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
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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 doneSlide15Slide16Slide17
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!