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Speeding Up MWU Based Approximation Schemes Speeding Up MWU Based Approximation Schemes

Speeding Up MWU Based Approximation Schemes - PowerPoint Presentation

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Speeding Up MWU Based Approximation Schemes - PPT Presentation

and Some Applications Chandra Chekuri Univ of Illinois UrbanaChampaign Based on joint work Nearlineartime approximation schemes for some implicit fractional packing problems   with ID: 616727

weights packing spanning edge packing weights edge spanning update log polylog tsp weight iteration problems mwu time trees metric

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Slide1

Speeding Up MWU Based Approximation Schemes and Some Applications

Chandra

Chekuri

Univ. of Illinois, Urbana-ChampaignSlide2

Based on joint workNear-linear-time approximation schemes for some implicit fractional packing problems

 

with

Kent

Quanrud

, SODA 2017

Approximating the Held-Karp Bound for Metric TSP in Nearly-Linear

Time

with

 

Kent

Quanrud

,

ArXiv

, 2017

Ongoing

work

with

Kent

Quanrud

On Multiplicative Weight Updates for Concave and

Submodular

Function Maximization

,

 with

Jayaram

Thathachar

and

Jan

Vondrak

, ITCS 2015.Slide3

(Pure) Packing LPn: dimension of problem (size of

x

)

m:

number of rows of

A (non-trivial constraints)N: number of non-zeroes in A

v, A non-negativeSlide4

Explicit Packing ProblemsHow fast can a

(1-

)

approximation be computed?

MWU based algorithms:

randomized O(N + (n+m

) log n/

)

[Koufogiannakis-Young’07]

deterministic O(N log n/) [Young’15]Accelerated gradient descent based algorithm: randomized O(N log n log (1//) [AllenZhu-Orrechia’15]

 Slide5

Implicit Packing ProblemsPacking matrix A

defined implicitly by a combinatorial structure

N

is “large” in terms of original inputSlide6

Packing Spanning Trees

Input:

graph

G=(V,E)

edge capacities

ce

Goal: find a max fractional packing of spanning trees

Slide7

Interval Packingn closed intervals I

1

, I

2

,

…, In: I

i = [ai, b

i

]

Ii has size di and value vim points p1, p2, …, pm on real line pj has capacity c

j

Slide8

Interval Packing LPSlide9

TSP and Metric-TSPTSP: Undir graph

G=(V,E)

,

edge

costs

cefind Hamiltonian Cycle in G of minimum cost

Metric-TSP: Undir graph G=(V,E), edge costs ce

find spanning

tour

in G of minimum costsame as Hamiltonian Cycle in metric completion of GSlide10

Subtour Elimination LP for TSPSlide11

2ECSS LP

For Metric-TSP solving 2ECSS LP is equivalent to solving

Subtour

LPSlide12

Faster Algorithms

Problem

Previous best

New

bound

Packing Spanning Trees

O(mn polylog n/

)

O(m

polylog n/)

Interval

Packing LP

O(

mn

polylog

/

)

O((

m+n

) polylog n/

)

Metric-TSP

LPO(m

2 log n/

)

O(m

polylog n/

)

(randomized)

Problem

Previous best

New

bound

Packing Spanning Trees

Interval

Packing LP

Metric-TSP

LP

Ideas extend to covering and some mixed packing covering problems

Several applications to LPs for combinatorial problemsSlide13

High-level IdeasSpeed up classical MWU based approximation schemesProblem-specific

integration

of

dynamic data structures

for two separate issues

oracle for MWUlazy weight updateImportant observation: matrix A is 0,1 or column restricted (all non-zero entries in each column is same)Slide14

MWUMaintain weights for constraints: w1, w2

,

…, w

m

In each iteration solve Lagrangean relaxation:

collapse m constraints into one constraintTake small step in direction of new solutionUpdate weights Iterate until doneSlide15
Slide16
Slide17

Simple AnalysisNumber of iterations is

O(m log m/

)

Potential function:

In each iteration at least one weight increases by a multiplicative

(1+

)

exp

(

)

factor

In each iteration

h

need to compute

best

coordinate

j

h

update all weights

 Slide18

Packing Spanning Trees

Input:

graph

G=(V,E)

edge capacities

ce

Goal: find a max fractional packing of spanning trees

Slide19

Applying MWU FrameworkMaintain weights

w(e)

for each edge

e

In each iteration

h compute MST Th wrt current

edge weights: this will be the single “coordinate” Update weights of edges in Th Runtime: O(m log m/

(m + n)) = O(m

2

log m/ Slide20

Improving run time via data structuresIn each iteration

h

compute MST

T

h

wrt edge current edge weights w(e)Do not compute MST from scratch: maintain MST via dynamic data structure

Maintain weights lazily, update only if weight increases by (1+ )

factor,

Total number of updates is

O(m log m/)MST update time per edge weight change is polylog(m) [Holm-Lichtenburg-Thorup’98/01] Slide21

Applying MWU FrameworkMaintain weights

w(e)

for each edge

e

In each iteration

h compute MST Th wrt edge current edge weights

w(e): this will be the single “coordinate” Update weights of edges in Th Runtime: O(m log m/

(

polylog

(m) + n)) = O(mn polylog(m)/Bottleneck is weight update!

 Slide22

Updating weights in each iterationSlide23

Updating weights in each iterationSlide24

Updating weights lazilyCan update weight lazily if it does not change much

maintain within a

(1

multiplicative actor

When column

j is updated rate of change of wi depends on

A

ij

if all Aij values are uniform then can charge weight update to potential function changeif Aij values are non-uniform delay updating small weight changes. How? Slide25

Updating weights lazilyBorrow ideas from [Young]

Deterministically:

Bucket

A

ij

values geometrically and lazily update using amortizationRandomization: touch/update wi in proportion to 1/

Aij. For efficiency pick r from [0,1] and correlate

Crucial in spanning trees:

all non-zero entries in each row are same because of 0,1 incidence matrix. Slide26

Packing Spanning Trees: Related Results

Can approximately decompose a point in a spanning tree polytope into a convex combination of spanning trees in near-linear time

compact representation of a

(1+

)

packing with O(m polylog n/

)

edges

Can find network strength and fractional packing # in O((m + n/)polylog) time

 Slide27

Interval Packing ProblemsSlide28

Main IdeaUse range tree data structuresMatrix A factorizes via range tree data structure into

BC

where

B

is row-sparse and

C is column-sparseSlide29

Metric-TSPMuch more involvedHeavily builds on Karger’s near-linear time randomized

mincut

algorithm

Almost lucky?Slide30

Open ProblemsMore applications for implicit packing/covering/mixed packing covering problems

Ongoing work for mixed packing and covering: randomized run time of

O

(N/

+ (

n+m)/

)

O(1/

)

dependence Slide31

Thank You!