An Introduction to Matroids & Greedy in Approximation Algorithms

Transcription

1 An Introduction to Matroids & Greedy in Approximation Algorithms (Juliàn Mestre, ESA 2006) CoReLab Monday seminar presentation: Evangelos Bampas 1/21

2 Subset systems CoReLab Monday seminar presentation: Evangelos Bampas 2/21

3 Subset systems Let E be a finite set. Let L be a non-empty family of subsets of E (independent sets). CoReLab Monday seminar presentation: Evangelos Bampas 2/21

4 Subset systems Let E be a finite set. Let L be a non-empty family of subsets of E (independent sets). Def. (E, L) is called a subset system if: A L, A A, A L [hereditary property] CoReLab Monday seminar presentation: Evangelos Bampas 2/21

5 Subset systems Let E be a finite set. Let L be a non-empty family of subsets of E (independent sets). Def. (E, L) is called a subset system if: A L, A A, A L [hereditary property] A (positive) weight function w defined on E induces a weight function defined on L: w (X) = e X w (e) CoReLab Monday seminar presentation: Evangelos Bampas 2/21

6 Picking a heaviest independent set Prob. Given a subset system (E, L) and a weight function w : E R +, pick a maximum-weight element of L. CoReLab Monday seminar presentation: Evangelos Bampas 3/21

7 Picking a heaviest independent set Prob. Given a subset system (E, L) and a weight function w : E R +, pick a maximum-weight element of L. Alg. Greedy: SOL for each e E in non-increasing order of w (e) if SOL + e L then SOL SOL + e return SOL CoReLab Monday seminar presentation: Evangelos Bampas 3/21

8 Picking a heaviest independent set Prob. Given a subset system (E, L) and a weight function w : E R +, pick a maximum-weight element of L. Alg. Greedy: SOL for each e E in non-increasing order of w (e) if SOL + e L then SOL SOL + e return SOL Thm. Greedy is optimal for any weight function on (E, L) iff (E, L) is a matroid. (Rado-Edmonds) CoReLab Monday seminar presentation: Evangelos Bampas 3/21

9 What on earth is a matroid? Def. A subset system (E, L) is a matroid if: A L, B L with A < B, z B \ A such that A + z L [augmentation property] CoReLab Monday seminar presentation: Evangelos Bampas 4/21

10 What on earth is a matroid? Def. A subset system (E, L) is a matroid if: A L, B L with A < B, z B \ A such that A + z L [augmentation property] A CoReLab Monday seminar presentation: Evangelos Bampas 4/21

11 What on earth is a matroid? Def. A subset system (E, L) is a matroid if: A L, B L with A < B, z B \ A such that A + z L [augmentation property] A B : A < B CoReLab Monday seminar presentation: Evangelos Bampas 4/21

12 What on earth is a matroid? Def. A subset system (E, L) is a matroid if: A L, B L with A < B, z B \ A such that A + z L [augmentation property] A z: A+z B : A < B CoReLab Monday seminar presentation: Evangelos Bampas 4/21

13 Matroid examples Ex. 1 Subsets of at most k elements. k N. E: finite set. L = {X E : X k}. CoReLab Monday seminar presentation: Evangelos Bampas 5/21

14 Matroid examples Ex. 1 Subsets of at most k elements. k N. E: finite set. L = {X E : X k}. non-empty: k L CoReLab Monday seminar presentation: Evangelos Bampas 5/21

15 Matroid examples Ex. 1 Subsets of at most k elements. k N. E: finite set. L = {X E : X k}. hereditary: if A L and A A, then: A A k A L CoReLab Monday seminar presentation: Evangelos Bampas 5/21

16 Matroid examples Ex. 1 Subsets of at most k elements. k N. E: finite set. L = {X E : X k}. augmentation: if A < B, then for arbitrary z B \ A: A + z = A + 1 B k thus, A + z L CoReLab Monday seminar presentation: Evangelos Bampas 5/21

17 Matroid examples Ex. 1 Subsets of at most k elements. k N. E: finite set. L = {X E : X k}. Cor. We can find a heaviest subset of k elements using the Greedy algorithm. CoReLab Monday seminar presentation: Evangelos Bampas 5/21

18 Matroid examples (cont d) Ex. 2 Column matroids. A: matrix with elements from a field. E = { x : x is a column of A}. L = {X E : X is linearly independent}. CoReLab Monday seminar presentation: Evangelos Bampas 6/21

19 Matroid examples (cont d) Ex. 2 Column matroids. A: matrix with elements from a field. E = { x : x is a column of A}. L = {X E : X is linearly independent}. non-empty: the empty set is vacuously linearly independent CoReLab Monday seminar presentation: Evangelos Bampas 6/21

20 Matroid examples (cont d) Ex. 2 Column matroids. A: matrix with elements from a field. E = { x : x is a column of A}. L = {X E : X is linearly independent}. hereditary: linear dependency cannot be introduced by removing vectors CoReLab Monday seminar presentation: Evangelos Bampas 6/21

21 Matroid examples (cont d) Ex. 2 Column matroids. A: matrix with elements from a field. E = { x : x is a column of A}. L = {X E : X is linearly independent}. augmentation: if z B \ A, A + z L, then each vector of B is linearly dependent on the vectors of A which implies B A CoReLab Monday seminar presentation: Evangelos Bampas 6/21

22 Matroid examples (cont d) Ex. 2 Column matroids. A: matrix with elements from a field. E = { x : x is a column of A}. L = {X E : X is linearly independent}. Cor. We can find a heaviest base among the vectors of A using the Greedy algorithm. CoReLab Monday seminar presentation: Evangelos Bampas 6/21

23 Matroid examples (cont d) Ex. 3 Cycle/Graphic matroids. G = (V, E): undirected graph. E = edge set of G. L = {X E : G X = (V, X) is a forest}. CoReLab Monday seminar presentation: Evangelos Bampas 7/21

24 Matroid examples (cont d) Ex. 3 Cycle/Graphic matroids. G = (V, E): undirected graph. E = edge set of G. L = {X E : G X = (V, X) is a forest}. non-empty: G = (V, ) is a forest CoReLab Monday seminar presentation: Evangelos Bampas 7/21

25 Matroid examples (cont d) Ex. 3 Cycle/Graphic matroids. G = (V, E): undirected graph. E = edge set of G. L = {X E : G X = (V, X) is a forest}. hereditary: any subset of a forest is a forest CoReLab Monday seminar presentation: Evangelos Bampas 7/21

26 Matroid examples (cont d) Ex. 3 Cycle/Graphic matroids. G = (V, E): undirected graph. E = edge set of G. L = {X E : G X = (V, X) is a forest}. augmentation: if A < B, then #trees in G A > #trees in G B CoReLab Monday seminar presentation: Evangelos Bampas 7/21

27 Matroid examples (cont d) Ex. 3 Cycle/Graphic matroids. G = (V, E): undirected graph. E = edge set of G. L = {X E : G X = (V, X) is a forest}. augmentation: G B u v CoReLab Monday seminar presentation: Evangelos Bampas 7/21

28 Matroid examples (cont d) Ex. 3 Cycle/Graphic matroids. G = (V, E): undirected graph. E = edge set of G. L = {X E : G X = (V, X) is a forest}. augmentation: G B u G A v u v CoReLab Monday seminar presentation: Evangelos Bampas 7/21

29 Matroid examples (cont d) Ex. 3 Cycle/Graphic matroids. G = (V, E): undirected graph. E = edge set of G. L = {X E : G X = (V, X) is a forest}. augmentation: G B u G A +(u,v) v u v CoReLab Monday seminar presentation: Evangelos Bampas 7/21

30 Matroid examples (cont d) Ex. 3 Cycle/Graphic matroids. G = (V, E): undirected graph. E = edge set of G. L = {X E : G X = (V, X) is a forest}. Cor. We can find a heaviest spanning tree of G using the Greedy algorithm. CoReLab Monday seminar presentation: Evangelos Bampas 7/21

31 k-extendible systems Def. A subset system (E, L) is k-extendible if: C L, x C with C + x L, D extension of C, Y D \ C such that Y k and D \ Y + x L CoReLab Monday seminar presentation: Evangelos Bampas 8/21

32 k-extendible systems Def. A subset system (E, L) is k-extendible if: C CoReLab Monday seminar presentation: Evangelos Bampas 8/21

33 k-extendible systems Def. A subset system (E, L) is k-extendible if: C x: C+x CoReLab Monday seminar presentation: Evangelos Bampas 8/21

34 k-extendible systems Def. A subset system (E, L) is k-extendible if: D C x: C+x CoReLab Monday seminar presentation: Evangelos Bampas 8/21

35 k-extendible systems Def. A subset system (E, L) is k-extendible if: D C Y x: C+x CoReLab Monday seminar presentation: Evangelos Bampas 8/21

36 k-extendible systems Def. A subset system (E, L) is k-extendible if: D C Y Y k x: C+x CoReLab Monday seminar presentation: Evangelos Bampas 8/21

37 k-extendible systems Def. A subset system (E, L) is k-extendible if: D C Y x: C+x Y k D\Y+x CoReLab Monday seminar presentation: Evangelos Bampas 8/21

38 Matroids 1-extendible systems Thm. (E, L) is a matroid iff (E, L) is 1-extendible. A z: A+z B : A < B D C Y x: C+x Y 1 D\Y+x CoReLab Monday seminar presentation: Evangelos Bampas 9/21

39 Matroids 1-extendible systems C D x: C+x CoReLab Monday seminar presentation: Evangelos Bampas 10/21

40 Matroids 1-extendible systems C D x: C+x If C + x = D, then D \ C = 1. CoReLab Monday seminar presentation: Evangelos Bampas 10/21

41 Matroids 1-extendible systems C Y D x: C+x D \ Y + x L. CoReLab Monday seminar presentation: Evangelos Bampas 10/21

42 Matroids 1-extendible systems C D z 0 x: C+x Applying the augmentation property, z 0 D \ C such that C + z 0 + x L. CoReLab Monday seminar presentation: Evangelos Bampas 10/21

43 Matroids 1-extendible systems C D z 0 z 1 Y z x: C+x Finally, we get a sequence z 0,..., z ρ such that C + z i + x L and D \ (C + z i ) = 1. CoReLab Monday seminar presentation: Evangelos Bampas 10/21

44 Matroids 1-extendible systems A B : A < B CoReLab Monday seminar presentation: Evangelos Bampas 11/21

45 Matroids 1-extendible systems A B : A < B x 0 Pick any x 0 A \ B. CoReLab Monday seminar presentation: Evangelos Bampas 11/21

46 Matroids 1-extendible systems A B : A < B x 0 Y 0 By 1-extendibility, Y 0 B \ A with Y 0 1 such that B \ Y 0 + x 0 L. CoReLab Monday seminar presentation: Evangelos Bampas 11/21

47 Matroids 1-extendible systems A x B : A < B Y x 0 Y 0 Keep picking x i s in A \ B until there are no more. Then B \ Y i + x i L. CoReLab Monday seminar presentation: Evangelos Bampas 11/21

48 Matroids 1-extendible systems A x B : A < B Y x 0 z Y 0 Moreover, A B \ Y i + x i, which implies A + z L for arbitrary z B \ Y i. CoReLab Monday seminar presentation: Evangelos Bampas 11/21

49 Greedy on k-extendible systems Thm. If (E, L) is k-extendible then Greedy obtains a 1 k-approximate solution for any weight function on (E, L). CoReLab Monday seminar presentation: Evangelos Bampas 12/21

50 Greedy on k-extendible systems Thm. If (E, L) is k-extendible then Greedy obtains a 1 k-approximate solution for any weight function on (E, L). Proof. x 1, x 2,..., x ρ : successive choices of Greedy = S 0, S 1, S 2,..., S ρ = SOL: successive partial solutions with S i = S i 1 + x i, i : 1 i ρ CoReLab Monday seminar presentation: Evangelos Bampas 12/21

51 Obtaining the approximation ratio Lemma For each i : 1 i ρ, w (OPT (S i 1 )) w (OPT (S i ))+(k 1) w (x i ) CoReLab Monday seminar presentation: Evangelos Bampas 13/21

52 Obtaining the approximation ratio Lemma For each i : 1 i ρ, w (OPT (S i 1 )) w (OPT (S i ))+(k 1) w (x i ) Apply lemma ρ times: OPT = w (OPT (S 0 )) w (OPT (S ρ )) + (k 1) w (S ρ ) = k SOL CoReLab Monday seminar presentation: Evangelos Bampas 13/21

53 Proof of lemma Lemma For each i : 1 i ρ, w (OPT (S i 1 )) w (OPT (S i ))+(k 1) w (x i ) CoReLab Monday seminar presentation: Evangelos Bampas 14/21

54 Proof of lemma Lemma For each i : 1 i ρ, w (OPT (S i 1 )) w (OPT (S i ))+(k 1) w (x i ) OPT(S i-1 ) S i-1 Y x i : S i-1 +x i =S i Y k OPT(S i-1 )\Y+x i CoReLab Monday seminar presentation: Evangelos Bampas 14/21

55 Proof of lemma Lemma For each i : 1 i ρ, w (OPT (S i 1 )) w (OPT (S i ))+(k 1) w (x i ) OPT(S i-1 ) S i-1 Y x i : S i-1 +x i =S i Y k OPT(S i-1 )\Y+x i w (OPT (S i 1 )) = w (OPT (S i 1 ) \ Y + x i ) + w (Y ) w (x i ) CoReLab Monday seminar presentation: Evangelos Bampas 14/21

56 Proof of lemma Lemma For each i : 1 i ρ, w (OPT (S i 1 )) w (OPT (S i ))+(k 1) w (x i ) OPT(S i-1 ) S i-1 Y x i : S i-1 +x i =S i Y k OPT(S i-1 )\Y+x i w (OPT (S i 1 )) w (OPT (S i ))+w (Y ) w (x i ) CoReLab Monday seminar presentation: Evangelos Bampas 14/21

57 Proof of lemma Lemma For each i : 1 i ρ, w (OPT (S i 1 )) w (OPT (S i ))+(k 1) w (x i ) OPT(S i-1 ) S i-1 Y x i : S i-1 +x i =S i Y k OPT(S i-1 )\Y+x i y Y j i 1, S j + y OPT (S i 1 ) j i 1, S j + y L w (y) w (x i ) CoReLab Monday seminar presentation: Evangelos Bampas 14/21

58 Example of a 2-extendible system Maximum-weight b-matching. G = (V, E): undirected graph, b : V N, w : E R. E: edge set of G. L = {M E : u V, deg M (u) b (u)}. CoReLab Monday seminar presentation: Evangelos Bampas 15/21

59 Example of a 2-extendible system Maximum-weight b-matching. G = (V, E): undirected graph, b : V N, w : E R. E: edge set of G. L = {M E : u V, deg M (u) b (u)}. non-empty: u V, deg (u) = 0 b (u) CoReLab Monday seminar presentation: Evangelos Bampas 15/21

60 Example of a 2-extendible system Maximum-weight b-matching. G = (V, E): undirected graph, b : V N, w : E R. E: edge set of G. L = {M E : u V, deg M (u) b (u)}. hereditary: if M L and M M, then for any u V : deg M (u) deg M (u) b (u) CoReLab Monday seminar presentation: Evangelos Bampas 15/21

61 Example of a 2-extendible system Maximum-weight b-matching. G = (V, E): undirected graph, b : V N, w : E R. E: edge set of G. L = {M E : u V, deg M (u) b (u)}. 2-extendible: CoReLab Monday seminar presentation: Evangelos Bampas 15/21

62 Example of a 2-extendible system Maximum-weight b-matching. G = (V, E): undirected graph, b : V N, w : E R. E: edge set of G. L = {M E : u V, deg M (u) b (u)}. 2-extendible: CoReLab Monday seminar presentation: Evangelos Bampas 15/21

63 Example of a 2-extendible system Maximum-weight b-matching. G = (V, E): undirected graph, b : V N, w : E R. E: edge set of G. L = {M E : u V, deg M (u) b (u)}. 2-extendible: CoReLab Monday seminar presentation: Evangelos Bampas 15/21

64 Example of a 2-extendible system Maximum-weight b-matching. G = (V, E): undirected graph, b : V N, w : E R. E: edge set of G. L = {M E : u V, deg M (u) b (u)}. Cor. Greedy is a 1 2-approximation algorithm for Maximum-weight b-matching. CoReLab Monday seminar presentation: Evangelos Bampas 15/21

65 Other k-extendible systems Maximum profit scheduling (2-extendible). Maximum asymmetric TSP (3-extendible). Intersection of k matroids (k-extendible). CoReLab Monday seminar presentation: Evangelos Bampas 16/21

66 Tradeoffs for b-matching Maximum-weight b-matching can be solved exactly in time O ( b (u) min ( m log n, n 2 )). CoReLab Monday seminar presentation: Evangelos Bampas 17/21

67 Tradeoffs for b-matching Maximum-weight b-matching can be solved exactly in time O ( b (u) min ( m log n, n 2 )). Therefore the Greedy algorithm should be regarded as a tradeoff: 1 2-approximation in time O (m log n). CoReLab Monday seminar presentation: Evangelos Bampas 17/21

68 Tradeoffs for b-matching Maximum-weight b-matching can be solved exactly in time O ( b (u) min ( m log n, n 2 )). Therefore the Greedy algorithm should be regarded as a tradeoff: 1 2-approximation in time O (m log n). Improvement: 1 2-approximation in time O (bm). CoReLab Monday seminar presentation: Evangelos Bampas 17/21

69 Tradeoffs for b-matching Maximum-weight b-matching can be solved exactly in time O ( b (u) min ( m log n, n 2 )). Therefore the Greedy algorithm should be regarded as a tradeoff: 1 2-approximation in time O (m log n). Improvement: 1 2-approximation in time O (bm). Further ( improvement: randomized 2 3 ɛ) -approximation in time O ( ) bm log 1 ɛ. CoReLab Monday seminar presentation: Evangelos Bampas 17/21

70 b-matching by greedy walks Alg. Find-Walk(u) b (u) b (u) 1 if deg (u) = 0 then return let (u, v) be the heaviest edge out of u remove (u, v) from G if b (u) = 0 then remove all edges incident to u return (u, v) + Find-Walk(v) CoReLab Monday seminar presentation: Evangelos Bampas 18/21

71 b-matching by greedy walks Alg. Find-Walk(u) b (u) b (u) 1 if deg (u) = 0 then return let (u, v) be the heaviest edge out of u remove (u, v) from G if b (u) = 0 then remove all edges incident to u return (u, v) + Find-Walk(v) while u V s.t. b (u) > 0 and deg (u) > 0 do M M+ Find-Walk(u) CoReLab Monday seminar presentation: Evangelos Bampas 18/21

72 b-matching by greedy walks Alg. Find-Walk(u) b (u) b (u) 1 if deg (u) = 0 then return let (u, v) be the heaviest edge out of u remove (u, v) from G if b (u) = 0 then remove all edges incident to u return (u, v) + Find-Walk(v) while u V s.t. b (u) > 0 and deg (u) > 0 do M M+ Find-Walk(u) Only guarantees that deg M (u) 2b (u). CoReLab Monday seminar presentation: Evangelos Bampas 18/21

73 1-approximation for b-matching 2 Split M into M 1 and M 2 by taking alternative edges of individual walks. Pick the heaviest of M 1, M 2. CoReLab Monday seminar presentation: Evangelos Bampas 19/21

74 1-approximation for b-matching 2 Split M into M 1 and M 2 by taking alternative edges of individual walks. Pick the heaviest of M 1, M 2. SOL w(m) 2 w(m OPT) 2. Why? CoReLab Monday seminar presentation: Evangelos Bampas 19/21

75 1-approximation for b-matching 2 Split M into M 1 and M 2 by taking alternative edges of individual walks. Pick the heaviest of M 1, M 2. SOL w(m) 2 w(m OPT) 2. Why? To each edge (u, v) picked by Find-Walk, assign some edge e M OPT. If (u, v) M OPT, assign it to itself. Otherwise, pick any e M OPT incident to u. CoReLab Monday seminar presentation: Evangelos Bampas 19/21

76 1-approximation for b-matching 2 Split M into M 1 and M 2 by taking alternative edges of individual walks. Pick the heaviest of M 1, M 2. SOL w(m) 2 w(m OPT) 2. Why? To each edge (u, v) picked by Find-Walk, assign some edge e M OPT. If (u, v) M OPT, assign it to itself. Otherwise, pick any e M OPT incident to u. Each edge in M OPT is assigned to a unique edge in M. Moreover, w (e) w (u, v). CoReLab Monday seminar presentation: Evangelos Bampas 19/21

77 Arms and pieces CoReLab Monday seminar presentation: Evangelos Bampas 20/21

78 Arms and pieces u An Arm out of node u CoReLab Monday seminar presentation: Evangelos Bampas 20/21

79 Arms and pieces u An Arm out of node u u A Piece about edge (u, v) v CoReLab Monday seminar presentation: Evangelos Bampas 20/21

80 A randomized algorithm for b- matching Alg. Linear-Random(G, w) M repeat k times pick a vertex u uniformly at random with probability deg M (u) do b pick (u, v) M uniformly at random find max-benefit compatible piece P about (u, v) M M P with probability b(u) deg M (u) do b find max-benefit compatible arm A out of u M M A CoReLab Monday seminar presentation: Evangelos Bampas 21/21