A Randomized Polynomial-Time Simplex Algorithm for

Linear ProgrammingCS3150 Course Presentation

Linear Programming

Example:･ Find the maximum value of p = 3x - 2y + 4z

･ subject to ･ 4x + 3y - z >= 3

･ x + 2y + z<=4 ･ x >= 0, y >= 0, z >= 0

Linear Programming

• Objective Functionmax zT x

• Constraintss.t. A x y

Simplex Method - Intuition

Objective:– Min C = 3x + 4y

Constraints:– ･ 3x - 4y <= 12, – ･ x + 2y >= 4 – ･ x >= 1, y >= 0.

Simplex Method - Intuition

max zT xs.t. A x y• Worst-Case:

exponential• Average-Case:

polynomial• Widely used in

practice

Shadow Vertices

Shadow vertex pivot rule

objective

start z

Complexity Landscape

Complexity Landscape of Perturbed Problem

Some issues

Problemmax zT xs.t. A x y

– The perturbed problem is no longer the original problem we want to solve!

Solution– Reduce the original

problem to another problem, where the perturb won’t affect solution.

– Is this set of constraints bounded?

A’w<=1

Intuition of the Algorithm

Since the right hand side won’t affect solution, we want to carefully choose it so that the Shadow-Vertex Simplex will run poly-time with high probability.

Intuition of the Proof

P is the polytope of A’w<=1

Case 1:– The polytope P is in k-near-isotropic

position

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B(0,1)⊂P ⊂B(0,k)

K-near-isotropic position

Intuition of the Proof

Case 1:– The polytope is in k-near-isotropic position–

Case 2:– The polytope is not in k-near-isotropic position

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B(0,1)⊂P ⊂B(0,k)

K-near-isotropic Case

Upper bound of total shadow length (Shadow Size).

Lower bound expected length of each edge.

Number of edges of the shadow is poly in size w.h.p.

K-near-isotropic position

Randomization

Each of vector is a independently exponentially distributed random variable with expectation

Project onto a random plane€

A'w ≤1+ r

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ri

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r

€

λ

None-K-near-isotropic Case

By Running the shadow vertex for a limited amount of time we can either:– Find the optimal– Or find a way to eliminate bad events

w.h.p.

None-K-near-isotropic Case

K-near-isotropic Case

Upper bound of the total shadow length.

Lower bound the expected length of each edge.

Number of edges of the shadow is poly in size w.h.p.

Upper Bound of Shadow Size

A’w<=1

A’w<=1+r

Shadow Size in Case 1

The expected shadow size is at most:

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2πk(1+ λ ln(ne))

Upper Bound of Shadow Size

K-near-isotropic Case

Upper bound of the total shadow length.

Lower bound the expected length of each edge.

Number of edges of the shadow is poly in size w.h.p.

Expected Edge Length

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λ6 dn

The Expected Edge Length is at least:

Case 1 Main Theorem

The expected number of edges is at most

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12πk(1+ λ ln(ne)) dn

λ

None-K-Isotropic Case

The expected shadow size inside any given ball is small

None-K-near-Isotropic Case

Upper bound of the total shadow length within the given ball.

Lower bound the expected length of each edge within the given ball.

Number of edges of the shadow is poly in size w.h.p.

Outline of the Algorithm

Run Shadow Vertex Simplex on the Randomized Input– If Find Optimal then halt

Else transform the coordinates and run shadow vertex simplex on the transformed inputsAlgorithm will halt in poly-time w.h.p.

Summery and Intuitions

Deterministic algorithms run exponential time on some “bad” inputsBy introducing some randomness into the algorithm fixed the problem. The Randomized algorithm run poly time on all inputs with high probability.Start with something strict, which is easy to prove the poly-bound, eliminate the bad events in poly-time.