2 of maximal additional weight. 3 And so on. Local Search 1 Start at some solution S 1,..., S n. 2 Replace some S i with some S that improves i total weight. 3 Repeat Step 2 while possible.

11 Failure of greedy Bad instance for Greedy A 1 = {x, ɛ} B = {x} A 2 = {y} F 1 F 2 w(x) = w(y) w(ɛ) Greedy chooses {A 1, B}, optimal is {A 2, B}. Resulting approximation ratio is only 1/2.

12 Failure of greedy Bad instance for Greedy A 1 = {x, ɛ} B = {x} A 2 = {y} F 1 F 2 w(x) = w(y) w(ɛ) Greedy chooses {A 1, B}, optimal is {A 2, B}. Resulting approximation ratio is only 1/2. Local search finds optimal solution.

13 Maybe local search? Bad instance for Local Search A 1 = {x, ɛ x } B 1 = {x} A 2 = {y} B 2 = {ɛ y } F 1 F 2 w(x) = w(y) w(ɛ x ) = w(ɛ y ) {A 1, B 2 } is local maximum. Optimum is {A 2, B 1 }.

14 Maybe local search? Bad instance for Local Search A 1 = {x, ɛ x } B 1 = {x} A 2 = {y} B 2 = {ɛ y } F 1 F 2 w(x) = w(y) w(ɛ x ) = w(ɛ y ) {A 1, B 2 } is local maximum. Optimum is {A 2, B 1 }. k-local search (on SBO matroids) has approx ratio k 1 2n k 1.

15 Local search fantasy A 1 = {x, ɛ x } B 1 = {x} A 2 = {y} B 2 = {ɛ y } Fantasy algorithm

16 Local search fantasy A 1 = {x, ɛ x } B 1 = {x} A 2 = {y} B 2 = {ɛ y } x 1 ɛ x 1 Greedy stage

17 Local search fantasy A 1 = {x, ɛ x } B 1 = {x} A 2 = {y} B 2 = {ɛ y } x 1 ɛ x 1 ɛ y 1 Greedy stage

18 Local search fantasy A 1 = {x, ɛ x } B 1 = {x} A 2 = {y} B 2 = {ɛ y } x 2 ɛ x 1 Local search stage We lose ɛ y but gain second appearance of x.

19 Local search fantasy A 1 = {x, ɛ x } B 1 = {x} A 2 = {y} B 2 = {ɛ y } x 1 y 1 Local search stage

20 Local search fantasy A 1 = {x, ɛ x } B 1 = {x} A 2 = {y} B 2 = {ɛ y } x 1 y 1 Done found optimal solution

21 Non-oblivious local search Idea Give more weight to duplicate elements. Use local search with auxiliary objective function (Alimonti 94; Khanna, Motwani, Sudan & U. Vazirani 98): f(s) = α #u (S)w(u). u U Change is considered beneficial if it improves f(s). Oblivious local search: α 0 = 0, α i = 1 for i 1.

22 Choosing the weights Consider what happens at termination. Setup: S 1,..., S n : local maximum. O 1,..., O n : optimal solution. Local optimality implies nf(s 1,..., S n ) n f(s 1,..., S i 1, O i, S i+1,..., S n ) i=1

23 Choosing the weights Parametrize situation using w l,c,g = total weight of elements which belong to l + c sets S i g + c sets O i c of the indices in common Each element of the universe occurs in some w l,c,g.

24 Choosing the weights Local optimality implies nf(s 1,..., S n ) n f(s 1,..., S i 1, O i, S i+1,..., S n ) i=1 In terms of w l,c,g, this is [l(α l+c α l+c 1 ) + g(α l+c α l+c+1 )] w l,c,g 0 l,c,g

25 Choosing the weights Local optimality translates to [(l + g)α l+c lα l+c 1 gα l+c+1 ] w l,c,g 0 l,c,g Also, w(o 1,..., O n ) = w(s 1,..., S n ) = g+c 1 l+c 1 w l,c,g w l,c,g

26 Choosing the weights Approximation ratio θ is given by s.t. max min w(s 1,..., S n ) α i w l,c,g w(o 1,..., O n ) = 1 nf(s 1,..., S n ) w l,c,g 0 n f(s 1,..., S i 1, O i, S i+1,..., S n ) i=1

27 Choosing the weights Approximation ratio θ is given by s.t. max min α i w l,c,g g+c 1 l+c 1 w l,c,g = 1 w l,c,g [(l + g)α l+c lα l+c 1 gα l+c+1 ] w l,c,g 0 l,c,g w l,c,g 0

28 Choosing the weights Dualize the inner LP: max θ α i,θ s.t. l(α l α l 1 ) 1 gα 1 θ (l + g)α l+c lα l+c 1 gα l+c+1 1 θ (c 1 or l,g 1)

29 Optimal weights Solution to LP is θ = 1 1/e and α 0 = 0, α 1 = θ, α l+1 = (l + 1)α l lα l 1 (1 θ). Sequence monotone concave, α l = 1 e log l + O(1).

30 Optimal weights Solution to LP is θ = 1 1/e and α 0 = 0, α 1 = θ, α l+1 = (l + 1)α l lα l 1 (1 θ). Sequence monotone concave, α l = 1 e For rank n, can replace e with log l + O(1). e [n] = n 1 k=0 1 k! + 1 (n 1) (n 1)! e + 1 (n + 2)!.

31 Optimal weights (normalized)

32 Recap Our Algorithm 1 Start with some solution S 1,..., S n. 2 Repeat while possible: Replace some S i with some S improving i f(s 1,..., S n ) = α #u (S 1,...,S n )w(u). u S 1 S n Main Theorem At the end of the algorithm, ( w(s 1,..., S n ) 1 1 ) w(o 1,..., O n ). e

33 Further work Our framework generalizes. Montone submodular maximization over a matroid Optimal combinatorial algorithm. Continuous algorithm by Calinescu, Chekuri, Pál and Vondrák (STOC 2008).

34 Further work Our framework generalizes. Montone submodular maximization over a matroid Optimal combinatorial algorithm.... with curvature constraint Optimal combinatorial algorithm. NP-hardness result (extending Feige 1998). Vondrák 2010: extended continuous algorithm, gave lower bound in oracle model.

35 Further work Our framework generalizes. Montone submodular maximization over a matroid Optimal combinatorial algorithm.... with curvature constraint Optimal combinatorial algorithm. NP-hardness result. Submodular maximization over bases of matroid 1 2/e combinatorial algorithm. Best prior result: 1/4 (Vondrák, FOCS 2009).

36 Thank you Questions?