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A Friendly Smoothed Analysis of the Simplex Method

Daniel Dadush (CWI) & Sophie Huiberts (CWI)

A Friendly Smoothed Analysis of the Simplex Method

Daniel Dadush (CWI) & Sophie Huiberts (CWI)

Linear Programming and the Simplex Method

maximize cTx

subject to Ax ≤ b

d variables

n constraints

How many pivot steps?

Practical experience: number of pivots nearly linear in n. R. Shamir ‘87.Worst case is exponential in n.

Smoothed Analysis (Spielman-Teng ’01)

︸ ︷︷ ︸Worst case, σ = 0

︸ ︷︷ ︸Smoothed analysis, σ variable

Smoothed Linear Program

c ∈ Rd, A ∈ Rn×d, b ∈ Rn. Rows of (A, b) have norm at most 1.

A, b: entries iid N(0, σ).

A = A + A, b = b + b.

Smoothed instance: maximize cTx

subject to Ax ≤ b.

Polynomial smoothed complexity: expected running time poly(n, d, 1/σ).

Results: Running Times

Works Expected Number of Pivots

Spielman-Teng ’04 O(n86d55(1 + σ−30))

Vershynin ’09 O(d3 ln3 n σ−4 + d9 ln7 n)

Dadush-H. ’18 O(d3√

lnnσ−2 + d3.5 ln3/2 n(1 + σ−1))

Dadush-H. ’18 O(d2√

lnnσ−2 + d5 ln3/2 n)

Shadow Vertex Pivot Rule

1. Start at vertex x optimizing an objective d ∈ Rd.

2. cλ := λc + (1− λ)d.

3. Increase λ from 0 to 1, tracking optimal vertex for cλ.

Gass, Saaty ’55.

Results: Shadow Bounds

Fixed 2D plane W , smoothed P := x : Ax ≤ 1. How many edges are in πW (P )?

Works Expected Number of Edges

Spielman-Teng ’04 O(d3nσ−6 + d6n ln3 n)

Deshpande-Spielman ’05 O(dn2 lnn σ−2 + d2n2 ln2 n)

Vershynin ’09 O(d3σ−4 + d5 ln2 n)

Dadush-H. ’18 O(d2√

lnn σ−2 + d2.5 ln3/2 n (1 + σ−1))

Polyhedral Duality

P :=x : aTi x ≤ 1 ∀i ≤ n Q : = ConvexHull(a1, . . . , an)

W

number of edges in πW (P ) = number of edges in Q ∩W.

Main Theorem

E[#edges] ≤ E[perimeter(Q ∩W )]

minB⊂a1,...,an,|B|=d

E[length(conv(ai ∈ B) ∩W ) | B forms an edge ]

≤ O(d1.5L

τ(1 + r)(1 + R))

Parameter Description Gaussian

r r = E[maxi≤n ‖πW (ai)‖] O(σ√

lnn)

R Pr[maxi≤n ‖ai‖ ≥ R] ≤ n−d O(σ√d lnn)

L L-log-Lipschitz probability density within radius R O(σ√d lnn)

τ Restricted to any line l the variance of each ai is at least τ 2 σ

Expected Perimeter

Convex polyhedron’s perimeter boundedby enclosing shape.

E[perimeter(Q ∩W )]

≤ 2πE[ maxx∈Q∩W

‖x‖]

≤ 2πE[max ‖πW (ai)‖]≤ 2π(1 + E[max ‖πW (ai)‖])≤ O(1 + r)

Expected Edge Length

H

W

l

a1

a2

a3

E[height of simplex along l]

≥ Ω(τ/√d)

E[

intersection length

height of simplex along l

]≥ Ω

(1

dL(1 + R)

)

Open Problems

1. Improve upper/lower bounds on expected shadow size.

2. Generalize to bounded error distributions.

3. Analyze different pivot rules (e.g. steepest edge).

4. Analyze sequences of related LPs.

5. Bound the diameter of smoothed polyhedra.