A Fast Approximation for Maximum Weight Matroid Intersection

Transcription

1 A Fast Approximation for Maximum Weight Matroid Intersection Chandra Chekuri Kent Quanrud University of Illinois at Urbana-Champaign UIUC Theory Seminar February 8, 2016

2 Max. weight bipartite matching Input: bipartite graph G = (L R, E) weights w : E R 0 Output: matching M E maximizing w(m) def = e M w(e)

3 Max. weight bipartite matching total weight = Input: bipartite graph G = (L R, E) weights w : E R 0 Output: matching M E maximizing w(m) def = e M w(e)

4 Max. weight bipartite matching total weight = Input: bipartite graph G = (L R, E) weights w : E R 0 Output: matching M E maximizing w(m) def = e M w(e) Approximate output: matching M E s.t. w(m) (1 ɛ)opt

5 Fast approximations: bipartite matching cardinality weighted exact O(m n) Hopcroft and Karp [1973] Õ(m 10/7 ) Mądry [2013] O(mn + n 2 log n) Fredman and Tarjan [1987] O(m n log W ) Duan and Su [2012] approximate O(m/ɛ) O(m log(1/ɛ)/ɛ) Duan and Pettie [2014] (m = edges, n = vertices, W = max weight)

6 Fast approximations: bipartite matching cardinality weighted exact O(m n) Hopcroft and Karp [1973] Õ(m 10/7 ) Mądry [2013] O(mn + n 2 log n) Fredman and Tarjan [1987] O(m n log W ) Duan and Su [2012] approximate O(m/ɛ) O(m log(1/ɛ)/ɛ) Duan and Pettie [2014] (m = edges, n = vertices, W = max weight) (!) For fixed ɛ, weighted approximation is faster than unweighted exact

7 Duan and Pettie [2014] 1. Primal-dual (extends to general matching) - Only approximates dual optimal conditions 2. Scaling reduces weighted problem to unweighted 3. Runs an approximate subroutine at each scale 4. Updates dual variables with small loss from approximation

8 Matroids M = (N, I) N : ground set of elements I 2 N : independent (feasible) sets

9 y x Matroids k red = 2 k blue = 3 k green = 1 - Empty set is independent - Subsets of independent sets are independent - Maximal independent sets have same cardinality max independent set is a base max cardinality is the rank - If A, B I and A < B, then there is b B \ A s.t. A + b I

10 Matroid Intersection (same ground set) M 1 = (N, I 1 ) M 2 = (N, I 2 ) M 1 M 2 (N, I 1 I 2 ) e.g. bipartite matchings, arborescences

11 Matroid intersection problems Maximum cardinality matroid intersection Input: matroids M 1 = (N, I 1 ), M 2 = (N, I 2 ) Output: S I 1 I 2 maximizing S

12 Matroid intersection problems Maximum cardinality matroid intersection Input: matroids M 1 = (N, I 1 ), M 2 = (N, I 2 ) Output: S I 1 I 2 maximizing S Maximum weight matroid intersection Input: matroids M 1 = (N, I 1 ), M 2 = (N, I 2 ), weights w : N R 0 Output: S I 1 I 2 maximizing w(s) def = e S w(s)

13 Matroid intersection problems Maximum cardinality matroid intersection Input: matroids M 1 = (N, I 1 ), M 2 = (N, I 2 ) Output: S I 1 I 2 maximizing S Maximum weight matroid intersection Input: matroids M 1 = (N, I 1 ), M 2 = (N, I 2 ), weights w : N R 0 Output: S I 1 I 2 maximizing w(s) def = e S w(s) Oracle Model Independence queries of the form Is S I?

14 Running times for matroid intersection

15 Running times for matroid intersection Cardinality O(nk 1.5 Q) Cunningham [1986] (n = N, k = rank(m 1 M 2 ), Q = cost of indep. query)

16 Running times for matroid intersection Cardinality O(nk 1.5 Q) Cunningham [1986] (n = N, k = rank(m 1 M 2 ), Q = cost of indep. query) Weighted (W = max{w(e) : e N }) O(nk 2 Q) Frank [1981], Brezovec et al. [1986], Schrijver [2003] O(n 2 k log(kw )Q) Fujishige and Zhang [1995] O(nk 1.5 W Q) Huang et al. [2014] O((n 2 log(n)q + n 3 polylog(n)) log(nw )) Lee et al. [2015]

17 Approximate matroid intersection Input: M 1 = (N, I 1 ), M 2 = (N, I 2 ), w : N R 0, ɛ > 0 Output: S I 1 I 2 s.t. w(s) (1 ɛ)opt (OPT = max{w(t ) : T I 1 I 2 }) Previous bound: O(nk 1.5 log(k)q/ɛ) Huang et al. [2014]

18 Approximate matroid intersection Input: M 1 = (N, I 1 ), M 2 = (N, I 2 ), w : N R 0, ɛ > 0 Output: S I 1 I 2 s.t. w(s) (1 ɛ)opt (OPT = max{w(t ) : T I 1 I 2 }) Previous bound: O(nk 1.5 log(k)q/ɛ) Huang et al. [2014] Main result: (1 ɛ)-approximation in time O(nkQ log 2 (ɛ)/ɛ 2 )

19 Fast approximations: matroid intersection exact approximate cardinality O(nk 1.5 Q) Cunningham [1986] O(nkQ/ɛ) weighted (nk 2 Q) Frank [1981] and others O(n 2 k log(kw )Q) Fujishige and Zhang [1995] O(nk 1.5 W Q) Huang et al. [2014] O((n 2 log(n)q + n 3 polylog(n)) log(nw )) Lee et al. [2015] O(nkQ log 2 (1/ɛ)/ɛ 2 ) (n = elements, k = rank, Q = indep. query, W = max weight)

20 Fast approximations: matroid intersection exact approximate cardinality O(nk 1.5 Q) Cunningham [1986] O(nkQ/ɛ) weighted (nk 2 Q) Frank [1981] and others O(n 2 k log(kw )Q) Fujishige and Zhang [1995] O(nk 1.5 W Q) Huang et al. [2014] O((n 2 log(n)q + n 3 polylog(n)) log(nw )) Lee et al. [2015] O(nkQ log 2 (1/ɛ)/ɛ 2 ) (n = elements, k = rank, Q = indep. query, W = max weight) (!) For fixed ɛ, weighted approximation is faster than unweighted exact

21 A Fast Approximation for Maximum Weight Bipartite Matching Chandra Chekuri Kent Quanrud University of Illinois at Urbana-Champaign UIUC Theory Seminar February 8, 2016

22 Max. weight bipartite matching total weight = Input: bipartite graph G = (L R, E) weights w : E R 0 Output: matching M E maximizing w(m) def = e M w(e) Approximate output: matching M E s.t. w(m) (1 ɛ)opt

23 Fast approximations: bipartite matching cardinality weighted exact O(m n) Hopcroft and Karp [1973] Õ(m 10/7 ) Mądry [2013] O(mn + n 2 log n) Fredman and Tarjan [1987] O(m n log W ) Duan and Su [2012] approximate O(m/ɛ) O(m log(1/ɛ)/ɛ) Duan and Pettie [2014] (m = edges, n = vertices, W = max weight) (!) For fixed ɛ, weighted approximation is faster than unweighted exact

24 Maximum cardinality matching - Augmenting paths faster Hopcroft-Karp - Augment in batch - Most paths are short faster APX Hopcroft-Karp weighted weighted weighted Hungarian algo. - primal-dual - combinatorial scaling faster Integral Hungarian - primal-dual - (small) integral weights - fixed scaling faster APX Hungarian - APX primal-dual - exponential scaling

25 Maximum cardinality matching - Augmenting paths 1 2 Hopcroft-Karp faster - Augment in batch - Most paths are short faster APX Hopcroft-Karp 3 weighted weighted weighted Hungarian algo. - primal-dual - combinatorial scaling 4 faster Integral Hungarian - primal-dual - (small) integral weights - fixed scaling 5 faster APX Hungarian - APX primal-dual - exponential scaling 6

26 Maximum cardinality matching - Augmenting paths 1 2 Hopcroft-Karp faster - Augment in batch - Most paths are short faster APX Hopcroft-Karp 3 weighted weighted weighted Hungarian algo. - primal-dual - combinatorial scaling 4 faster Integral Hungarian - primal-dual - (small) integral weights - fixed scaling 5 faster APX Hungarian - APX primal-dual - exponential scaling 6

27 Augmenting paths

28 Augmenting paths = Exposed vertices

29 Augmenting paths = Exposed vertices

30 Augmenting paths = Exposed vertices

31 Augmenting paths = Exposed vertices

32 Augmenting paths = Exposed vertices

33 Augmenting paths = Exposed vertices

34 Augmenting paths

35 Max. matching no aug. paths

36 Max. matching no aug. paths

37 Max. matching no aug. paths

38 Finding augmenting paths = Exposed vertices

39 Finding augmenting paths = Exposed vertices

40 Finding augmenting paths = Exposed vertices = Exposed vertices

41 Finding augmenting paths = Exposed vertices = Exposed vertices

42 M ; does M have an augmenting path? M M4P nope! yes, P return M

43 M ; O(m) does M have an augmenting path? M M4P nope! yes, P apple n paths return M O(mn)

44 Maximum cardinality matching König Augmenting paths weighted 1 2 Hopcroft-Karp faster O(mn) - Augment in batch - Most paths are short weighted faster APX Hopcroft-Karp 3 weighted Hungarian algo. - primal-dual - combinatorial scaling 4 faster Integral Hungarian - primal-dual - (small) integral weights - fixed scaling 5 faster APX Hungarian - APX primal-dual - exponential scaling 6

45 Hopcroft-Karp Most augmenting paths are short Augment in batch and increase shortest aug. path length

46 Matching M, k def = M < OPT Lemma. M has aug. path P of length k P OPT k

47 Matching M, k def = M < OPT Lemma. M has aug. path P of length k P OPT k P M

48 Matching M, k def = M < OPT Lemma. M has aug. path P of length k P OPT k P M

49 Matching M, k def = M < OPT Lemma. M has aug. path P of length k P OPT k P 2 p n n p n M

50 batch-augment

51 batch-augment

52 batch-augment L 0 L 1 L 2 L 3 L 4

53 batch-augment L 0 L 1 L 2 L 3 L 4

54 batch-augment L 0 L 1 L 2 L 3 L 4

55 M ; O(m) does M have an augmenting path? M M4P nope! yes, P apple n paths return M O(mn)

56 M ; O(m) does M have an augmenting path? nope! yes batch-augment O( p n)times return M O(m p n)

57 Maximum cardinality matching König Augmenting paths weighted Hungarian algo. - primal-dual - combinatorial scaling 1 2 Hopcroft-Karp faster O(mn) 4 faster Hopcroft and Karp Augment in batch - Most paths are short weighted Integral Hungarian O(m p n) - primal-dual - (small) integral weights - fixed scaling 5 faster faster APX Hopcroft-Karp weighted APX Hungarian 3 - APX primal-dual - exponential scaling 6

58 Matching M, k def = M < OPT Lemma. M has aug. path P of length k P OPT k P 2 p n n p n M

59 Matching M, k def = M < OPT Lemma. M has aug. path P of length k P OPT k P 1/ 2 p n (1 )n n p n M

60 M ; O(m) repeat O(1/ ) times batch-augment nope! return M O(m/ )

61 Maximum cardinality matching König Augmenting paths weighted Hungarian algo. - primal-dual - combinatorial scaling 1 2 Hopcroft-Karp faster O(mn) 4 faster Hopcroft and Karp Augment in batch - Most paths are short weighted Integral Hungarian O(m p n) - primal-dual - (small) integral weights - fixed scaling 5 faster faster APX Hopcroft-Karp weighted APX Hungarian 3 O(m/ ) - APX primal-dual - exponential scaling 6

62 Weighted matchings

63 Weighted matchings

64 Primal max x s. t. w(e) x(e) e E e δ(v) x 0 x(e) 1 v L R, e E

65 Primal max x s. t. Dual min y w(e) x(e) e E e δ(v) x 0 v L R x(e) 1 v L R, y(v) e E s. t. y(l) + y(r) w({l, r}) {l, r} E y(v) 0 v L R

66 max x s. t. Primal X w(e) x(e) e2e X e2 (v) x(e) apple 1 8v, x 0 8e x min y Dual X v2l[r y(v) s. t. y(`)+ y(r) w(e) 8e = {`, r} y(v) 0 8v y

67 max x s. t. Primal X w(e) x(e) e2e X e2 (v) x(e) apple 1 8v, x 0 8e min y Dual X v2l[r y(v) s. t. y(`)+ y(r) w(e) 8e = {`, r} y(v) 0 8v x y x, y optimal if: Orthogonality x({`,r}) > 0 ) y(`)+y(r) =w({`,r}) y(`) > 0 ) X x(e) =1 e2 (`) y(r) > 0 ) x(e) =1 {`,r} 2 E ` 2 L r 2 R

68 Primal Dual y(`)+y(r) w({`,r}) {`,r} 2 E y(v) 0 v 2 L [ R M y Orthogonality {`,r} 2 M ) y(`)+y(r) =w({`,r}) y(`) > 0 ) ` 2 V (M) y(r) > 0 ) r 2 V (M) {`,r} 2 E ` 2 L r 2 R w(m) =OPT

69 Dual y(`)+y(r) w({`,r}) {`,r} 2 E y(v) 0 v 2 L [ R Orthogonality y(`)+y(r) =w({`,r}) 8{`,r} 2 M ` 2 V (M) 8` : y(`) > 0 r 2 V (M) 8r : y(r) > 0

70 M ; y(`) max e2e w(e) 8` 2 L y(r) 0 8r 2 R Dual y(`)+y(r) w({`,r}) {`,r} 2 E y(v) 0 v 2 L [ R Orthogonality y(`)+y(r) =w({`,r}) 8{`,r} 2 M ` 2 V (M) 8` : y(`) > 0 r 2 V (M) 8r : y(r) > 0

71 M ; y(`) max e2e w(e) 8` 2 L y(r) 0 8r 2 R Dual y(`)+y(r) w({`,r}) {`,r} 2 E y(v) 0 v 2 L [ R Orthogonality y(`)+y(r) =w({`,r}) 8{`,r} 2 M ` 2 V (M) 8` : y(`) > 0 r 2 V (M) 8r : y(r) > 0

72 M ; y(`) max e2e w(e) 8` 2 L y(r) 0 8r 2 R GOAL push y(`) # 0 8 exposed ` 2 L Dual y(`)+y(r) w({`,r}) {`,r} 2 E y(v) 0 v 2 L [ R Orthogonality y(`)+y(r) =w({`,r}) {`,r} 2 M ` 2 V (M) 8` : y(`) > 0 r 2 V (M) 8r : y(r) >

73 M ; y(`) max w(e)8` 2 L e2e y(r) 0 8r 2 R y(`) =0 for all exposed ` 2 L???? yes no return M

74 GOAL push y(`) # 0 8 exposed ` 2 L Dual y(`)+y(r) w({`,r}) {`,r} 2 E y(v) 0 v 2 L [ R Orthogonality y(`)+y(r) =w({`,r}) 8{`,r} 2 M ` 2 V (M) 8` : y(`) > 0 r 2 V (M) 8r : y(r) > 0

75 we can add tight edges to our matching GOAL push y(`) # 0 8 exposed ` 2 L Dual y(`)+y(r) w({`,r}) {`,r} 2 E y(v) 0 v 2 L [ R Orthogonality y(`)+y(r) =w({`,r}) 8{`,r} 2 M ` 2 V (M) 8` : y(`) > 0 r 2 V (M) 8r : y(r) > 0

76 M ; y(`) max w(e)8` 2 L e2e y(r) 0 8r 2 R??? y(`) =0 for all exposed ` 2 L? yes no augment yes is there a tight aug. path? no return M

77 y(`)+y(r) w({`,r})

78 y(`)+y(r) w({`,r})

79 y(`)+y(r) w({`,r})

80 y(`)+y(r) w({`,r})

81 y(`)+y(r) w({`,r})

82 y(`)+y(r) w({`,r})

83 y(`)+y(r) w({`,r})

84 exposed weights go down y(`)+y(r) w({`,r})

85 exposed weights go down y(`)+y(r) w({`,r}) tight edges stay covered

86 exposed weights go down y(`)+y(r) w({`,r}) new tight edges tight edges stay covered

87 M ; y(`) max w(e)8` 2 L e2e y(r) 0 8r 2 R shift weight y(`) =0 for all exposed ` 2 L? yes no augment yes is there a tight aug. path? no return M

88 M ; y(`) max w(e)8` 2 L e2e y(r) 0 8r 2 R shift weight O(m) ' O(m) ' n times n times y(`) =0 for all exposed ` 2 L? yes return M no augment yes is there a tight aug. path? no O(mn)

89 Using Egerváry s reduction and Konig s maximum matching alorithm, in the fall of 1953 I solved several 12 by 12 assignment problems (with 3-digit integers as data) by hand. Each of these examples took under two hours to solve and I was convinced that the combined algorithm was good. This must have been one of the last times when pencil and paper could beat the largest and fastest electronic computer in the world. - Harold Kuhn, On the origin of the Hungarian method

90 Jacobi Maximum cardinality matching König Augmenting paths Hungarian algo primal-dual - combinatorial scaling Egerváry 1931 Kuhn 1955 weighted 1 2 Hopcroft-Karp faster O(mn) 4 O(mn) faster Hopcroft and Karp Augment in batch - Most paths are short weighted Integral Hungarian O(m p n) - primal-dual - (small) integral weights - fixed scaling 5 faster faster APX Hopcroft-Karp weighted APX Hungarian 3 O(m/ ) - APX primal-dual - exponential scaling 6

91 What if the weights are small and integral? w(e) {1,..., W } e E

92 M ; y(`) max w(e)8` 2 L e2e y(r) 0 8r 2 R shift weight y(`) =0 for all exposed ` 2 L? yes no augment yes is there a tight aug. path? no return M

93 M ; y(`) max w(e)8` 2 L e2e y(r) 0 8r 2 R y(`) =0 for all exposed ` 2 L? yes no shift weight no is there a tight aug. path? augment yes return M

94 M ; y(`) W 8` 2 L y(r) 0 8r 2 R y(`) =0 for all exposed ` 2 L? yes no shift weight no is there a tight aug. path? augment yes return M

95 M ; y(`) W 8` 2 L shift 1 unit y(r) 0 8r 2 R of weight y(`) =0 for all exposed ` 2 L? yes no no is there a tight aug. path? augment yes return M

96 M ; y(`) W 8` 2 L y(r) 0 8r 2 R repeat W times shift 1 unit of weight no is there a tight aug. path? augment yes return M

97 M ; y(`) W 8` 2 L y(r) 0 8r 2 R shift 1 unit of weight repeat W times Hopcroft-Karp return M

98 M ; y(`) W 8` 2 L y(r) 0 8r 2 R shift 1 unit of weight O(m p n) repeat W times Hopcroft-Karp return M O(m p nw )

99 Jacobi Maximum cardinality matching König Augmenting paths Hungarian algo primal-dual - combinatorial scaling Egerváry 1931 Kuhn 1955 weighted 1 2 Hopcroft-Karp faster O(mn) 4 O(mn) faster Hopcroft and Karp Augment in batch - Most paths are short weighted Integral Hungarian O(m p n) - primal-dual - (small) integral weights - fixed scaling 5 O(m p nw ) faster faster APX Hopcroft-Karp weighted APX Hungarian 3 O(m/ ) - APX primal-dual - exponential scaling 6

100 Goal: beat O(m n) 1. Approximate primal-dual conditions 2. Exponential scaling 3. Approximate cardinality at each scale 4. Lossy weight shift

101 M ; y(`) W 8` 2 L y(r) 0 8r 2 R shift 1 unit of weight O(m p n) repeat W times Hopcroft-Karp return M O(m p nw )

102 M ; y(`) W 8` 2 L y(r) 0 8r 2 R shift 1 unit of weight O(m p n) repeat W times Hopcroft-Karp return M O(m p nw ) Õ(m)

103 M ; y(`) W 8` 2 L y(r) 0 8r 2 R shift 1 unit of weight O(m p n) repeat W times Hopcroft-Karp return M O(m p nw ) Õ(m)

104 M ; y(`) W 8` 2 L y(r) 0 8r 2 R shift 1 unit of weight eo(m/ ) repeat W times APX Hopcroft-Karp return M O(mW/ ) Õ(m)

105 M ; y(`) W 8` 2 L y(r) 0 8r 2 R shift 1 unit of weight eo(m/ ) repeat W times APX Hopcroft-Karp return M O(mW/ ) Õ(m)

106 y(`)+y(r) w({`,r})

107 y(`)+y(r) w({`,r}) y(r) =0 8r 2 R \ V (M)

108 y(`)+y(r) w({`,r}) y(r) =0 8r 2 R \ V (M) never touch exposed weights

109 y(`)+y(r) w({`,r}) y(r) =0 8r 2 R \ V (M) bad news bears

110 M ; y(`) W 8` 2 L y(r) 0 8r 2 R repeat W times shift 1 unit of weight aug. paths are long APX Hopcroft-Karp eo(m/ ) return M O(mW/ ) Õ(m)

111 y(`)+y(r) w({`,r}) y(r) =0 8r 2 R \ V (M) O(1/ ) layers

112 y(`)+y(r) w({`,r}) y(r) =0 8r 2 R \ V (M) O(1/ ) layers

113 y(`)+y(r) w({`,r}) y(r) =0 8r 2 R \ V (M) O(1/ ) layers no longer tight

114 y(`)+y(r) w({`,r}) y(r) =0 8r 2 R \ V (M) O(1/ ) layers artificially increase weight

115 Primal Dual y(`)+y(r) w({`,r}) {`,r} 2 E M y(v) 0 v 2 L [ R y Orthogonality y(`)+y(r) =w({`,r}) 8{`,r} 2 M ` 2 V (M) 8` : y(`) > 0 r 2 V (M) 8r : y(r) > 0 w(m) =OPT

116 Primal Dual y(`)+y(r) w({`,r}) {`,r} 2 E M y(v) 0 v 2 L [ R y : E! R excess y(`)+y(r) apple w({`,r})+ ({`,r}) 8{`,r} 2 M X e2m APX Orthogonality ` 2 V (M) 8` : y(`) > 0 r 2 V (M) 8r : y(r) > 0 (e) apple w(m) w(m) apple (1 + )OPT

117 y(`)+y(r) w({`,r}) y(r) =0 8r 2 R \ V (M) O(1/ ) layers artificially increase weight

118 y(`)+y(r) w({`,r}) y(r) =0 8r 2 R \ V (M) O(1/ ) layers add weight to (e)

119 y(`)+y(r) w({`,r}) y(r) =0 8r 2 R \ V (M) O(1/ ) layers add weight to (e) choose layer that cuts -fraction of edges

120 M ; y(`) W 8` 2 L y(r) 0 8r 2 R repeat W times shift weight and cheat aug. paths are long APX Hopcroft-Karp eo(m/ ) return M O(mW/ ) Õ(m)

121 M ; y(`) W 8` 2 L y(r) 0 8r 2 R repeat W times shift weight and cheat aug. paths are long APX Hopcroft-Karp eo(m/ ) return M O(mW/ ) Õ(m)

122 M ; y(`) W 8` 2 L y(r) 0 8r 2 R repeat W times shift weight and cheat (a) too slow aug. paths are long APX Hopcroft-Karp eo(m/ ) return M O(mW/ ) Õ(m)

123 M ; y(`) W 8` 2 L y(r) 0 8r 2 R repeat W times shift weight and cheat (a) too slow (b) too much cheating aug. paths are long APX Hopcroft-Karp eo(m/ ) return M O(mW/ ) Õ(m)

124 Approximate scaling 1. Round up edge weights to powers of (1 + ɛ): w(e) = (1 + ɛ) log (1+ɛ) w(e) W = max w(e) e E e E (1 ɛ)-apx w/r/t w (1 O(ɛ))-APX w/r/t w

125 Approximate scaling 1. Round up edge weights to powers of (1 + ɛ): w(e) = (1 + ɛ) log (1+ɛ) w(e) W = max w(e) e E e E (1 ɛ)-apx w/r/t w (1 O(ɛ))-APX w/r/t w 2. Operate in units of ɛ(1 + ɛ) i 1 for scales i = log 1+ɛ W,..., log1+ɛ ɛ

126 M ; y(`) f W 8` 2 L y(r) 0 8r 2 R shift (1 + ) i 1 weight and cheat eo(m/ ) 8 log >< 1+ W, f i =. >: blog 1+ c aug. paths are long APX Hopcroft-Karp return M y(`) (1 + ) i 8` 2 L y(`) =(1+ ) i 8 exposed ` 2 L O(m log 2 (W/ )/ 2 ) Õ(m)

127 M ; y(`) f W 8` 2 L y(r) 0 8r 2 R shift (1 + ) i 1 weight and cheat eo(m/ ) 8 log >< 1+ W, f i =. >: blog 1+ c apple ew(e) 8e 2 M because aug. paths are long APX Hopcroft-Karp return M y(`) (1 + ) i 8` 2 L y(`) =(1+ ) i 8 exposed ` 2 L O(m log 2 (W/ )/ 2 ) Õ(m)

128 M ; y(`) f W 8` 2 L y(r) 0 8r 2 R shift (1 + ) i 1 weight and cheat eo(m/ ) 8 log >< 1+ W, f i =. >: blog 1+ c aug. paths are long APX Hopcroft-Karp return M y(`) (1 + ) i 8` 2 L y(`) =(1+ ) i 8 exposed ` 2 L O(m log 2 (W/ )/ 2 ) A few more low level details O(m log 2 (1/ )/ ) 2

129 Jacobi Maximum cardinality matching König Augmenting paths Hungarian algo primal-dual - combinatorial scaling Egerváry 1931 Kuhn 1955 weighted 1 2 Hopcroft-Karp faster O(mn) 4 O(mn) faster Hopcroft and Karp Augment in batch - Most paths are short weighted Integral Hungarian O(m p n) - primal-dual - (small) integral weights - fixed scaling 5 O(m p nw ) faster faster APX Hopcroft-Karp weighted APX Hungarian 3 - APX primal-dual - exponential scaling Duan and Pettie 2014 O(m/ ) 6 Õ(m/ )

130 exact algorithm Primal Orthogonality {`,r} 2 M ) y(`)+y(r) =w({`,r}) y(`) > 0 ) ` 2 V (M) y(r) > 0 ) r 2 V (M) Dual y(`)+y(r) w({`,r}) {`,r} 2 E y(v) 0 v 2 L [ R M y w(m) =OPT {`,r} 2 E ` 2 L r 2 R y(`) M ; max e2e w(e)8` 2 L y(r) 0 8r 2 R y(`) =0 for all exposed ` 2 L? yes return M no augment shift weight yes is there a tight aug. path? O(m) ' no O(m) ' O(mn) n times n times

131 approximate algorithm Primal Dual y(`)+y(r) w({`,r}) {`,r} 2 E y(v) 0 v 2 L [ R M y : E! R excess APX Orthogonality y(`)+y(r) apple w({`,r})+ ({`,r}) 8{`,r} 2 M ` 2 V (M) 8` : y(`) > 0 r 2 V (M) 8r : y(r) > 0 X (e) apple w(m) e2m w(m) apple (1 + )OPT M ; y(`) f W 8` 2 L y(r) 0 8r 2 R 8 log >< 1+ W, f i =. >: blog 1+ c return M shift (1 + ) i 1 weight and cheat y(`) (1 + ) i 8` 2 L y(`) =(1+ ) i 8 exposed ` 2 L O(m log 2 (W/ )/ 2 ) aug. paths are long eo(m/ ) APX Hopcroft-Karp A few more low level details O(m log 2 (1/ )/ ) 2

132 Bibliography I C. Brezovec, G. Cornuéjols, and F. Glover. Two algorithms for weighted matroid intersection. Math. Prog., 36(1):39 53, W. H. Cunningham. Improved bounds for matroid partition and intersection algorithms. SIAM J. Comput., 15(4): , R. Duan and S. Pettie. Linear-time approximation for maximum weight matching. J. Assoc. Comput. Mach., 61(1), January R. Duan and H. Su. A scaling algorithm for maximum weight matching in bipartite graphs. In Proc. 23rd ACM-SIAM Sympos. Discrete Algs. (SODA), pages , A. Frank. A weighted matroid intersection algorithm. J. Algorithms, 2: , M. L. Fredman and R. E. Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. J. Assoc. Comput. Mach., 34(3): , S. Fujishige and X. Zhang. An efficient cost scaling algorithm for the independent assignment problem. J. Oper. Res. Soc. Japan, 38(1): , 1995.

133 Bibliography II J. Hopcroft and R. Karp. An n 5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Comput., pages , C. Huang, N. Kakimura, and N. Kamiyama. Exact and approximation algorithms for weighted matroid intersection. MI Preprint Series, Math. for Industry, Kyushu University, November To appear in Proc. 27th ACM-SIAM Symp. Discrete Algs. (SODA), Y.T. Lee, A. Sidford, and S.C. Wong. A faster cutting plane method and its implications for combinatorial and convex optimization. In Proc. 56th Annu. IEEE Symp. Found. Comput. Sci. (FOCS), A. Mądry. Navigating central path with electrical flows: from flows to matchings, and back. In Proc. 54th Annu. IEEE Symp. Found. Comput. Sci. (FOCS), pages , A. Schrijver. Combinatorial Optimization: Polyhedra and Efficiency, volume 24 of Algorithms and Combinatorics. Springer, 2003.