Algorithms (2IL15) Lecture 12 Linear Programming 1 TU/e - - PowerPoint PPT Presentation

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Algorithms (2IL15) – Lecture 12

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Algorithms (2IL15) – Lecture 12 Linear Programming

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Algorithms (2IL15) – Lecture 12

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Summary of previous lecture

• ρ-approximation algorithm:

algorithm for which computed solution is within factor ρ from OPT

• to prove approximation ratio we usually need lower bound on OPT

(or, for maximization, upper bound)

• PTAS = polynomial-time approximation scheme

= algorithm with two parameters, input instance and ε > 0, such that

– approximation ratio is1+ ε – running time is polynomial in n for constant ε

• FPTAS = PTAS whose running time is polynomial in 1/ ε
• some problems are even hard to approximate

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Today: linear programming (LP)

• most used and most widely studied optimization method
• can be solved in polynomial time (input size measured in bits)
• can be used to model many problems
• also used in many approximation algorithms (integer LP + rounding)

we will only have a very brief look at LP …

• what is LP? what are integer LP and 0/1-LP ?
• how can we model problems as an LP ?

… and not study algorithms to solve LP’s

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Example problem: running a chocolate factory Assortment Cost and availability of ingredients caca

• mil

k hazelnut s retail price (100 g) Pure Black 1 1.99 Creamy Milk 0.6 0.4 1.49 Hazelnut Delight 0.6 0.2 0.2 1.69 Super Nuts 0.5 0.1 0.4 1.79

cost (kg) available (kg) cacao 2.1 50 milk 0.35 50 hazelnuts 1.9 30

How much should we produce of each product to maximize our profit ?

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Modeling the chocolate-factory problem

cacao milk hazelnuts price (100 g) Pure Black 1 1.99 Creamy Milk 0.6 0.4 1.49 Hazelnut Delight 0.6 0.2 0.2 1.69 Super Nuts 0.5 0.1 0.4 1.79 cost (kg) available (kg) cacao 2.1 50 milk 0.35 50 hazelnuts 1.9 30

variables we want to determine: production (kg) of the products b = production of Pure Black m = production of Creamy Milk h = production of Hazelnut Delight s = production of Super Nuts

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Modeling the chocolate-factory problem (cont’d)

cacao milk hazelnuts price (100 g) Pure Black 1 1.99 Creamy Milk 0.6 0.4 1.49 Hazelnut Delight 0.6 0.2 0.2 1.69 Super Nuts 0.5 0.1 0.4 1.79 cost (kg) available (kg) cacao 2.1 50 milk 0.35 50 hazelnuts 1.9 30

profits (per kg) of the products Pure Black 19.9 – 2.1 = 17.8 Creamy Milk 14.9 – 0.6 x 2.1 + 0.4 x 0.35 = 13.5 Hazelnut Delight 15.19 Super Nuts: 16.055 total profit: 17.8 b + 13.5 m + 15.19 h + 16.055 s

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Modeling the chocolate-factory problem

cacao milk hazelnuts price (100 g) Pure Black 1 1.99 Creamy Milk 0.6 0.4 1.49 Hazelnut Delight 0.6 0.2 0.2 1.69 Super Nuts 0.5 0.1 0.4 1.79 cost (kg) available (kg) cacao 2.1 50 milk 0.35 50 hazelnuts 1.9 30

we want to maximize the total profit 17.8 b + 13.5 m + 15.19 h + 16.055 s under the constraints b + 0.6 m + 0.6 h + 0.5 s ≤ 50 (cacao availability) 0.4 m + 0.2 h + 0.1 s ≤ 50 (milk availability) 0.2 h + 0.4 s ≤ 30 (hazelnut availability) This is a linear program: optimize linear function, under set of linear constraints

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1 −3 y ≥ − x + 3 y ≥ 2x − 4 y ≤ ½ x + 2 y ≥ 0

m constraints: linear function ≤ constant

• r ≥ or =, > and < not allowed

n variables; here n=2, in chocolate example n=4, but often n is large

• bjective function; must be

linear function in the variables goal: maximize (or minimize) x y Linear programming Find values of real variables x, y such that

• x − 3 y is maximized
• subject to the constraints

− 2 x + y ≥ − 4 x + y ≥ 3 − ½ x + y ≤ 2 y ≥ 0

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y = − x + 3 y = 2x − 4 y = ½ x + 2 1 −3 y = 0

Linear programming Find values of real variables x, y such that

• x − 3 y is maximized
• subject to the constraints

− 2 x + y ≥ − 4 x + y ≥ 3 − ½ x + y ≤ 2 y ≥ 0 feasible region = region containing feasible solutions = region containing solutions satisfying all constraints feasible region is convex polytope in n-dim space

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Linear programming: Find values of real variables x1, …, xn such that

• given linear function c1 x1 + c2 x2 + … + cn xn is maximized (or: minimized)
• and given linear constraints on the variables are satisfied

constraints: equalities or inequalities using ≥ or ≤, cannot use < and > Possible outcomes:

• unique optimal solution: vertex of feasible region

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Linear programming: Find values of real variables x1, …, xn such that

• given linear function c1 x1 + c2 x2 + … + cn xn is maximized (or: minimized)
• and given linear constraints on the variables are satisfied

constraints: equalities or inequalities using ≥ or ≤, cannot use < and > Possible outcomes:

• unique optimal solution: vertex of feasible region
• no solution: feasible region in empty

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Linear programming: Find values of real variables x1, …, xn such that

• given linear function c1 x1 + c2 x2 + … + cn xn is maximized (or: minimized)
• and given linear constraints on the variables are satisfied

constraints: equalities or inequalities using ≥ or ≤, cannot use < and > Possible outcomes:

• no solution: feasible region in empty
• unique optimal solution: vertex of feasible region
• bounded optimal solution, but not unique

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Linear programming: Find values of real variables x1, …, xn such that

• given linear function c1 x1 + c2 x2 + … + cn xn is maximized (or: minimized)
• and given linear constraints on the variables are satisfied

constraints: equalities or inequalities using ≥ or ≤, cannot use < and > Possible outcomes:

• no solution: feasible region in empty
• unique optimal solution: vertex of feasible region
• bounded optimal solution, but not unique
• unbounded optimal solution

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n-dimensional vectors m x n matrix c, A, b are input, x must be computed Linear programming: standard form Maximize c1 x1 + c2 x2 + … + cn xn Subject to a1,1 x1 + a1,2 x2 + … + a1,n xn ≤ b1

a2,1 x1 + a2,2 x2 + … + a2,n xn ≤ b2 .... am,1 x1 + am,2 x2 + … + am,n xn ≤ bm

x1 ≥ 0 x2 ≥ 0 … xn ≥ 0 Maximize c∙x subject to A x ≤ b and non-negativity constraints on all xi non-negativity constraints for each variable

• nly “≤”

(no “=“ and no “≥”) not: minimize

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Lemma: Any LP with n variables and m constraints can be rewritten as an equivalent LP in standard form with 2n variables and 2n+2m constraints.

• Proof. LP may not be in standard form because
• minimization instead of maximization

− negate objective function: minimize 2 x1 − x2 + 4x3  maximize −2 x1 + x2 − 4 xn

• some constraints are ≥ or = instead of ≤

− getting rid of =: replace 3 x + x2 − x3 = 5 by 3 x + x2 − x3 ≤ 5 3 x + x2 − x3 ≥ 5 − changing ≥ to ≤: negate constraint 3 x + x2 − x3 ≥ 5  − 3 x − x2 + x3 ≤ − 5

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Lemma: Any LP with n variables and m constraints can be rewritten as an equivalent LP in standard form with 2n variables and 2n+2m constraints. Proof (cont’d). LP may not be in standard form because

• minimization instead of maximization
• some constraints are ≥ or = instead of ≤
• variables without non-negativity constraint

− for each such variable xi introduce two new variables ui and vi − replace each occurrence of xi by (ui − vi) − add non-negativity constraints ui ≥ 0 and vi ≥ 0

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Lemma: Any LP with n variables and m constraints can be rewritten as an equivalent LP in standard form with 2n variables and 2n+2m constraints. Proof (cont’d).

• variables without non-negativity constraint

− for each such variable xi introduce two new variables ui and vi − replace each occurrence of xi by (ui − vi) − add non-negativity constraints ui ≥ 0 and vi ≥ 0 new problem is equivalent to original problem : for any original solution there is new solution with same value − if xi ≥ 0 then ui = xi and vi = 0, otherwise ui = 0 and vi = − xi and vice versa − set xi = (ui − vi)

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Lemma: Any LP with n variables and m constraints can be rewritten as an equivalent LP in standard form with 2n variables and 2n+2m constraints. Instead of standard form, we can also get so-called slack form:

– non-negativity constraint for each variable – all other constraints are =, not ≥ or ≤

• Standard form (or slack form): convenient for developing LP algorithms
• When modeling a problem: just use general form

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Algorithms for solving LP’s simplex method − worst-case running time is exponential − fast in practice interior-point methods − worst-case running time is polynomial in input size in bits − some are slow in practice, others are competitive with simplex method LP when dimension (=number of variables) is constant − can be solved in linear time (see course Advanced Algorithms)

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Modeling a problem as an LP

• decide what the variables are (what are the choices to be made?)
• write the objective function to be optimized (should be linear)
• write the constraints on the variables (should be linear)

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Example: Max Flow

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Flow: function f : V x V → R satisfying

• capacity constraint: 0 ≤ f (u,v) ≤ c(u,v ) for all nodes u,v
• flow conservation: for all nodes u ≠ s, t we have flow in = flow out:

∑v in V f (v,u) = ∑v in V f (u,v) value of flow: |f | = ∑v in V f (s,v) − ∑v in V f (v,s) source sink 10 2 5 3 2 5 1 3 2 s t 3 2 / 3 / 4 / 2 / 2 / 1 / 1 / 2 / 1 / 0 / flow = 1, capacity = 5

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Modeling Max Flow as an LP

• decide what the variables are (what are the choices to be made?)

for each edge (u,v) introduce variable xuv ( xuv represents f(u,v) )

• write the objective function to be optimized (should be linear)

maximize ∑v in V xsv − ∑v in V xvs (note: linear function)

• write the constraints on the variables (should be linear)

xuv ≥ 0 for all pairs of nodes u,v xuv ≤ c(u,v) for all pairs of nodes u,v ∑v in V xvu − ∑v in V xuv = 0 for all nodes u ≠ s, t (note: linear functions)

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Modeling Max Flow as an LP Now write it down nicely maximize ∑v in V xsv − ∑v in V xv,s subject to xuv ≥ 0 for all pairs of nodes u,v xuv ≤ c (u,v) for all pairs of nodes u,v ∑v in V xvu − ∑v in V xuv = 0 for all nodes u ≠ s, t Conclusion: Max Flow can trivially be written as an LP (but dedicated max-flow algorithm are faster than using LP algorithms)

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Example: Shortest Paths

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Shortest paths weighted, directed graph G = (V,E )

• weight (or: length) of a path

= sum of edge weights

• δ (u,v)

= distance from u to v = min weight of any path from u to v

• shortest path from u to v

= any path from u to v of weight δ (u,v) v2 v4 v7 v5 v6 v3 v1

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weight = 2 weighted, directed graph δ(v1,v5) = 2 Is δ (u,v) always well defined? No, not if there are negative-weight cycles.

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Modeling single-source single-target shortest path as an LP Problem: compute distance δ(s,t) from given source s to given target t

• decide what the variables are (what are the choices to be made?)

for each vertex v introduce variable xv ( xv represents δ(s,v) )

• write the objective function to be optimized (should be linear)

minimize xt

• write the constraints on the variables (should be linear)

xv ≤ xu + w(u,v) for all edges (u,v) in E xs = 0 maximize xt

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Modeling single-source single-target shortest path as an LP variables: for each vertex v we have a variable xv LP: maximize xt subject to xv ≤ xu + w(u,v) for all edges (u,v) in E xs = 0 Lemma: optimal solution to LP = δ(s,t).

• Proof. (assume for simplicity that δ(s,t) is bounded)

≥ : consider solution where we set xv = δ(s,v) for all v − solution is feasible and has value dist(s,t)  opt solution ≥ δ(s,t) ≤ : consider opt solution, and shortest path s = v0,v1,…,vk,vk+1 = t − prove by induction that xi ≤ δ(s,vi)  opt solution ≤ δ(s,t)

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Example: Vertex Cover

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G = (V,E) is undirected graph vertex cover in G: subset C V such that for each edge (u,v) in E we have u in C or v in C (or both) Vertex Cover (optimization version) Input: undirected graph G = (V,E) Problem: compute vertex cover for G with minimum number of vertices

• Vertex Cover is NP-hard.
• there is a 2-approximation algorithm running in linear time.

∩

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Modeling Vertex Cover as an LP

• decide what the variables are (what are the choices to be made?)

for vertex v introduce variable xv

( idea: xv = 1 if v in cover, xv = 0 if v not in cover)

• write the objective function to be optimized (should be linear)

minimize ∑v in V xv (note: linear function)

• write the constraints on the variables (should be linear)

− for each edge (u,v) write constraint xu + xv ≥ 1 (NB: linear function) − for each vertex v write constraint xu in {0,1} not a linear constraint

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integrality constraint: “xi must be integral” 0/1-constraint: “xi must 0 or 1” integer LP: LP where all variables have integrality constraint 0/1-LP: LP where all variables have 0/1-constraint (of course there are also mixed versions)

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Theorem: 0/1-LP is NP-hard.

• Proof. Consider decision problem: is there feasible solution to given 0/1-LP?

Which problem do we use in reduction? Need to transform 3-SAT formula into instance of 0/1-LP maximize y1 (not relevant for decision problem, pick arbitrary function) subject to y1 + y2 + (1−y3) ≥ 1 y2 + (1−y4) + (1−y5 ) ≥ 1 (1 − y2) + y3 + y5 ≥ 1 yi in {0,1} for all i

( x1 V x2 V ¬x3) Λ (x2 V ¬x4 V ¬x5) Λ (¬x2 V x3 V x5 )

already saw reduction from Vertex Cover; let’s do another one: 3-SAT variable yi for each Boolean xi yi = 1 if xi = TRUE and yi = 0 if xi = FALSE

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Theorem: 0/1-LP is NP-hard. problem can be modeled as “normal” LP  problem can be solved using LP algorithms  problem can be solved efficiently problem can be modeled as integer LP (or 0/1-LP)  problem can be solved using integer LP (or 0/1-LP) algorithms  does not mean that problem can be solved efficiently (sometimes can get approximation algorithms by relaxation and rounding see course Advanced Algorithms)  there are solvers (software) for integer LPs that in practice are quite efficient

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Summary

• what is an LP? what are integer LP and 0/1-LP?
• any LP can be written in standard form (or in slack form)
• normal (that is, not integer) LP can be solved in polynomial time

(with input size measured in bits)

• integer LP and 0/1-LP are NP-hard
• when modeling a problem as an LP

− define variables and how they relate to the problem − describe objective function (should be linear) − describe constraints (should be linear, < and > not allowed) − no need to use standard or slack form, just use general form