Constant-factor approximation algorithms for the minmax regret - - PowerPoint PPT Presentation

Constant-factor approximation algorithms for the minmax regret problem

Juan Pablo Fern´ andez G. Universidad de Medell´ ın Cra 87 No 30 - 65, Colombia e-mail: jpfernandez@udem.edu.co Adviser: Eduardo Conde Universidad de Sevilla, Espa˜ na. Doc-course Mayo 21, 2010

1

Introduction and existing results The minmax regret approach for parameter optimization problems. Results from general complexity Some approximated algorithms of constant factor

2

General Result of 2-approximation

3

Applications The sequencing problem n/1//F. Finding compromise solution in a linear multiple objective problem. Facility Location under uncertain demand.

Bibliography

1

Introduction and existing results The minmax regret approach for parameter optimization problems. Results from general complexity Some approximated algorithms of constant factor

2

General Result of 2-approximation

3

Applications The sequencing problem n/1//F. Finding compromise solution in a linear multiple objective problem. Facility Location under uncertain demand.

Bibliography

1

Introduction and existing results The minmax regret approach for parameter optimization problems. Results from general complexity Some approximated algorithms of constant factor

2

General Result of 2-approximation

3

Applications The sequencing problem n/1//F. Finding compromise solution in a linear multiple objective problem. Facility Location under uncertain demand.

Bibliography

Definition

Given an optimization problem with parameter cost function Opt (w) = min

x∈X F(x, w)

where the parameter w ∈ W an hyperrectangle in Rn and X ⊆ Rn is a compact feasible set. How can i choose x under unknown scenario w?

Definition

Definition (minmax regret criterion) The minimization of the maximum absolute regret problem can be expressed as min

x∈X Z(x)

where Z (x) = max

w∈W R (x, w) the worst-case regret and

R (x, w) = F (x, w) − min

y∈X F (y, w) the regret assigned to the feasible

solution x under scenario w.

Robust?

1 Let x⋆ be a minmax regret solution. 2 if w H was the scenario that take place after the decision x⋆ has been

implemented.

3 Let y H be the solution of Opt

• w H

then F(x⋆, w H) − F(y H, w H) ≤ ǫ where ǫ = Z(x⋆).

Minmax regret complexity I

In [4], it is described one of the classical combinatorial problem as Definition Elements of relative robust shortest path problem (RRSPP): G = (V , A), directed arc weighted graph. V , node set, |V | = n. A, arc set, |A| = m.

• lij, lij
• , (i, j) ∈ A. Lengths (weights) of the arcs are intervals which

express ranges of possible realizations of lengths. No probability distribution is assumed for arc lengths.

Minmax regret complexity I

Definition Elements of RRSPP: Length lw

ij ∈

• lij, lij
• is assigned for each (i, j) ∈ A, is called a

scenario w, where lw

ij denotes the length of arc (i, j) in scenario w.

P, denotes the set of all the paths in G from o to d. lw

p =

• (i,j)∈p

lw

ij denotes the length of a path p ∈ P in scenario w.

W , denote the set of possible scenarios.

Minmax regret complexity I

applying the minmax regret concept: Definition Element of RRSPP: dw

p = lw p − lw p⋆(w) the regret for the path p in scenario w, where

p⋆ (w) ∈ P is the shortest path in scenario w. Zp = max

w∈W dw p is the maximum regret.

min

p∈P Zp = min p∈P max w∈W lw p − lw p⋆(w)

(1) can equivalently define the problem RRSPP.

Minmax regret complexity I

For the problem (1), it is proved: Theorem

1 (1) is NP-hard. 2 Decision-(1) is NP-complete, even if G is restricted to a planar

acyclic graph with node degree three.

3 (1) is NP-hard, even if G is restricted to a planar acyclic graph with

node degree three.

Minmax regret complexity II

In [6], it is described another combinatorial problem as Definition Elements of minimizing the total flow time in a scheduling problem with interval data (MTFT) via minmax regret criterion: J, |J| = n, n ≥ 2, set of jobs that have to be processed on a single machine. The machine cannot process more than one job at any time.

• pk =
• pk, pk
• , Jk ∈ J. Then the processing times are intervals which

express ranges of possible processing times for the jobs. pw

k ∈

pk processing time of jobs Jk ∈ J is called a scenario.

Minmax regret complexity II

Definition Elements of MTFT via minmax regret criterion: W being the Cartesian product of all

• pk. The set of all scenarios.

π = (π (1) , . . . , π (n)), a schedule of job. Π, the set of all feasible schedules. The total flow time in π under w is F (π, w) =

n

• k=1

(n − k + 1) pw

π(k).

(2)

Minmax regret complexity II

applying the minmax regret concept: Definition Element of MTFT: R (π, w) = F (π, w) − F ⋆ (w) the regret assigned to the schedule π in scenario w, where F ⋆ (w) = min

y∈Π F (y, w) is the flow for the

shortest processing time schedule under the scenario w. Z (π) = max

w∈W R (π, w) is the maximum regret.

min

π∈Π Z (π)

(3) The minmax regret version of Problem MTFT.

Minmax regret complexity II

Definition (Problem ROB1) Problem ROB1 is the special case of problem (3) where all intervals of uncertainty have the same center, that is,

pk+pk 2

is the same for all Jk ∈ J. Definition Let Jl, Jk ∈ J be jobs. Job Jl is wider than job Jk if pk ⊂ pl.

Minmax regret complexity II

Definition For any job Jk ∈ J and schedule π ∈ Π, let q (π, Jk) = min {n − π (k) , π (k) − 1} A permutation π ∈ Π is called uniform if for any Jl, Jk ∈ J, if Jl is wider than Jk, then q (π, Jl) ≥ q (π, Jk).

Minmax regret complexity II

Theorem

1 if the number of jobs n is even, then any uniform permutation is an

• ptimal solution to problem ROB1 (and therefore problem ROB1

with even number of jobs is solvable in O (n log n) time).

2 Problem ROB1 with odd number of jobs is NP-hard. 3 Problem (3) is NP-hard; it remains NP-hard even if the number of

jobs is even.

Some approximated algorithms of constant factor

Definition (Elements of the problem.) E = {e1, e2, . . . , en} a finite set. Φ ⊆ 2E a feasible solutions set.

• ce = [ce, ce], e ∈ E a range of possible values of the cost.

w = (cw

e )e∈E a particular vector assignment of costs cw e to elements

e ∈ E is called scenario. W being the Cartesian product of all

• ck. The set of all scenarios.

Special combinatorial optimization

Definition (Problem formulation.) F (χ, w) =

e∈χ

cw

e . Its cost function for a given solution χ ∈ Φ,

under a fixed scenario w ∈ W . R (χ, w) = F (χ, w) − F ⋆ (w) the regret assigned to feasible solution χ in scenario w, where F ⋆ (w) = min

y∈Φ F (y, w) is the value of the

cost of the optimal solution under scenario w. Z (χ) = max

w∈W R (χ, w) is the maximum regret.

min

χ∈Φ Z (χ)

(4)

Special combinatorial optimization

Using the worst case characterization, we obtain bound for Z (χ) and then Theorem Let M be the solution of minx∈Φ F(x, w) where w =

• ce+ce

2

• e∈E. Then

for every χ ∈ Φ it holds Z (M) ≤ 2Z (χ). In particular, if χ⋆ is the solution of (4), then Z (M) ≤ 2Z (χ⋆) . M is known as the mid-point solution, and this w is the mid-point scenario.

Classical formulation of sequencing

We return to the problems of n jobs to be processed on a single machine, but now, we consider it with precedence constrains. Definition It is used the following notation. n jobs for being processing in only one

• machine. The subscripts i refers to job Ji. The subscripts k refers to

position which is processed a particular job. The following data pertain to job Ji.

1 pi the processing time of the job Ji. 2 xik =

• 1

if Ji is proccesed in the position th-k

• therwise

Classical formulation of sequencing

Definition

3 Ci is the time to finish the processing of the job Ji. 4 Ci(k) time of completion of the job Ji in the th-k process.

calculating The completion time of the job Ji is Ci(k−1) +

n

• i=1
• pixik. And

Classical formulation of sequencing: The above 2-approximation results can not be applied.

Integer programming min

n

• k=1

k

• j=1

n

• i=1

pixij subject to

n

• k=1

xik = 1 for i = 1, . . . , n. xqk −

k−1

• j=1

xpj ≤ 0 for p, q such that job Jp precedes job Jq. xik ∈ {0, 1} for i, k = 1, . . . , n.

Sequencing optimization problem

For the sake of simplicity, we will denote by iπ the position occupying by job Ji in the schedule π. So, the total flow time function becomes F (π, w) =

n

• i=1

(n − iπ + 1) pw

i

and

Sequencing optimization problem

Property For any two feasible schedules π, σ and scenario w ∈ W ,

1

F (π, w) − F (σ, w) =

n

• i=1

(iσ − iπ) pw

i . 2

Z (π) ≥

• {i:iσ>iπ}

(iσ − iπ) pi +

• {i:iσ<iπ}

(iσ − iπ) pi

3

Z (σ) ≤ Z (π) +

• {i:iπ>iσ}

(iπ − iσ) pi +

• {i:iπ<iσ}

(iπ − iσ) pi

Sequencing optimization problem

Theorem Let w be the mid-point scenario, and let σ be an optimal schedule under

• w. Then for every feasible schedule π it holds that Z (σ) ≤ 2Z (π)

Corollary If the deterministic 1 |prec| Ci problem, for some particular structure of the precedence constraints, is polynomially solvable, then the minmax regret version of the problem with interval processing times is approximable within 2.

2-approximation for general linear optimization.

• bservation 1

we will see that [1] extend the result that we have already presented of [3] and [2]. Furthermore, we have already seen the reason for they wrote two papers, but this cases will not be necessary distinguish between them.

• bservation 2

Now, consider X as a general compact set of feasible solution in Rn, and a linear cost function w, x, where w ∈ W is a given cost scenario and W is a hyperrectangle of Rn.

2-approximation for general linear optimization.

Regret version The problem is mathematically described as Z ⋆ = min

x∈X Z (x)

(5) where Z (x) = max

w∈W R (x, w), and R (x, w) = w, x − min y∈X w, y.

2-approximation for general linear optimization.

element of the problem Let w j, j = 1, . . . , 2n be the extreme points of the set of possible corner of the hyperrectangle W . The function R (x, w) is a convex function on w. δ⋆ (χ|W ) = max {0, w, χ : j = 1, . . . , 2n} (6) Property Using (6), it is verified that Z (x) = max

y∈X δ⋆ (x − y|W ).

2-approximation for general linear optimization.

Property Given x, y ∈ X one has

1

Z (x) ≥

• w +

i max{xi − yi, 0} −

• w −

i max{yi − xi, 0} 2

Z (x) ≤ Z (y) + δ⋆ (x − y|W )

2-approximation for general linear optimization.

Theorem Let x be an optimal solution under the mid-point cost scenario, that is, an optimum of the problem min

x∈X w, x

(7) where w i = 1

2

• w −

i

+ w +

i

• for each i = 1, . . . , n then

Z (x) ≤ 2Z (x∗) where x∗ is any optimal solution of the problem (5).

The sequencing problem n/1//F.

sequencing without constrain Recalling that function (2), constrain to the minmax regret problem is NP-hard. We obtain for n/1//F problem, the total flow time has a 2-approximation for the minmax regret version by sequencing such that

• pπ(1) ≤

pπ(2) ≤ . . . ≤ pπ(n) where pπ(i) =

pπ(i)+pπ(i) 2

is the mid-point scenario.

The minmax regret problem for weight linear cost function.

Decision problem with linear cost functions. Suppose that in an optimization problem (for example, multiobjective

• ptimization problem), the decision maker has to decide what value shall

be given to the weight ti, i = 1, . . . , m for m linear cost functions. Instead of this, he shall be interesting in a range for each weight, as,

• ti = [ti, ti], i = 1, . . . , m. We must solve the optimization problem

min

x∈X m

• i=1

ti ai, x (8) where ti ∈ ti, X ⊂ Rn and m ≤ n.

The minmax regret problem for weight linear cost function.

Transformation. Taking Am×n the matrix which rows are the vector ai. Applying the transform y = Ax, we obtain min

y∈Y m

• i=1

tiyi where Y = AX. For this version, we can apply the 2-approximation theorem for the regret version, obtaining a robust approximation Y ⋆. After that, one has an affine space of 2-approximated solutions of the

• riginal problem given by the system AX = Y ⋆.

Facility Location under uncertain demand.

Localization of a facility.

Definition Elements of the problem: Population ai, i = 1, . . . , n. x localization of the facility. X the compact feasible set of possible position of the service. d

• x, ai

a given metric. wi, weights (population, demand...) are intervals which express ranges of possible values, denote by w.

Localization of a facility.

Definition (Problem formulation.) F (x, w) =

n

• i=1

wid

• x, ai

. Its cost function for a given solution x ∈ X, under a fixed scenario w ∈ W . R (x, w) = F (x, w) − F ⋆ (w) the regret assigned to feasible solution x in scenario w, where F ⋆ (w) = min

y∈X F (y, w) is the value of the

cost of the optimal solution under scenario w. Z (x) = max

w∈W R (x, w) is the maximum regret.

min

x∈X Z (X)

(9)

Localization of a facility.

What is the problem? No linear parameter function on x is appearing on it. transformation Taking ui = d

• x, ai

for all i = 1, . . . , n, taking u = (u1, . . . , un) and U =

• u ∈ Rn : ∃x ∈ X and ui = d
• x, ai

for all i = 1, . . . , n

• the

parameter function become F (u, w) =

n

• i=1

wiui, and (9) become min

u∈U Z (u)

where max

w∈W n

• i=1

wiui − min

y∈U n

• i=1

wiyi.

Conde, E., A 2-approximation for minmax regret problems via a mid-point scenario optimal solution, Operations Research Letters, preprint. Kasperski, A., Zieli´ nski, P., A 2approximation algorithm for interval data minmax regret sequencing problems with the total flow time criterion, Operations Research Letters, 36 pag. 343-344, 2008, doi: 10.1016/j.orl.2007.11.04. Kasperski, A., Zieli´ nski, P., An approximation algorithm for interval data minmax regret combinatorial optimization problems, Information Processing Letters, 97, pag. 177-180, 2006, doi:10.1016/j.ipl.2005.11.001.

Beginning

Zieli´ nski, P., The coputational complexity of the relative robust shortest path problem with interval data, European journal of operational research, 158, pag. 570-576, 2004, doi: 10.1016/S0377-2217(03)00373-4. French, S., Sequencing and scheduling An introduction to the mathematics of the Job-Shop, Ellis Horwoord Series, Jhon Wiley and Son, pag. 37-39, 51-53, 1982. Lebedev, V., Averbakh, I., Complexity of minimizing the total flow time with interval data and minmax regret criterion, Discrete applied mathematics, 154, pag 2167 - 2177, 2006, doi: 10.1016/j.dam.2005.04.015.

Bienvenidos a Medell´ ın.

acknowledgment

Funded by IMUS,University of Sevilla and University of Medell´ ın. Special acknowledge for my adviser who made this possible.

4

Supplement of Doc-course Speech.

5

Worst-case scenario for uncertain processing time.

Minmax regret complexity II

For the problem (3), has been proved: Property if the same constant is added to all numbers pk, pk, Jk ∈ J, value Z (π) does not change for any π ∈ Π. Remark The previous Property allows us do not assume number pk, pk, Jk ∈ J to be nonnegative, even, this does not have any practical sense because processing times of job are always nonnegative.

Minmax regret complexity II

Definition A scenario w = {pw

k , Jk ∈ J} such that pw k ∈

• pk, pk
• for all k is

called an extreme scenario. A worst-case scenario for x which is also an extreme scenario will be called a worst-case extreme scenario for x. the jobs Jl, Jk which hold pl ≤ pk and pl ≤ pk, we say that job Jl dominates job Jk.

Minmax regret complexity II

Corollary

1 For any permutation π ∈ Π, there always exist a worst-case extreme

scenario.

2 For any π ∈ Π, value Z (π) can be obtained in polynomial time (by

matching techniques).

3 Suppose that job Jl dominates job Jk, π ∈ Π is an optimal

permutation for problem (3), and Jk precedes Jl in π. Then switching the positions of jobs Jl and Jk will result in another

• ptimal permutation for problem (3).