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Linear Programming

$$\matrix{A} \times \matrix{X} = \matrix{B} $$

Coeffienent matrix

A => n * n, x vector of unknowns, B right hand side

Linear system of equations.


$$A \times X ≥ B$$

Objective function

$$C^T \times X $$

linear

Goal : Minimize the objective function

E.g.: $$ \cases{x1 - x2 ≥ 0 \
x_1 ≥ 0 \
x_2 ≥ 0 \
x1 + x2 ≤ 4 }$$

Maximize $x1 + 2x_2$

###Simplex method Only look at the vertex

Start of one vertex, go clockwise, find the max before the value going down

###Weighted vertex cover problem

for G = (V, E), S $\in$ V in a set such that each edge has at least one end in S $W_i ≥ 0$ for each i $\in$ V $$W(S) = \sum i \in S w_i$$

Objective: Minimize W(S)

Model this as an LP

x_i is a decision valuable for each node i \in V

$$ \cases{ x_1 - x_2 ≥ 0 \
x_1 ≥ 0 \
x_2 ≥ 0 \
x_1 + x_2 ≤ 4 }$$

$$ \cases{ x_i = 0 & i \notin S \
x_i = 1 & i \in S } $$

x_i + x_j ≥ 1 for each edge

Minimize \sigma w_ix_i Subject to $x_i + x_j ≥ 1 for (i, j) \in E$ $x_i \in {0,1} for i \in V$ <= discreate

Integer Programming

Linear programming (continues variables) Integer Programming (discrete variables) mixed integer programming

Drop the requirement that x_i \in {0,1} and solve the LP in poly time and find {x_i^*} between 0 and 1

so that x_i^* + x_j^* ≥ 1 for each edge

W_{LP} = \sigma w_ix_i^*

S^* is the opt vertex cover set W(S^*) = weight of the opt solution

W(S^*) ≥ W_{LP}

S is our approx. solution

W(S) ≤ 2 * W{LP} W{LP} ≤ 2 * W(s^*)

W(S) ≤ 2 * W(S^*)

Solve the max. flow problem using LP. Variable are flow over edges.

Maximize \sigma f{e} subject to 0 ≤ f(e) ≤ c_e foe each edge e \in E \sigma f(e) - \sigma f(e) = 0 for v \in V

A = B A - B ≥ 0 B - A ≥ 0

Max flow with lower bounds on flow over the edges objective function stays same conservation of the flow stays same

Cap constraint: l_e≤f(e)≤c_e for each edge e \in E

###Multi commodity flow f_i(e): flow of commodity i over edge e \alpha_i: is the profit associated with one unit of flow for commodity i.

We have m commodities

Objective: maximize profit

Maximize $\sigma{l=i}^m \sigma{eoutofS} \alpha_i f_i(e)$

subject to 0 ≤ \sigma_{i=1}^m fi{e} ≤c{e} for each e \in E

\sigma_{i=1}^m fi{e} = \sigma{i=1}^m f_i{e} for each node v \in V and for each i = 1 to m

###Shortest path using LP Shortest distance from V to t is d(v) for each node V For each node V $$ d(t) ≤ d(y) + c{yt} \
d(t) ≤ d(w) + c
{wt} \
d(t) ≤ d(x) + c_{xt} \
$$

d{v} ≤ d{u} + w(u, v) for each edge (u, v) \in E d(s) = 0

Objective function: Minimize d(t)

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Linear Programming