CS 473: Algorithms Chandra Chekuri Ruta Mehta University of - PowerPoint PPT Presentation

CS 473: Algorithms Chandra Chekuri Ruta Mehta University of Illinois, Urbana-Champaign Fall 2016 Chandra & Ruta (UIUC) CS473 1 Fall 2016 1 / 43

CS 473: Algorithms, Fall 2016 Introduction to Linear Programming Lecture 18 October 26, 2016 Chandra & Ruta (UIUC) CS473 2 Fall 2016 2 / 43

Part I Introduction to Linear Programming Chandra & Ruta (UIUC) CS473 3 Fall 2016 3 / 43

A Factory Example Problem Suppose a factory produces two products 1 and 2 using resources A , B , C . Making a unit of 1 requires a unit each of A and C . 1 A unit of 2 requires one unit of B and C . 2 We have 200 units of A , 300 units of B , and 400 units of C . 3 Product 1 can be sold for $1 and product 2 for $6 . 4 How many units of product 1 and product 2 should the factory manufacture to maximize profit? Chandra & Ruta (UIUC) CS473 4 Fall 2016 4 / 43

A Factory Example Problem Suppose a factory produces two products 1 and 2 using resources A , B , C . Making a unit of 1 requires a unit each of A and C . 1 A unit of 2 requires one unit of B and C . 2 We have 200 units of A , 300 units of B , and 400 units of C . 3 Product 1 can be sold for $1 and product 2 for $6 . 4 How many units of product 1 and product 2 should the factory manufacture to maximize profit? Solution: Chandra & Ruta (UIUC) CS473 4 Fall 2016 4 / 43

A Factory Example Problem Suppose a factory produces two products 1 and 2 using resources A , B , C . Making a unit of 1 requires a unit each of A and C . 1 A unit of 2 requires one unit of B and C . 2 We have 200 units of A , 300 units of B , and 400 units of C . 3 Product 1 can be sold for $1 and product 2 for $6 . 4 How many units of product 1 and product 2 should the factory manufacture to maximize profit? Solution: Formulate as a linear program. Chandra & Ruta (UIUC) CS473 4 Fall 2016 4 / 43

A Factory Example Problem Suppose a factory produces two products 1 and 2 , using resources A , B , C . Making a unit of 1 : Req. one 1 unit of A , C . Making unit of 2 : Req. one 2 unit of B , C . Have A : 200, B : 300 , and 3 C : 400. Price of 1 : $1 , and 2 : $6 . 4 How many units of 1 and 2 to manufacture to max profit? Chandra & Ruta (UIUC) CS473 5 Fall 2016 5 / 43

A Factory Example Problem Suppose a factory produces two products 1 and 2 , using resources A , B , C . max x 1 + 6x 2 Making a unit of 1 : Req. one s.t. x 1 ≤ 200 (A) 1 unit of A , C . x 2 ≤ 300 (B) Making unit of 2 : Req. one 2 x 1 + x 2 ≤ 400 (C) unit of B , C . x 1 ≥ 0 Have A : 200, B : 300 , and 3 x 2 ≥ 0 C : 400. Price of 1 : $1 , and 2 : $6 . 4 How many units of 1 and 2 to manufacture to max profit? Chandra & Ruta (UIUC) CS473 5 Fall 2016 5 / 43

Linear Programming Formulation Let us produce x 1 units of product 1 and x 2 units of product 2. Our profit can be computed by solving maximize x 1 + 6x 2 subject to x 1 ≤ 200 x 2 ≤ 300 x 1 + x 2 ≤ 400 x 1 , x 2 ≥ 0 Chandra & Ruta (UIUC) CS473 6 Fall 2016 6 / 43

Linear Programming Formulation Let us produce x 1 units of product 1 and x 2 units of product 2. Our profit can be computed by solving maximize x 1 + 6x 2 subject to x 1 ≤ 200 x 2 ≤ 300 x 1 + x 2 ≤ 400 x 1 , x 2 ≥ 0 What is the solution? Chandra & Ruta (UIUC) CS473 6 Fall 2016 6 / 43

Maximum Flow in Network 9 3 6 10 15 10 4 15 5 8 10 s 2 5 t 15 10 4 15 6 30 1 4 Chandra & Ruta (UIUC) CS473 7 Fall 2016 7 / 43

Maximum Flow in Network Need to compute values 9 3 6 f s1 , f s2 , . . . f 25 , . . . f 5t , f 6t such that 10 15 10 4 15 5 8 10 s 2 5 t 15 10 4 15 6 30 1 4 Chandra & Ruta (UIUC) CS473 7 Fall 2016 7 / 43

Maximum Flow in Network Need to compute values 9 3 6 f s1 , f s2 , . . . f 25 , . . . f 5t , f 6t such that 10 15 10 4 15 f s1 ≤ 15 f s2 ≤ 5 f s3 ≤ 10 5 8 10 f 14 ≤ 30 f 21 ≤ 4 f 25 ≤ 8 s 2 5 t f 32 ≤ 4 f 35 ≤ 15 f 36 ≤ 9 15 10 4 15 6 f 42 ≤ 6 f 4t ≤ 10 f 54 ≤ 15 30 f 5t ≤ 10 f 65 ≤ 15 f 6t ≤ 10 1 4 Chandra & Ruta (UIUC) CS473 7 Fall 2016 7 / 43

Maximum Flow in Network Need to compute values 9 3 6 f s1 , f s2 , . . . f 25 , . . . f 5t , f 6t such that 10 15 10 4 15 f s1 ≤ 15 f s2 ≤ 5 f s3 ≤ 10 5 8 10 f 14 ≤ 30 f 21 ≤ 4 f 25 ≤ 8 s 2 5 t f 32 ≤ 4 f 35 ≤ 15 f 36 ≤ 9 15 10 4 15 6 f 42 ≤ 6 f 4t ≤ 10 f 54 ≤ 15 30 f 5t ≤ 10 f 65 ≤ 15 f 6t ≤ 10 1 4 and f s1 + f 21 = f 14 f s2 + f 32 = f 21 + f 25 f s3 = f 32 + f 35 + f 36 f 14 + f 54 = f 42 + f 4t f 25 + f 35 + f 65 = f 54 + f 5t f 36 = f 65 + f 6t Chandra & Ruta (UIUC) CS473 7 Fall 2016 7 / 43

Maximum Flow in Network Need to compute values 9 3 6 f s1 , f s2 , . . . f 25 , . . . f 5t , f 6t such that 10 15 10 4 15 f s1 ≤ 15 f s2 ≤ 5 f s3 ≤ 10 5 8 10 f 14 ≤ 30 f 21 ≤ 4 f 25 ≤ 8 s 2 5 t f 32 ≤ 4 f 35 ≤ 15 f 36 ≤ 9 15 10 4 15 6 f 42 ≤ 6 f 4t ≤ 10 f 54 ≤ 15 30 f 5t ≤ 10 f 65 ≤ 15 f 6t ≤ 10 1 4 and f s1 + f 21 = f 14 f s2 + f 32 = f 21 + f 25 f s3 = f 32 + f 35 + f 36 f 14 + f 54 = f 42 + f 4t f 25 + f 35 + f 65 = f 54 + f 5t f 36 = f 65 + f 6t f s1 ≥ 0 f s2 ≥ 0 f s3 ≥ 0 · · · f 4t ≥ 0 f 5t ≥ 0 f 6t ≥ 0 Chandra & Ruta (UIUC) CS473 7 Fall 2016 7 / 43

Maximum Flow in Network Need to compute values 9 3 6 f s1 , f s2 , . . . f 25 , . . . f 5t , f 6t such that 10 15 10 4 15 f s1 ≤ 15 f s2 ≤ 5 f s3 ≤ 10 5 8 10 f 14 ≤ 30 f 21 ≤ 4 f 25 ≤ 8 s 2 5 t f 32 ≤ 4 f 35 ≤ 15 f 36 ≤ 9 15 10 4 15 6 f 42 ≤ 6 f 4t ≤ 10 f 54 ≤ 15 30 f 5t ≤ 10 f 65 ≤ 15 f 6t ≤ 10 1 4 and f s1 + f 21 = f 14 f s2 + f 32 = f 21 + f 25 f s3 = f 32 + f 35 + f 36 f 14 + f 54 = f 42 + f 4t f 25 + f 35 + f 65 = f 54 + f 5t f 36 = f 65 + f 6t f s1 ≥ 0 f s2 ≥ 0 f s3 ≥ 0 · · · f 4t ≥ 0 f 5t ≥ 0 f 6t ≥ 0 and f s1 + f s2 + f s3 is maximized. Chandra & Ruta (UIUC) CS473 7 Fall 2016 7 / 43

Maximum Flow as a Linear Program For a general flow network G = (V , E) with capacities c e on edge e ∈ E , we have variables f e indicating flow on edge e � Maximize f e e out of s subject to f e ≤ c e for each e ∈ E � � f e − f e = 0 ∀ v ∈ V \ { s , t } e out of v e into v f e ≥ 0 for each e ∈ E . Chandra & Ruta (UIUC) CS473 8 Fall 2016 8 / 43

Maximum Flow as a Linear Program For a general flow network G = (V , E) with capacities c e on edge e ∈ E , we have variables f e indicating flow on edge e � Maximize f e e out of s subject to f e ≤ c e for each e ∈ E � � f e − f e = 0 ∀ v ∈ V \ { s , t } e out of v e into v f e ≥ 0 for each e ∈ E . Number of variables: m , one for each edge. Number of constraints: m + n − 2 + m . Chandra & Ruta (UIUC) CS473 8 Fall 2016 8 / 43

Minimum Cost Flow with Lower Bounds ... as a Linear Program For a general flow network G = (V , E) with capacities c e , lower bounds ℓ e , and costs w e , we have variables f e indicating flow on edge e . Suppose we want a min-cost flow of value at least v . � Minimize w e f e e ∈ E � subject to f e ≥ v e out of s f e ≤ c e f e ≥ ℓ e for each e ∈ E � � f e − f e = 0 for each v ∈ V − { s , t } e out of v e into v f e ≥ 0 for each e ∈ E . Chandra & Ruta (UIUC) CS473 9 Fall 2016 9 / 43

Minimum Cost Flow with Lower Bounds ... as a Linear Program For a general flow network G = (V , E) with capacities c e , lower bounds ℓ e , and costs w e , we have variables f e indicating flow on edge e . Suppose we want a min-cost flow of value at least v . � Minimize w e f e e ∈ E � subject to f e ≥ v e out of s f e ≤ c e f e ≥ ℓ e for each e ∈ E � � f e − f e = 0 for each v ∈ V − { s , t } e out of v e into v f e ≥ 0 for each e ∈ E . Number of variables: m , one for each edge Number of constraints: 1 + m + m + n − 2 + m = 3m + n − 1 . Chandra & Ruta (UIUC) CS473 9 Fall 2016 9 / 43

Linear Programs Problem Find a vector x ∈ R d that � d maximize/minimize j=1 c j x j � d subject to j=1 a ij x j ≤ b i for i = 1 . . . p � d j=1 a ij x j = b i for i = p + 1 . . . q � d j=1 a ij x j ≥ b i for i = q + 1 . . . n Chandra & Ruta (UIUC) CS473 10 Fall 2016 10 / 43

Linear Programs Problem Find a vector x ∈ R d that � d maximize/minimize j=1 c j x j � d subject to j=1 a ij x j ≤ b i for i = 1 . . . p � d j=1 a ij x j = b i for i = p + 1 . . . q � d j=1 a ij x j ≥ b i for i = q + 1 . . . n Input is matrix A = (a ij ) ∈ R n × d , column vector b = (b i ) ∈ R n , and row vector c = (c j ) ∈ R d Chandra & Ruta (UIUC) CS473 10 Fall 2016 10 / 43

Canonical Form of Linear Programs Canonical Form A linear program is in canonical form if it has the following structure � d maximize j=1 c j x j � d subject to j=1 a ij x j ≤ b i for i = 1 . . . n Chandra & Ruta (UIUC) CS473 11 Fall 2016 11 / 43

Canonical Form of Linear Programs Canonical Form A linear program is in canonical form if it has the following structure � d maximize j=1 c j x j � d subject to j=1 a ij x j ≤ b i for i = 1 . . . n Conversion to Canonical Form Replace � j a ij x j = b i by � j a ij x j ≤ b i and − � j a ij x j ≤ − b i 1 Replace � j a ij x j ≥ b i by − � j a ij x j ≤ − b i 2 Chandra & Ruta (UIUC) CS473 11 Fall 2016 11 / 43