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Smoothed Analysis of Algorithms Why The Simplex

Method Usually Takes Polynomial Time

Gaussian Perturbation with variance s2

- Shang-Hua Teng
- Boston University/Akamai

Joint work with Daniel Spielman (MIT)

Remarkable Algorithms and Heuristics

- Work well in practice, but
- Worst case bad,
- exponential,
- contrived.
- Average case good,
- polynomial,
- meaningful?

Random is not typical

Smoothed Analysis of Algorithms

worst case maxx T(x) average case avgr

T(r) smoothed complexity maxx avgr T(xer)

Smoothed Analysis of Algorithms

- Interpolate between Worst case and Average Case.
- Consider neighborhood of every input instance
- If low, have to be unlucky to find bad input

instance

Complexity Landscape

Classical Example Simplex Method for Linear

Programming

max zT x s.t. A x y

- Worst-Case exponential
- Average-Case polynomial
- Widely used in practice

The Diet Problem

Minimize 30 x1 80 x2 20 x3 s.t. 30x1

10 x2 6 x3 ? 300 5x1

9x2 8x3 ? 50 1.5x1 2.5 x2

18 x3 ? 70 10x1

6 x3 ? 100

x1, x2, x3 ? 0

Linear Programming

max zT x s.t. A x y

Max x1 x2 s.t x1 1

x2 1 -x1 - 2x2 1

Smoothed Analysis of Simplex Method

G is Gaussian

Smoothed Analysis of Simplex Method

- Worst-Case exponential
- Average-Case polynomial
- Smoothed Complexity polynomial

max zT x s.t. aiT x 1, ai 1

max zT x s.t. (aisgi )T x 1

Perturbation yields Approximation

- For polytope of good aspect ratio

But, combinatorially

The Simplex Method

History of Linear Programming

- Simplex Method (Dantzig, 47)
- Exponential Worst-Case (Klee-Minty 72)
- Avg-Case Analysis (Borgwardt 77, Smale 82,

Haimovich, Adler, Megiddo, Shamir, Karp, Todd) - Ellipsoid Method (Khaciyan, 79)
- Interior-Point Method (Karmarkar, 84)
- Randomized Simplex Method (mO(?d) )
- (Kalai 92, Matousek-Sharir-Welzl

92)

Shadow Vertices

Another shadow

Shadow vertex pivot rule

start

z

objective

Theorem For every plane, the expected size of

the shadow of the perturbed tope is poly(m,d,1/s )

Theorem For every z, two-Phase Algorithm runs in

expected time poly(m,d,1/s )

z

A Local condition for optimality

z

a2

z

0

Vertex on a1,,ad maximizes z iff z Î

cone(a1,,ad )

a1

Primal a1T x ? 1 a2T x ? 1 amT x ?

1

Polar ConvexHull(a1, a2, am)

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Polar Linear Program

z

max ? ?z Î ConvexHull(a1, a2, ..., am)

Initial Simplex

Opt Simplex

Shadow vertex pivot rule

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Count facets by discretizingto N directions, N ??

Count pairs in different facets

So, expect c Facets

Expect cone of large angle

Angle

Distance

Isolate on one Simplex

Integral Formulation

Example For a and b Gaussian

distributed points, given that ab

intersects x-axis Prob? lt

e O(e2)

a

?

b

a

?

b

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a

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Change of variables

u

a

z

?

v

b

da db (uv)sin(q) du dv dz dq

Analysis For e lt e0, Pe lt e2

Slight change in q has little effect on ni

for all but very rare u,v,z

Distance Gaussian distributed corners

Idea fix by perturbation

Trickier in 3d

Future Research Simplex Method

- Smoothed analysis of other pivot rules
- Analysis under relative perturbations.
- Trace solutions as un-perturb.
- Strongly polynomial algorithm for linear

programming?

A Theory Closer to Practice

- Optimization algorithms and heuristics, such as

Newtons Method, Conjugate Gradient, Simulated

Annealing, Differential Evolution, etc. - Computational Geometry, Scientific Computing and

Numerical Analysis - Heuristics solving instances of NP-Hard

problems. - Discrete problems?
- Shrink intuition gap between theory and practice.