Streaming Algorithms for Submodular Function Maximization

Transcription

1 Streaming Algorithms for Submodular Function Maximization Chandra Chekuri Shalmoli Gupta Kent Quanrud University of Illinois at Urbana-Champaign October 6, 2015

2 Submodular functions f : 2 N R if S T N, and e N \ T, then f(s + e) f(s) f(t + e) f(e) we will abbreviate f S (e) def = f(s + e) f(s)

3 Types of submodular f Monotone S T f(s) f(t ) Nonnegative f(s) 0

4 Coverage N = subsets of points, f(s) = s S s

5 Coverage N = subsets of points, f(s) = s S s

6 Coverage N = subsets of points, f(s) = s S s

7 Coverage N = subsets of points, f(s) = s S s

8 Coverage N = subsets of points, f(s) = s S s

9 Coverage N = subsets of points, f(s) = s S s

10 Coverage N = subsets of points, f(s) = s S s

11 Coverage N = subsets of points, f(s) = s S s

12 Directed edge cuts N = V, f(s) = E(S, V \ S)

13 Directed edge cuts N = V, f(s) = E(S, V \ S)

14 Directed edge cuts N = V, f(s) = E(S, V \ S)

15 Directed edge cuts N = V, f(s) = E(S, V \ S)

16 Directed edge cuts +3 arcs N = V, f(s) = E(S, V \ S)

17 Directed edge cuts +3 arcs +1 arc N = V, f(s) = E(S, V \ S)

18 Directed edge cuts N = V, f(s) = E(S, V \ S)

19 Directed edge cuts +2 arcs N = V, f(s) = E(S, V \ S)

20 Directed edge cuts +2 arcs 1 arc N = V, f(s) = E(S, V \ S)

21 Canonical problem Pick (up to) k elements e 1,..., e k N maximizing f({e 1,..., e k })

22 Streaming model N = {e 1, e 2,...} presented one at a time Arbitrary order Our main constraint is space (ideally, Õ(k)) e i+3 e i+2 e i+1 e i e i 1 e i 2 e i 3 Õ(k) space

23 Oracle model Black box access for: (a) Evaluating f(s) e i+3 e i+2 e i+1 e i e i 1 e i 2 e i 3 Õ(k) space Oracle for f

24 Oracle model Black box access for: (a) Evaluating f(s) (b) Checking if S is feasible (for combinatorial constraints) e i+3 e i+2 e i+1 e i e i 1 e i 2 e i 3 Õ(k) space feasibility oracle Oracle for f

25 Monotone f in streams O(k log(k)/ɛ) space Oracle for f Constraint: cardinality Approximation ratio: 1 2 ɛ Badanidiyuru, Mirzasoleiman, Karbasi, and Krause KDD 2014

26 Monotone f in streams e i+3 e i+2 e i+1 e i e i 1 e i 2 e i 3 O(k) space feasibility oracle Oracle for f Constraint: Matroids, matchings, matroid intersection Approximation ratio: 1 4p for p matroids Chakrabarti and Kale IPCO 2014

27 Nonnegative f in streams (our result) e i+3 e i+2 e i+1 e i e i 1 e i 2 e i 3 O(k log(k)/ɛ 2 ) space Oracle for f Constraint: Cardinality Approximation ratio: 1 ɛ 2+e

28 Nonnegative f in streams (our result) e i+3 e i+2 e i+1 e i e i 1 e i 2 e i 3 O(k log(k)/ɛ 2 ) space feasibility oracle Oracle for f Constraint: p-matchoid M = (N, I) Approximation ratio: Ω ( 1 p )

29 Monotone Nonnegative Cardinality p matroids 1 ɛ 2 1 4p 1 ɛ 2+ɛ (1 ɛ)(p 1) 5p 2 4p

30 Monotone submodular maximization

31 Nemhauser, Wolsey, Fisher 1978 greedy 1 1/e approximation for cardinality constraint

32 greedy S for i = 1,..., k e i arg max e N f S (e) S S + e i N N e i return S (recall f S (e) = f(s + e) f(s))

33 Monotone submodular maximization in streams Õ(k) space feasibility oracle Oracle for f

34 Setup? You have a running solution S. ( S k) Oracle for f The stream gives you an element e. S Should you add e to S?

35 Thresholding Badanidiyuru, Mirzasoleiman, Karbasi, and Krause if S < k if f S (e) OPT/2k S S + e

36 Guessing OPT OP T = (1 + ɛ) i 2 O(k) space OP T = (1 + ɛ) i 1 O(k) space OP T = (1 + ɛ) i O(k) space OP T = (1 + ɛ) i+1 O(k) space OP T = (1 + ɛ) i+2 O(k) space

37 Thresholding Badanidiyuru, Mirzasoleiman, Karbasi, and Krause if S < k if f S (e) OPT/2k S S + e

38 Thresholding S < k? Oracle for f Yes No Thresholding S

39 Exchange-based algorithm Chakrabarti and Kale if S < k then S S + e else if d S s.t. exchanging d for e is a good enough exchange then S S d + e

40 Exchanges S < k? Oracle for f Yes No Exchanges S

41 Nonnegative submodular maximization

42 greedy does not work

43 Something like greedy works Gupta, Roth, Schoenebeck, Talwar WINE 2010 iterated-greedy Ω(1/p) approximation for p systems Buchbinder, Feldman, Naor, Schwartz SODA 2014 randomized-greedy 1/e approximation for cardinality

44 greedy? Let S be greedy solution, and T an optimal solution greedy gets you f(s) c f(s T ) for some constant c without invoking monotonicity if f is monotone, then f(s T ) f(t ) = OPT

45 If f is not monotone, then f(s T ) what?

46 Randomization lemma if S is a random set with P[e S] p for all e, then E[f(S T )] (1 p)f(t ). Buchbinder, Feldman, Naor, Schwartz

47 randomized-greedy S for i = 1,..., k let e 1,..., e k maximize f S (e) pick e j {e 1,..., e k } randomly S S + e j N N e j return S Buchbinder, Feldman, Naor, Schwartz SODA 2014

48 Nonnegative submodular maximization in streams Õ(k) space feasibility oracle Oracle for f

49 Randomized-Streaming-Greedy S < k? Oracle for f Yes Thresholding No Exchanges Randomized Buffer S

50 Randomized-Streaming-Greedy S, B for each element e in the stream if Is-Good(S,e) B B + e if B is full // B = Θ(k) pick e B randomly add or exchange e into S clean up B S Offline(f,B) return better of S and S

51 Is-Good(S,e) if S < k if f S (e) Ω(OPT/k) then return GOOD else // S = k if d S such that f S (e) 2ν(f, S, d) + Ω(OPT/k) then return GOOD return BAD

52 Magic value ν(f, S, d) ν(f, S, d) should: if d S such that f S (e) 2ν(f, S, d) + Ω(OPT/k) then return GOOD

53 Magic value ν(f, S, d) ν(f, S, d) should: if d S such that f S (e) 2ν(f, S, d) + Ω(OPT/k) then return GOOD Account for the value originally added by d to S.

54 Magic value ν(f, S, d) ν(f, S, d) should: if d S such that f S (e) 2ν(f, S, d) + Ω(OPT/k) then return GOOD Account for the value originally added by d to S. Adapt dynamically to changing S.

55 Magic value ν(f, S, d) ν(f, S, d) should: if d S such that f S (e) 2ν(f, S, d) + Ω(OPT/k) then return GOOD Account for the value originally added by d to S. Adapt dynamically to changing S. Ensure that exchanging S S d + e increases f(s) substantially.

56 Incremental value Let S = {d 1,..., d k } in order of insertion The incremental value of d i is defined as ν(f, S, d i ) def = f(d 1,..., d i ) f(d 1,..., d i 1 ).

57 Incremental value Let S = {d 1,..., d k } in order of insertion The incremental value of d i is defined as ν(f, S, d i ) def = f(d 1,..., d i ) f(d 1,..., d i 1 ). Property 1: When we add an element e to the running solution S d, ν(f, S d + e, e) = f S d (e)

58 Incremental value Let S = {d 1,..., d k } in order of insertion The incremental value of d i is defined as ν(f, S, d i ) def = f(d 1,..., d i ) f(d 1,..., d i 1 ). Property 2: ν telescopes. ν(f, S, d) = f(s) d S

59 Incremental value Let S = {d 1,..., d k } in order of insertion The incremental value of d i is defined as ν(f, S, d i ) def = f(d 1,..., d i ) f(d 1,..., d i 1 ). Property 3: For fixed d S, its incremental value ν(f, S, d) only increases over the course of the algorithm.

60 Is-Good(S,e) if S < k if f S (e) Ω(OPT/k) then return GOOD else // S = k if d S such that f S (e) 2ν(f, S, d) + Ω(OPT/k) then return GOOD return BAD

61 Randomized-Streaming-Greedy S, B for each element e in the stream if Is-Good(S,e) B B + e if B is full // B = Θ(k) pick e B randomly add or exchange e into S clean up B S Offline(f,B) return better of S and S

62 Conclusion

63 Technical result Constant factor approximations Nonnegative submodular maximization 1-pass streams Broad class of combinatorial constraints

64 Main techniques Randomized buffer Greedy w/r/t incremental value Post-processing

65 Open questions Modeling Lower bounds

66 thanks