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A Fast Approximation forMaximum Weight

Matroid Intersection

Chandra Chekuri Kent Quanrud

University of Illinois at Urbana-Champaign

UIUC Theory SeminarFebruary 8, 2016

Max. weight bipartite matching

Input:

bipartite graph G = (LtR,E)

weights w : E → R≥0

Output:

matching M ⊆ E maximizing

w(M)def=∑

e∈Mw(e)

Approximate output:

matching M ⊆ E s.t. w(M) ≥ (1− ε)OPT

total weight = 355

Input:

Output:

w(M)def=∑

e∈Mw(e)

Approximate output:

total weight = 355

Input:

Output:

w(M)def=∑

e∈Mw(e)

Approximate output:

Fast approximations: bipartite matching

exact approximate

cardinality O(m√n) Hopcroft and Karp [1973]

O(m10/7) Mądry [2013]O(m/ε)

weighted O(mn+ n2 log n) Fredman and Tarjan [1987]

O(m√n logW ) Duan and Su [2012]

O(m log(1/ε)/ε)Duan and Pettie [2014]

(m = edges, n = vertices, W = max weight)

(!) For fixed ε, weighted approximation is faster thanunweighted exact

exact approximate

O(m10/7) Mądry [2013]O(m/ε)

Duan and Pettie [2014]

(extends to general matching)

1. Primal-dual

- 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 fromapproximation

Matroids

M = (N , I)N : ground set of elements

I ⊆ 2N : independent (feasible) sets

Matroidskred = 2kblue = 3kgreen = 1

- Empty set is independent

- Subsets of independent setsare independent

- Maximal independent setshave same cardinalitymax 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

Matroid Intersection

M1 = (N , I1) M2 = (N , I2)

M1 ∩M2 , (N , I1 ∩ I2)

(same ground set)

e.g. bipartite matchings, arborescences

Matroid intersection problems

Maximum cardinality matroid intersection

Input: matroidsM1 = (N , I1),M2 = (N , I2)Output: S ∈ I1 ∩ I2 maximizing |S|

Maximum weight matroid intersection

Input: matroids M1 = (N , I1), M2 = (N , I2),weights w : N → R≥0

Output: S ∈ I1 ∩I2 maximizing w(S) def=∑

e∈S w(s)

Oracle Model

Independence queries of the form “Is S ∈ I?”

e∈S w(s)

Oracle Model

e∈S w(s)

Oracle Model

Running times for matroid intersection

CardinalityO(nk1.5Q) Cunningham [1986]

(n = |N |, k = rank(M1 ∩M2), Q = cost of indep. query)

Weighted (W = max{w(e) : e ∈ N})O(nk2Q) Frank [1981], Brezovec et al. [1986], Schrijver [2003]

O(n2√k log(kW )Q) Fujishige and Zhang [1995]

O(nk1.5WQ) Huang et al. [2014]

O((n2 log(n)Q+ n3 polylog(n)) log(nW ))Lee et al. [2015]

Approximate matroid intersection

Input: M1 = (N , I1),M2 = (N , I2),w : N → R≥0, ε > 0

Output: S ∈ I1 ∩ I2 s.t. w(S) ≥ (1− ε)OPT(OPT = max{w(T ) : T ∈ I1 ∩ I2})

Previous bound:O(nk1.5 log(k)Q/ε) Huang et al. [2014]

Main result: (1− ε)-approximation in time

O(nkQ log2(ε)/ε2)

Approximate matroid intersection

Input: M1 = (N , I1),M2 = (N , I2),w : N → R≥0, ε > 0

Output: S ∈ I1 ∩ I2 s.t. w(S) ≥ (1− ε)OPT(OPT = max{w(T ) : T ∈ I1 ∩ I2})

Previous bound:O(nk1.5 log(k)Q/ε) Huang et al. [2014]

Main result: (1− ε)-approximation in time

O(nkQ log2(ε)/ε2)

Fast approximations: matroid intersection

exact approximate

cardinality O(nk1.5Q) Cunningham [1986] O(nkQ/ε)

weighted

(nk2Q) Frank [1981] and others

O(n2√k log(kW )Q)

Fujishige and Zhang [1995]

O((n2 log(n)Q+ n3 polylog(n)) log(nW ))

Lee et al. [2015]

O(nkQ log2(1/ε)/ε2)

(n = elements, k = rank, Q = indep. query, W = max weight)

Fast approximations: matroid intersection

exact approximate

cardinality O(nk1.5Q) Cunningham [1986] O(nkQ/ε)

weighted

(nk2Q) Frank [1981] and others

O(n2√k log(kW )Q)

Fujishige and Zhang [1995]

O((n2 log(n)Q+ n3 polylog(n)) log(nW ))

Lee et al. [2015]

O(nkQ log2(1/ε)/ε2)

(n = elements, k = rank, Q = indep. query, W = max weight)

A Fast Approximation forMaximum WeightBipartite Matching

Chandra Chekuri Kent Quanrud

University of Illinois at Urbana-Champaign

UIUC Theory SeminarFebruary 8, 2016

total weight = 355

Input:

bipartite graphG = (LtR,E)weights w : E → R≥0

Output:

w(M)def=∑

e∈Mw(e)

Approximate output:

exact approximate

O(m10/7) Mądry [2013]O(m/ε)

faster

Maximum cardinality matching

- Augmenting paths

Hopcroft-Karp

- Augment in batch- Most paths are short

faster

Integral Hungarian

- primal-dual- (small) integral weights

- fixed scaling

Hungarian algo.

- primal-dual- combinatorial scaling

APX Hungarian

- APX primal-dual- exponential scaling

weighted

APX Hopcroft-Karpfaster

weighted

faster

weighted

faster

- Augmenting paths

Hopcroft-Karp

faster

Integral Hungarian

- fixed scaling

Hungarian algo.

APX Hungarian

weighted

faster

weighted

faster

- Augmenting paths

Hopcroft-Karp

faster

Integral Hungarian

- fixed scaling

Hungarian algo.

APX Hungarian

weighted

faster

weighted

Augmenting paths

= Exposed vertices

Augmenting paths

= Exposed vertices

Augmenting paths

= Exposed vertices

Augmenting paths

= Exposed vertices

Augmenting paths

= Exposed vertices

Augmenting paths

= Exposed vertices

Augmenting paths

Max. matching ⇔ no aug. paths

Finding augmenting paths

= Exposed vertices

does M have anaugmenting path? M M4P

yes, P

return M

yes, P

return M

n paths

faster

- Augmenting paths

Hopcroft-Karp

faster

Integral Hungarian

- fixed scaling

Hungarian algo.

APX Hungarian

weighted

faster

weighted

König 1931

Hopcroft-Karp

Most augmenting paths are short

Augment in batch and increase shortest aug. pathlength

MatchingM, k def= |M| < OPT

Lemma. M has aug. path P of length

|P | ≤ 2

OPT− k

⌋+ 1

|P | ≤ 2

OPT− k

⌋+ 1

|P | ≤ 2

OPT− k

⌋+ 1

|P | ≤ 2

OPT− k

⌋+ 1

n �pn

batch-augment

yes, P

return M

n paths

does M have anaugmenting path?

return M

batch-augment

n) times

faster

- Augmenting paths

Hopcroft-Karp

faster

Integral Hungarian

- fixed scaling

Hungarian algo.

APX Hungarian

weighted

faster

weighted

König 1931

O(mn)Hopcroft and Karp

|P | ≤ 2

OPT− k

⌋+ 1

n �pn

|P | ≤ 2

OPT− k

⌋+ 1

n �pn

1/✏(1 � ✏)n

return M

batch-augmentrepeat O(1/✏)times

O(m/✏)

faster

- Augmenting paths

Hopcroft-Karp

faster

Integral Hungarian

- fixed scaling

Hungarian algo.

APX Hungarian

weighted

faster

weighted

König 1931

n)O(m/✏)

Weighted matchings

Primal

w(e) · x(e)

s. t.∑

e∈δ(v)

x(e) ≤ 1 v ∈ L ∪R,

x ≥ 0 e ∈ E

v∈L∪R

s. t. y(`) + y(r) ≥ w({`, r}) {`, r} ∈ Ey(v) ≥ 0 v ∈ L ∪R

Primal

w(e) · x(e)

s. t.∑

e∈δ(v)

x(e) ≤ 1 v ∈ L ∪R,

x ≥ 0 e ∈ E

34Dual

v∈L∪R

s. t. y(`) + y(r) ≥ w({`, r}) {`, r} ∈ Ey(v) ≥ 0 v ∈ L ∪R

w(e) · x(e)

s. t.X

e2�(v)

x(e) 1 8v,

x � 0 8e

s. t. y(`) + y(r) � w(e) 8e = {`, r}y(v) � 0 8v

Primal Dual

x, y optimal if:

x({`, r}) > 0 ) y(`) + y(r) = w({`, r}) {`, r} 2 E

y(`) > 0 )X

e2�(`)

x(e) = 1 ` 2 L

y(r) > 0 ) x(e) = 1 r 2 R

Orthogonality

w(e) · x(e)

s. t.X

e2�(v)

x(e) 1 8v,

x � 0 8e

s. t. y(`) + y(r) � w(e) 8e = {`, r}y(v) � 0 8v

Primal Dual

x, y optimal if:

x({`, r}) > 0 ) y(`) + y(r) = w({`, r}) {`, r} 2 E

y(`) > 0 )X

e2�(`)

x(e) = 1 ` 2 L

y(r) > 0 ) x(e) = 1 r 2 R

Orthogonality

{`, r} 2 M ) y(`) + y(r) = w({`, r}) {`, r} 2 E

y(`) > 0 ) ` 2 V (M) ` 2 L

y(r) > 0 ) r 2 V (M) r 2 R

Dualy(`) + y(r) � w({`, r}) {`, r} 2 E

y(v) � 0 v 2 L [ R

Primal

w(M) = OPT

Orthogonality

y(`) + y(r) = w({`, r}) 8{`, r} 2 M` 2 V (M) 8` : y(`) > 0

r 2 V (M) 8r : y(r) > 0

Orthogonality

y(`) + y(r) � w({`, r}) {`, r} 2 E

y(v) � 0 v 2 L [ R

y(`) + y(r) = w({`, r}) 8{`, r} 2 M` 2 V (M) 8` : y(`) > 0

r 2 V (M) 8r : y(r) > 0

Orthogonality

y(`) + y(r) � w({`, r}) {`, r} 2 E

y(v) � 0 v 2 L [ R

M ;y(`) max

e2Ew(e)

8` 2 L

y(r) 0 8r 2 R

y(`) + y(r) = w({`, r}) 8{`, r} 2 M` 2 V (M) 8` : y(`) > 0

r 2 V (M) 8r : y(r) > 0

Orthogonality

y(`) + y(r) � w({`, r}) {`, r} 2 E

y(v) � 0 v 2 L [ R

M ;y(`) max

e2Ew(e)

8` 2 L

y(r) 0 8r 2 R

y(`) + y(r) = w({`, r}) 8{`, r} 2 M` 2 V (M) 8` : y(`) > 0

r 2 V (M) 8r : y(r) > 0

Orthogonality

y(`) + y(r) � w({`, r}) {`, r} 2 E

y(v) � 0 v 2 L [ R

M ;y(`) max

e2Ew(e)

8` 2 L

y(r) 0 8r 2 R

GOALpush y(`) # 0

8 exposed ` 2 L

return M

y(`) maxe2E

w(e)8` 2 L

y(r) 0 8r 2 R

y(`) = 0for all exposed

` 2 L?

y(`) + y(r) = w({`, r}) 8{`, r} 2 M` 2 V (M) 8` : y(`) > 0

r 2 V (M) 8r : y(r) > 0

Orthogonality

y(`) + y(r) � w({`, r}) {`, r} 2 E

y(v) � 0 v 2 L [ R

GOALpush y(`) # 0

8 exposed ` 2 L

y(`) + y(r) = w({`, r}) 8{`, r} 2 M` 2 V (M) 8` : y(`) > 0

r 2 V (M) 8r : y(r) > 0

Orthogonality

y(`) + y(r) � w({`, r}) {`, r} 2 E

y(v) � 0 v 2 L [ R

GOALpush y(`) # 0

8 exposed ` 2 L

we can add“tight” edges

to our matching

return M

y(`) maxe2E

w(e)8` 2 L

y(r) 0 8r 2 R

` 2 L?

is there atight aug. path?

augment

y(`) + y(r) � w({`, r})

exposed weights go down

y(`) + y(r) � w({`, r})

tight edges stay covered

y(`) + y(r) � w({`, r})

tight edges stay covered

new tight edges

y(`) + y(r) � w({`, r})

return M

y(`) maxe2E

w(e)8` 2 L

y(r) 0 8r 2 R

` 2 L?

augment

shift weight

return M

y(`) maxe2E

w(e)8` 2 L

y(r) 0 8r 2 R

` 2 L?

augment

shift weight

O(m)' n times

n times

Using Egerváry’s reduction and Konig’s maximummatching alorithm, in the fall of 1953 I solved several12 by 12 assignment problems (with 3-digit integers asdata) by hand. Each of these examples took undertwo hours to solve and I was convinced that thecombined algorithm was ‘good’.

This must have been one of the last times when penciland paper could beat the largest and fastest electroniccomputer in the world.

- Harold Kuhn,On the origin of the Hungarian method

faster

- Augmenting paths

Hopcroft-Karp

faster

Integral Hungarian

- fixed scaling

Hungarian algo.

APX Hungarian

weighted

faster

weighted

König 1931

Egerváry 1931

Kuhn 1955

O(m/✏)

Jacobi 1890

What if the weights are small and integral?

w(e) ∈ {1, . . . ,W} ∀e ∈ E

return M

y(`) maxe2E

w(e)8` 2 L

y(r) 0 8r 2 R

` 2 L?

augment

shift weight

return M

y(`) maxe2E

w(e)8` 2 L

y(r) 0 8r 2 R

` 2 L?

augment

shift weight

return M

y(r) 0 8r 2 R

` 2 L?

augment

shift weight

y(`) W 8` 2 L

return M

y(r) 0 8r 2 R

` 2 L?

augmentno

y(`) W 8` 2 L shift 1 unitof weight

return M

y(r) 0 8r 2 R

augment

y(`) W 8` 2 L

repeatW times

shift 1 unitof weight

return M

y(r) 0 8r 2 R

y(`) W 8` 2 L

repeatW times

Hopcroft-Karp

return M

y(r) 0 8r 2 R

y(`) W 8` 2 L

repeatW times

Hopcroft-Karp

faster

- Augmenting paths

Hopcroft-Karp

faster

Integral Hungarian

- fixed scaling

Hungarian algo.

APX Hungarian

weighted

faster

weighted

König 1931

O(mn) Hopcroft and Karp

Egerváry 1931

Kuhn 1955

O(m/✏)

Jacobi 1890

Goal: beat O(m√n)

1. Approximate primal-dual conditions

2. Exponential scaling

3. Approximate cardinality at each scale

4. Lossy weight shift

return M

y(r) 0 8r 2 R

y(`) W 8` 2 L

repeatW times

Hopcroft-Karp

return M

y(r) 0 8r 2 R

y(`) W 8` 2 L

repeatW times

Hopcroft-Karp

nW ) O(m)

return M

y(r) 0 8r 2 R

y(`) W 8` 2 L

repeatW times

Hopcroft-Karp

nW ) O(m)

return M

y(r) 0 8r 2 R

y(`) W 8` 2 L

repeatW times

APX Hopcroft-Karp

eO(m/✏)

O(mW/✏)

return M

y(r) 0 8r 2 R

y(`) W 8` 2 L

repeatW times

APX Hopcroft-Karp

eO(m/✏)

O(mW/✏)

y(`) + y(r) � w({`, r})

y(r) = 0 8r 2 R \ V (M)

y(`) + y(r) � w({`, r})

y(r) = 0 8r 2 R \ V (M)

never touchexposed weights

y(`) + y(r) � w({`, r})

y(r) = 0 8r 2 R \ V (M)

bad news bears

return M

y(r) 0 8r 2 R

y(`) W 8` 2 L

repeatW times

APX Hopcroft-Karp

eO(m/✏)

O(mW/✏)

aug. pathsare long

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

O(1/✏)layers

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

O(1/✏)layers

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

O(1/✏)layers

no longer tight

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

O(1/✏)layers

artificially increase weight

Dualy(`) + y(r) � w({`, r}) {`, r} 2 E

y(v) � 0 v 2 L [ R

Primal

w(M) = OPT

y(`) + y(r) = w({`, r}) 8{`, r} 2 M` 2 V (M) 8` : y(`) > 0

r 2 V (M) 8r : y(r) > 0

Orthogonality

Dualy(`) + y(r) � w({`, r}) {`, r} 2 E

y(v) � 0 v 2 L [ R

Primal

M � : E ! R“excess”

y(`) + y(r) w({`, r}) + �({`, r}) 8{`, r} 2 M` 2 V (M) 8` : y(`) > 0

r 2 V (M) 8r : y(r) > 0X

e2M�(e) ✏ · w(M)

w(M) (1 + ✏) OPT

APX Orthogonality

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

O(1/✏)layers

artificially increase weight

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

O(1/✏)layers

addweightto �(e)

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

O(1/✏)layers

addweightto �(e)

choose layer that cuts✏-fraction of edges

return M

y(r) 0 8r 2 R

y(`) W 8` 2 L

repeatW times

APX Hopcroft-Karp

eO(m/✏)

O(mW/✏)

aug. pathsare long

shift weightand cheat

return M

y(r) 0 8r 2 R

y(`) W 8` 2 L

repeatW times

APX Hopcroft-Karp

eO(m/✏)

O(mW/✏)

aug. pathsare long

return M

y(r) 0 8r 2 R

y(`) W 8` 2 L

repeatW times

APX Hopcroft-Karp

eO(m/✏)

O(mW/✏)

aug. pathsare long

(a) too slow

return M

y(r) 0 8r 2 R

y(`) W 8` 2 L

repeatW times

APX Hopcroft-Karp

eO(m/✏)

O(mW/✏)

aug. pathsare long

(a) too slow(b) too much cheating

Approximate scaling

1. Round up edge weights to powers of (1 + ε):

w(e) = (1 + ε)dlog(1+ε) w(e)e e ∈ EW = max

e∈Ew(e)

(1− ε)-APX w/r/t w ⇒ (1−O(ε))-APX w/r/t w

2. Operate in units of ε(1 + ε)i−1 for “scales”

i = log1+ε W , . . . , blog1+ε εc

Approximate scaling

1. Round up edge weights to powers of (1 + ε):

w(e) = (1 + ε)dlog(1+ε) w(e)e e ∈ EW = max

e∈Ew(e)

(1− ε)-APX w/r/t w ⇒ (1−O(ε))-APX w/r/t w

2. Operate in units of ε(1 + ε)i−1 for “scales”

i = log1+ε W , . . . , blog1+ε εc

return M

y(r) 0 8r 2 R

APX Hopcroft-Karp

eO(m/✏)

aug. pathsare long

8>><>>:

log1+✏fW,

blog1+✏ ✏c

shift ✏(1 + ✏)i�1

weight and cheaty(`) fW 8` 2 L

y(`) � (1 + ✏)i 8` 2 L

y(`) = (1 + ✏)i

8 exposed ` 2 L

O(m log2(W/✏)/✏2)

return M

y(r) 0 8r 2 R

APX Hopcroft-Karp

eO(m/✏)

aug. pathsare long

8>><>>:

log1+✏fW,

blog1+✏ ✏c

shift ✏(1 + ✏)i�1

y(`) � (1 + ✏)i 8` 2 L

y(`) = (1 + ✏)i

8 exposed ` 2 L

✏ · ew(e) 8e 2 Mbecause

O(m)O(m log2(W/✏)/✏2)

return M

y(r) 0 8r 2 R

APX Hopcroft-Karp

eO(m/✏)

aug. pathsare long

8>><>>:

log1+✏fW,

blog1+✏ ✏c

shift ✏(1 + ✏)i�1

y(`) � (1 + ✏)i 8` 2 L

y(`) = (1 + ✏)i

8 exposed ` 2 L

O(m log2(W/✏)/✏2)

A few more low level details

O(m log2(1/✏)/✏)2

faster

- Augmenting paths

Hopcroft-Karp

faster

Integral Hungarian

- fixed scaling

Hungarian algo.

APX Hungarian

weighted

faster

weighted

König 1931

Egerváry 1931

Kuhn 1955

O(mn) Duan and Pettie 2014

O(m/✏)

Jacobi 1890

exact algorithm

{`, r} 2 M ) y(`) + y(r) = w({`, r}) {`, r} 2 E

y(`) > 0 ) ` 2 V (M) ` 2 L

y(r) > 0 ) r 2 V (M) r 2 R

Dualy(`) + y(r) � w({`, r}) {`, r} 2 E

y(v) � 0 v 2 L [ R

Primal

w(M) = OPT

Orthogonality

return M

y(`) maxe2E

w(e)8` 2 L

y(r) 0 8r 2 R

` 2 L?

augment

shift weight

O(m)' n times

n times

approximate algorithm

Dualy(`) + y(r) � w({`, r}) {`, r} 2 E

y(v) � 0 v 2 L [ R

Primal

M � : E ! R“excess”

y(`) + y(r) w({`, r}) + �({`, r}) 8{`, r} 2 M` 2 V (M) 8` : y(`) > 0

r 2 V (M) 8r : y(r) > 0X

e2M�(e) ✏ · w(M)

w(M) (1 + ✏) OPT

APX Orthogonality

return M

y(r) 0 8r 2 R

APX Hopcroft-Karp

eO(m/✏)

aug. pathsare long

8>><>>:

log1+✏fW,

blog1+✏ ✏c

shift ✏(1 + ✏)i�1

y(`) � (1 + ✏)i 8` 2 L

y(`) = (1 + ✏)i

8 exposed ` 2 L

O(m log2(W/✏)/✏2)

A few more low level details

O(m log2(1/✏)/✏)2

Bibliography I

C. Brezovec, G. Cornuéjols, and F. Glover. Two algorithms for weightedmatroid intersection. Math. Prog., 36(1):39–53, 1986.

W. H. Cunningham. Improved bounds for matroid partition andintersection algorithms. SIAM J. Comput., 15(4):948–957, 1986.

R. Duan and S. Pettie. Linear-time approximation for maximum weightmatching. J. Assoc. Comput. Mach., 61(1), January 2014.

R. Duan and H. Su. A scaling algorithm for maximum weight matching inbipartite graphs. In Proc. 23rd ACM-SIAM Sympos. Discrete Algs.(SODA), pages 1413–1424, 2012.

A. Frank. A weighted matroid intersection algorithm. J. Algorithms, 2:328–336, 1981.

M. L. Fredman and R. E. Tarjan. Fibonacci heaps and their uses inimproved network optimization algorithms. J. Assoc. Comput. Mach.,34(3):596–615, 1987.

S. Fujishige and X. Zhang. An efficient cost scaling algorithm for theindependent assignment problem. J. Oper. Res. Soc. Japan, 38(1):124–136, 1995.

Bibliography II

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