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On the performance of Smiths rule in single-machine scheduling with nonlinear cost Wiebke H ohn Technische Universit at Berlin Tobias Jacobs NEC Laboratories Europe 18th Combinatorial Optimization Workshop Aussois 2014 Generalized

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SLIDE 1

On the performance of Smith’s rule in single-machine scheduling with nonlinear cost

Wiebke H¨

  • hn

Technische Universit¨ at Berlin

Tobias Jacobs

NEC Laboratories Europe

18th Combinatorial Optimization Workshop Aussois 2014

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SLIDE 2

Generalized min-sum scheduling

time

Cj

wj pj

Given: jobs j = 1, . . . , n with weight wj > 0 processing time pj > 0

  • W. H¨
  • hn and T. Jacobs
slide-3
SLIDE 3

Generalized min-sum scheduling

time

Cj

wj pj wj pj

Given: jobs j = 1, . . . , n with weight wj > 0 processing time pj > 0 Task: compute sequence with minimum cost

j

wjf (Cj) Cj completion time of job j non-decreasing, non-negative cost function f

  • W. H¨
  • hn and T. Jacobs
slide-4
SLIDE 4

Motivation

priorities and fairness

Lk-norms/monomials compromise on worst and average case

  • W. H¨
  • hn and T. Jacobs
slide-5
SLIDE 5

Motivation

priorities and fairness

Lk-norms/monomials compromise on worst and average case linear cost

j

wjC (s)

j

but non-uniform speed s

  • W. H¨
  • hn and T. Jacobs
slide-6
SLIDE 6

Motivation

priorities and fairness

Lk-norms/monomials compromise on worst and average case linear cost

j

wjC (s)

j

but non-uniform speed s

speed time 1

  • W. H¨
  • hn and T. Jacobs
slide-7
SLIDE 7

Motivation

priorities and fairness

Lk-norms/monomials compromise on worst and average case linear cost

j

wjC (s)

j

but non-uniform speed s

speed time 1

  • W. H¨
  • hn and T. Jacobs
slide-8
SLIDE 8

Motivation

priorities and fairness

Lk-norms/monomials compromise on worst and average case linear cost

j

wjC (s)

j

but non-uniform speed s

speed time 1

  • W. H¨
  • hn and T. Jacobs
slide-9
SLIDE 9

Motivation

priorities and fairness

Lk-norms/monomials compromise on worst and average case linear cost

j

wjC (s)

j

but non-uniform speed s

speed time 1

  • W. H¨
  • hn and T. Jacobs
slide-10
SLIDE 10

Motivation

priorities and fairness

Lk-norms/monomials compromise on worst and average case linear cost

j

wjC (s)

j

but non-uniform speed s

speed time 1

  • W. H¨
  • hn and T. Jacobs
slide-11
SLIDE 11

Motivation

priorities and fairness

Lk-norms/monomials compromise on worst and average case linear cost

j

wjC (s)

j

but non-uniform speed s

speed time 1

  • W. H¨
  • hn and T. Jacobs
slide-12
SLIDE 12

Motivation

priorities and fairness

Lk-norms/monomials compromise on worst and average case linear cost

j

wjC (s)

j

but non-uniform speed s

speed time 1

C (s)

j

  • W. H¨
  • hn and T. Jacobs
slide-13
SLIDE 13

Motivation

priorities and fairness

Lk-norms/monomials compromise on worst and average case linear cost

j

wjC (s)

j

but non-uniform speed s

speed time 1

C (s)

j

C(s)

j

s(t) dt =

  • i≤j

pj

  • W. H¨
  • hn and T. Jacobs
slide-14
SLIDE 14

Motivation

priorities and fairness

Lk-norms/monomials compromise on worst and average case linear cost

j

wjC (s)

j

but non-uniform speed s

speed time 1

C (s)

j

C(s)

j

s(t) dt =

  • i≤j

pj = C (1)

j

  • W. H¨
  • hn and T. Jacobs
slide-15
SLIDE 15

Motivation

priorities and fairness

Lk-norms/monomials compromise on worst and average case linear cost

j

wjC (s)

j

but non-uniform speed s

speed time 1

C (s)

j

S

  • C (s)

j

  • :=

C(s)

j

s(t) dt =

  • i≤j

pj = C (1)

j

  • W. H¨
  • hn and T. Jacobs
slide-16
SLIDE 16

Motivation

priorities and fairness

Lk-norms/monomials compromise on worst and average case linear cost

j

wjC (s)

j

but non-uniform speed s

speed time 1

C (s)

j

S

  • C (s)

j

  • :=

C(s)

j

s(t) dt =

  • i≤j

pj = C (1)

j

⇔ C (s)

j

= S−1 C (1)

j

  • W. H¨
  • hn and T. Jacobs
slide-17
SLIDE 17

Motivation

priorities and fairness

Lk-norms/monomials compromise on worst and average case linear cost

j

wjC (s)

j

but non-uniform speed s

speed time 1

C (s)

j

S

  • C (s)

j

  • :=

C(s)

j

s(t) dt =

  • i≤j

pj = C (1)

j

⇔ C (s)

j

= S−1 C (1)

j

  • increasing speed s ↔ concave cost f

decreasing speed s ↔ convex cost f

  • W. H¨
  • hn and T. Jacobs
slide-18
SLIDE 18

Motivation

priorities and fairness

Lk-norms/monomials compromise on worst and average case linear cost

j

wjC (s)

j

but non-uniform speed s

speed time 1

C (s)

j

S

  • C (s)

j

  • :=

C(s)

j

s(t) dt =

  • i≤j

pj = C (1)

j

⇔ C (s)

j

= S−1 C (1)

j

  • increasing speed s ↔ concave cost f

decreasing speed s ↔ convex cost f Our main focus: convex / concave cost functions

  • W. H¨
  • hn and T. Jacobs
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SLIDE 19

Outline

1 Analysis of Smith’s rule for convex (and concave) cost 2 Exact algorithms for monomials

  • W. H¨
  • hn and T. Jacobs
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SLIDE 20

Related work & complexity status

linear in P

[Smith 1956]

  • W. H¨
  • hn and T. Jacobs
slide-21
SLIDE 21

Related work & complexity status

linear in P

[Smith 1956]

exponential in P

[Rothkopf 1966]

  • W. H¨
  • hn and T. Jacobs
slide-22
SLIDE 22

Related work & complexity status

linear in P

[Smith 1956]

exponential in P

[Rothkopf 1966]

general PTAS

[Megow, Verschae 2012]

strongly NP-hard [H., Jacobs 2012]

  • W. H¨
  • hn and T. Jacobs
slide-23
SLIDE 23

Related work & complexity status

linear in P

[Smith 1956]

exponential in P

[Rothkopf 1966]

general piece-wise linear PTAS

[Megow, Verschae 2012]

strongly NP-hard [H., Jacobs 2012]

  • W. H¨
  • hn and T. Jacobs
slide-24
SLIDE 24

Related work & complexity status

linear in P

[Smith 1956]

exponential in P

[Rothkopf 1966]

general piece-wise linear convex weakly NP-hard

[Yuan ’92]

PTAS

[Megow, Verschae 2012]

strongly NP-hard [H., Jacobs 2012]

  • W. H¨
  • hn and T. Jacobs
slide-25
SLIDE 25

Related work & complexity status

linear in P

[Smith 1956]

exponential in P

[Rothkopf 1966]

general piece-wise linear convex weakly NP-hard

[Yuan ’92]

strongly NP-hard ? FPTAS ? weakly NP-hard

[Yuan ’92]

strongly NP-hard ? PTAS

[Megow, Verschae 2012]

strongly NP-hard [H., Jacobs 2012]

  • W. H¨
  • hn and T. Jacobs
slide-26
SLIDE 26

Related work & complexity status

linear in P

[Smith 1956]

exponential in P

[Rothkopf 1966]

general piece-wise linear convex weakly NP-hard

[Yuan ’92]

strongly NP-hard ? FPTAS ? weakly NP-hard

[Yuan ’92]

strongly NP-hard ? concave in P / FPTAS ? (strongly) NP-hard ? PTAS

[Megow, Verschae 2012]

strongly NP-hard [H., Jacobs 2012]

  • W. H¨
  • hn and T. Jacobs
slide-27
SLIDE 27

Related work & complexity status

linear in P

[Smith 1956]

exponential in P

[Rothkopf 1966]

general piece-wise linear convex weakly NP-hard

[Yuan ’92]

strongly NP-hard ? FPTAS ? weakly NP-hard

[Yuan ’92]

strongly NP-hard ? concave in P / FPTAS ? (strongly) NP-hard ? monomials tk in P / FPTAS ? (strongly) NP-hard ? PTAS

[Megow, Verschae 2012]

strongly NP-hard [H., Jacobs 2012]

  • W. H¨
  • hn and T. Jacobs
slide-28
SLIDE 28

Related work & complexity status

linear in P

[Smith 1956]

exponential in P

[Rothkopf 1966]

general piece-wise linear convex weakly NP-hard

[Yuan ’92]

strongly NP-hard ? FPTAS ? weakly NP-hard

[Yuan ’92]

strongly NP-hard ? concave in P / FPTAS ? (strongly) NP-hard ? monomials tk in P / FPTAS ? (strongly) NP-hard ? piece-wise linear,

  • const. # pieces

FPTAS

[Megow, Verschae ’12]

weakly NP-hard

[Yuan ’92]

PTAS

[Megow, Verschae 2012]

strongly NP-hard [H., Jacobs 2012]

  • W. H¨
  • hn and T. Jacobs
slide-29
SLIDE 29

Analysis of Smith’s rule

Smith’s rule

Schedule jobs in non-increasing order of their density wj

pj .

  • W. H¨
  • hn and T. Jacobs
slide-30
SLIDE 30

Analysis of Smith’s rule

Smith’s rule

Schedule jobs in non-increasing order of their density wj

pj .

How good is this simple algorithm for a fixed convex/concave cost function?

  • W. H¨
  • hn and T. Jacobs
slide-31
SLIDE 31

Analysis of Smith’s rule

Smith’s rule

Schedule jobs in non-increasing order of their density wj

pj .

How good is this simple algorithm for a fixed convex/concave cost function? Smith’s rule is a

√ 3+1 2

  • approximation for

[Stiller & Wiese ’10]

any concave cost function f

  • W. H¨
  • hn and T. Jacobs
slide-32
SLIDE 32

Analysis of Smith’s rule

Smith’s rule

Schedule jobs in non-increasing order of their density wj

pj .

How good is this simple algorithm for a fixed convex/concave cost function? Smith’s rule is a

√ 3+1 2

  • approximation for

[Stiller & Wiese ’10]

any concave cost function f

Theorem

The tight approximation ratio of Smith’s rule for fixed convex f is sup

0<q,p q

0 f (t)dt+p·f (q+p)

p·f (p)+ p+q

p

f (t)dt .

  • W. H¨
  • hn and T. Jacobs
slide-33
SLIDE 33

Analysis of Smith’s rule

Smith’s rule

Schedule jobs in non-increasing order of their density wj

pj .

How good is this simple algorithm for a fixed convex/concave cost function? Smith’s rule is a

√ 3+1 2

  • approximation for

[Stiller & Wiese ’10]

any concave cost function f

Theorem

The tight approximation ratio of Smith’s rule for fixed convex f is sup

0<q,p q

0 f (t)dt+p·f (q+p)

p·f (p)+ p+q

p

f (t)dt .

holds with inverse ratio for concave cost function

  • W. H¨
  • hn and T. Jacobs
slide-34
SLIDE 34

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

  • W. H¨
  • hn and T. Jacobs
slide-35
SLIDE 35

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that:

  • W. H¨
  • hn and T. Jacobs
slide-36
SLIDE 36

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that:

  • 1. wj = pj for all jobs j
  • W. H¨
  • hn and T. Jacobs
slide-37
SLIDE 37

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that: merge density classes inductively

  • 1. wj = pj for all jobs j
  • W. H¨
  • hn and T. Jacobs
slide-38
SLIDE 38

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that: merge density classes inductively

  • 1. wj = pj for all jobs j

OPT ALG

  • 2. Smith’s Rule chooses

non-decreasing order, OPT the opposite

  • W. H¨
  • hn and T. Jacobs
slide-39
SLIDE 39

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that: merge density classes inductively

  • 1. wj = pj for all jobs j

exchange argument OPT ALG

  • 2. Smith’s Rule chooses

non-decreasing order, OPT the opposite

  • W. H¨
  • hn and T. Jacobs
slide-40
SLIDE 40

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that: merge density classes inductively

  • 1. wj = pj for all jobs j

exchange argument OPT ALG

  • 2. Smith’s Rule chooses

non-decreasing order, OPT the opposite OPT ALG

  • 3. one big job & several

very small jobs

  • W. H¨
  • hn and T. Jacobs
slide-41
SLIDE 41

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that:

  • 1. wj = pj for all jobs j

OPT ALG

  • 2. Smith’s Rule chooses

non-decreasing order, OPT the opposite OPT ALG

  • 3. one big job & several

very small jobs

  • W. H¨
  • hn and T. Jacobs
slide-42
SLIDE 42

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that:

  • 1. wj = pj for all jobs j

OPT ALG

  • 2. Smith’s Rule chooses

non-decreasing order, OPT the opposite OPT ALG

  • 3. one big job & several

very small jobs

  • W. H¨
  • hn and T. Jacobs
slide-43
SLIDE 43

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that:

  • 1. wj = pj for all jobs j

OPT ALG

  • 2. Smith’s Rule chooses

non-decreasing order, OPT the opposite OPT ALG

  • 3. one big job & several

very small jobs

  • W. H¨
  • hn and T. Jacobs
slide-44
SLIDE 44

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that:

  • 1. wj = pj for all jobs j

OPT ALG

  • 2. Smith’s Rule chooses

non-decreasing order, OPT the opposite OPT ALG

  • 3. one big job & several

very small jobs

  • W. H¨
  • hn and T. Jacobs
slide-45
SLIDE 45

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that:

  • 1. wj = pj for all jobs j

OPT ALG

  • 2. Smith’s Rule chooses

non-decreasing order, OPT the opposite OPT ALG

  • 3. one big job & several

very small jobs

  • W. H¨
  • hn and T. Jacobs
slide-46
SLIDE 46

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that:

  • 1. wj = pj for all jobs j

OPT ALG

  • 2. Smith’s Rule chooses

non-decreasing order, OPT the opposite OPT ALG

  • 3. one big job & several

very small jobs

f cost

  • W. H¨
  • hn and T. Jacobs
slide-47
SLIDE 47

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that:

  • 1. wj = pj for all jobs j

OPT ALG

  • 2. Smith’s Rule chooses

non-decreasing order, OPT the opposite

f (Cj ) wj f (Cj )

pj = wj

Cj

OPT ALG

  • 3. one big job & several

very small jobs

f cost

  • W. H¨
  • hn and T. Jacobs
slide-48
SLIDE 48

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that:

  • 1. wj = pj for all jobs j

OPT ALG

  • 2. Smith’s Rule chooses

non-decreasing order, OPT the opposite

f (Cj ) wj f (Cj )

pj = wj

Cj

OPT ALG

  • 3. one big job & several

very small jobs

f cost

  • W. H¨
  • hn and T. Jacobs
slide-49
SLIDE 49

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that:

  • 1. wj = pj for all jobs j

OPT ALG

  • 2. Smith’s Rule chooses

non-decreasing order, OPT the opposite OPT ALG

  • 3. one big job & several

very small jobs

f cost

  • W. H¨
  • hn and T. Jacobs
slide-50
SLIDE 50

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that:

  • 1. wj = pj for all jobs j

OPT ALG

  • 2. Smith’s Rule chooses

non-decreasing order, OPT the opposite OPT ALG

  • 3. one big job & several

very small jobs

f cost

  • W. H¨
  • hn and T. Jacobs
slide-51
SLIDE 51

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that:

  • 1. wj = pj for all jobs j

OPT ALG

  • 2. Smith’s Rule chooses

non-decreasing order, OPT the opposite OPT ALG

  • 3. one big job & several

very small jobs

f cost

  • W. H¨
  • hn and T. Jacobs
slide-52
SLIDE 52

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that:

  • 1. wj = pj for all jobs j

OPT ALG

  • 2. Smith’s Rule chooses

non-decreasing order, OPT the opposite

ALG OPT ≤

OPT ALG

  • 3. one big job & several

very small jobs

f cost

  • W. H¨
  • hn and T. Jacobs
slide-53
SLIDE 53

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that:

  • 1. wj = pj for all jobs j

OPT ALG

  • 2. Smith’s Rule chooses

non-decreasing order, OPT the opposite

ALG OPT ≤

OPT ALG

  • 3. one big job & several

very small jobs

f cost

  • W. H¨
  • hn and T. Jacobs
slide-54
SLIDE 54

Analysis of Smith’s rule

Narrow space of worst-case instances for convex cost:

pj wj

We can assume w.l.o.g. that:

  • 1. wj = pj for all jobs j

OPT ALG

  • 2. Smith’s Rule chooses

non-decreasing order, OPT the opposite

ALG OPT ≤

OPT ALG

  • 3. one big job & several

very small jobs

f cost

  • W. H¨
  • hn and T. Jacobs
slide-55
SLIDE 55

Analysis of Smith’s rule

Theorem

The tight approximation ratio of Smith’s rule for fixed convex f is sup

0<q,p q

0 f (t)dt+p·f (q+p)

p·f (p)+ p+q

p

f (t)dt .

  • W. H¨
  • hn and T. Jacobs
slide-56
SLIDE 56

Analysis of Smith’s rule

Theorem

The tight approximation ratio of Smith’s rule for fixed convex f is sup

0<q,p q

0 f (t)dt+p·f (q+p)

p·f (p)+ p+q

p

f (t)dt .

Corollary

If f is a polynomial of degree k with non-negative coefficients then the tight approximation ratio is αk := max

0.5≤p<1 (1−p)k+1 + (k+1)p kpk+1 + 1

.

  • W. H¨
  • hn and T. Jacobs
slide-57
SLIDE 57

Tight approximation ratios for polynomials

1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 0.25 0.5 0.75 1 1.25 1.5 degree of monomial degree approx. 20 40 60 80 100 20 40 60 80 100 degree of monomial degree approx.

cost function ratio square root 1.07 degree 2 polynomials 1.31 degree 3 polynomials 1.76 degree 4 polynomials 2.31 degree 5 polynomials 2.93 degree 6 polynomials 3.60 degree 10 polynomials 6.58 degree 20 polynomials 15.04 exponential ∞

  • W. H¨
  • hn and T. Jacobs
slide-58
SLIDE 58

Tight approximation ratios for polynomials

1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 0.25 0.5 0.75 1 1.25 1.5 degree of monomial degree approx. 20 40 60 80 100 20 40 60 80 100 degree of monomial degree approx. 0.5 0.55 0.6 0.65 0.7 0.75 0.8 1 2 3 4 5 6 degree of monomial approx./degree p 0.5 0.6 0.7 0.8 0.9 1 20 40 60 80 100 degree of monomial approx./degree p

Observation:

αk k ≈ pk

length of big job

corresponding to αk

  • W. H¨
  • hn and T. Jacobs
slide-59
SLIDE 59

Bounding the approximation ratio

Theorem

For cost function f (t) = tk, the tight approximation factor αk of Smith’s ruler observes the following for k ≥ 4: lim

k→∞

  • pk −

k+1

  • 1

k2

  • = 0,
  • W. H¨
  • hn and T. Jacobs
slide-60
SLIDE 60

Bounding the approximation ratio

Theorem

For cost function f (t) = tk, the tight approximation factor αk of Smith’s ruler observes the following for k ≥ 4: lim

k→∞

  • pk −

k+1

  • 1

k2

  • = 0,

lim

k→∞

  • αk − k

k−1 k+1

  • = 0,
  • W. H¨
  • hn and T. Jacobs
slide-61
SLIDE 61

Bounding the approximation ratio

Theorem

For cost function f (t) = tk, the tight approximation factor αk of Smith’s ruler observes the following for k ≥ 4: lim

k→∞

  • pk −

k+1

  • 1

k2

  • = 0,

lim

k→∞

  • αk − k

k−1 k+1

  • = 0,

k − αk ≥ ln k − 1

2k .

  • W. H¨
  • hn and T. Jacobs
slide-62
SLIDE 62

Related computational results

Approximation ratio in experiments:

  • W. H¨
  • hn and T. Jacobs
slide-63
SLIDE 63

Related computational results

Approximation ratio in experiments:

0.1 0.3 0.5 0.7 0.9 1 1.02

t2

tight ratio 1.31

0.1 0.3 0.5 0.7 0.9 1 1.02

t3

tight ratio 1.76

0.1 0.3 0.5 0.7 0.9 1 1.02

t4

tight ratio 2.31

0.1 0.3 0.5 0.7 0.9 1 1.02

t5

tight ratio 2.93 x-value correlation of wj and pj

  • W. H¨
  • hn and T. Jacobs
slide-64
SLIDE 64

Related computational results

Approximation ratio in experiments:

0.1 0.3 0.5 0.7 0.9 1 1.02

t2

tight ratio 1.31

0.1 0.3 0.5 0.7 0.9 1 1.02

t3

tight ratio 1.76

0.1 0.3 0.5 0.7 0.9 1 1.02

t4

tight ratio 2.31

0.1 0.3 0.5 0.7 0.9 1 1.02

t5

tight ratio 2.93 x-value correlation of wj and pj

experimental performance much better than worst-case

  • W. H¨
  • hn and T. Jacobs
slide-65
SLIDE 65

Related computational results

Approximation ratio in experiments:

0.1 0.3 0.5 0.7 0.9 1 1.02

t2

tight ratio 1.31

0.1 0.3 0.5 0.7 0.9 1 1.02

t3

tight ratio 1.76

0.1 0.3 0.5 0.7 0.9 1 1.02

t4

tight ratio 2.31

0.1 0.3 0.5 0.7 0.9 1 1.02

t5

tight ratio 2.93 x-value correlation of wj and pj

experimental performance much better than worst-case more realistic analysis for processing times 1, 2, . . . , pmax and given pj

  • W. H¨
  • hn and T. Jacobs
slide-66
SLIDE 66

Parametrized analysis of Smith’s rule

Theorem

The tight approximation ratio of Smith’s rule for convex f and fixed parameters pmax and

j pj is

sup

  • INC(p, pmax,

j pj)

DEC(p, pmax,

j pj)

  • p = 0, 1, 2, . . . ,
  • j

pj

  • .

1 2 3 4 5 6 100 200 300 400 500 approximation factor P t2 t5 t10

  • W. H¨
  • hn and T. Jacobs
slide-67
SLIDE 67

Parametrized analysis of Smith’s rule

Theorem

The tight approximation ratio of Smith’s rule for convex f and fixed parameters pmax and

j pj is

sup

  • INC(p, pmax,

j pj)

DEC(p, pmax,

j pj)

  • p = 0, 1, 2, . . . ,
  • j

pj

  • .

proof follows same idea as unparametrized analysis

1 2 3 4 5 6 100 200 300 400 500 approximation factor P t2 t5 t10

  • W. H¨
  • hn and T. Jacobs
slide-68
SLIDE 68

Parametrized analysis of Smith’s rule

Theorem

The tight approximation ratio of Smith’s rule for convex f and fixed parameters pmax and

j pj is

sup

  • INC(p, pmax,

j pj)

DEC(p, pmax,

j pj)

  • p = 0, 1, 2, . . . ,
  • j

pj

  • .

proof follows same idea as unparametrized analysis valuable lower bound for exact computations

1 2 3 4 5 6 100 200 300 400 500 approximation factor P t2 t5 t10

pmax = 50

  • W. H¨
  • hn and T. Jacobs
slide-69
SLIDE 69

Exact algorithms for quadratic cost

Approach proposed for quadratic cost: best first graph search based on A∗

[Sen et al. ’96, Kaindl et al. ’01]

  • W. H¨
  • hn and T. Jacobs
slide-70
SLIDE 70

Exact algorithms for quadratic cost

Approach proposed for quadratic cost: best first graph search based on A∗

[Sen et al. ’96, Kaindl et al. ’01]

local and global comparability

[Schild, Fredman ’62, Sen et al. ’90]

  • W. H¨
  • hn and T. Jacobs
slide-71
SLIDE 71

Exact algorithms for quadratic cost

Approach proposed for quadratic cost: best first graph search based on A∗

[Sen et al. ’96, Kaindl et al. ’01]

local and global comparability

[Schild, Fredman ’62, Sen et al. ’90]

global comparability: not matter where scheduled

  • W. H¨
  • hn and T. Jacobs
slide-72
SLIDE 72

Exact algorithms for quadratic cost

Approach proposed for quadratic cost: best first graph search based on A∗

[Sen et al. ’96, Kaindl et al. ’01]

local and global comparability

[Schild, Fredman ’62, Sen et al. ’90]

global comparability: not matter where scheduled local comparability:

  • nly if scheduled consecutively
  • W. H¨
  • hn and T. Jacobs
slide-73
SLIDE 73

Exact algorithms for quadratic cost

Approach proposed for quadratic cost: best first graph search based on A∗

[Sen et al. ’96, Kaindl et al. ’01]

local and global comparability

[Schild, Fredman ’62, Sen et al. ’90]

global comparability: not matter where scheduled local comparability:

  • nly if scheduled consecutively

k( k(

weight processing time

pj wj

i ≺g j i ≺ℓ j j ≺g i j ≺ℓ i

  • W. H¨
  • hn and T. Jacobs
slide-74
SLIDE 74

Exact algorithms for quadratic cost

Approach proposed for quadratic cost: best first graph search based on A∗

[Sen et al. ’96, Kaindl et al. ’01]

local and global comparability

[Schild, Fredman ’62, Sen et al. ’90]

global comparability: not matter where scheduled local comparability:

  • nly if scheduled consecutively

k( k(Conjecture Mondal, Sen (2000)

weight processing time

pj wj

i ≺g j i ≺ℓ j j ≺g i j ≺ℓ i

  • W. H¨
  • hn and T. Jacobs
slide-75
SLIDE 75

Exact algorithms for quadratic cost

Approach proposed for quadratic cost: best first graph search based on A∗

[Sen et al. ’96, Kaindl et al. ’01]

local and global comparability

[Schild, Fredman ’62, Sen et al. ’90]

global comparability: not matter where scheduled local comparability:

  • nly if scheduled consecutively

k( k(H., Jacobs

weight processing time

pj wj

i ≺g j j ≺g i j ≺ℓ i

  • W. H¨
  • hn and T. Jacobs
slide-76
SLIDE 76

Exact algorithms for quadratic cost

Approach proposed for quadratic cost: best first graph search based on A∗

[Sen et al. ’96, Kaindl et al. ’01]

local and global comparability

[Schild, Fredman ’62, Sen et al. ’90]

global comparability: not matter where scheduled local comparability:

  • nly if scheduled consecutively

k( k(D¨ urr, Vasquez

weight processing time

pj wj

i ≺g j j ≺g i

  • W. H¨
  • hn and T. Jacobs
slide-77
SLIDE 77

Exact algorithms for quadratic cost

Approach proposed for quadratic cost: best first graph search based on A∗

[Sen et al. ’96, Kaindl et al. ’01]

local and global comparability

[Schild, Fredman ’62, Sen et al. ’90]

global comparability: not matter where scheduled local comparability:

  • nly if scheduled consecutively

k( k(D¨ urr, Vasquez

weight processing time

pj wj

i ≺g j j ≺g i

Approaches tested by us: different graph searches with integrated comparabilities

  • W. H¨
  • hn and T. Jacobs
slide-78
SLIDE 78

Exact algorithms for quadratic cost

Approach proposed for quadratic cost: best first graph search based on A∗

[Sen et al. ’96, Kaindl et al. ’01]

local and global comparability

[Schild, Fredman ’62, Sen et al. ’90]

global comparability: not matter where scheduled local comparability:

  • nly if scheduled consecutively

k( k(D¨ urr, Vasquez

weight processing time

pj wj

i ≺g j j ≺g i

Approaches tested by us: different graph searches with integrated comparabilities quadratic IP with integrated comparabilities (Cplex 12.4)

  • W. H¨
  • hn and T. Jacobs
slide-79
SLIDE 79

Exact algorithms for quadratic cost

Approach proposed for quadratic cost: best first graph search based on A∗

[Sen et al. ’96, Kaindl et al. ’01]

local and global comparability

[Schild, Fredman ’62, Sen et al. ’90]

global comparability: not matter where scheduled local comparability:

  • nly if scheduled consecutively

k( k(D¨ urr, Vasquez

weight processing time

pj wj

i ≺g j j ≺g i

Approaches tested by us: different graph searches with integrated comparabilities quadratic IP with integrated comparabilities (Cplex 12.4) major numerical problems

  • W. H¨
  • hn and T. Jacobs
slide-80
SLIDE 80

Exact algorithms for monomial cost

Feasible for monomial cost tk: best first graph search based on A∗

  • W. H¨
  • hn and T. Jacobs
slide-81
SLIDE 81

Exact algorithms for monomial cost

Feasible for monomial cost tk: best first graph search based on A∗ local and global comparability

k( k(

weight processing time

pj wj

i ≺g j i ≺ℓ j j ≺g i j ≺ℓ i

  • W. H¨
  • hn and T. Jacobs
slide-82
SLIDE 82

Exact algorithms for monomial cost

Feasible for monomial cost tk: best first graph search based on A∗ local and global comparability

k( k(

weight processing time

pj wj

i ≺g j i ≺ℓ j j ≺g i j ≺ℓ i

Constraint programming approach:

(joint work with Jens Schulz & Daniela Luft)

start time based formulations with disjunctive constraint and domain propagation (SCIP 2.1.1)

  • W. H¨
  • hn and T. Jacobs
slide-83
SLIDE 83

Exact algorithms for monomial cost

Feasible for monomial cost tk: best first graph search based on A∗ local and global comparability

k( k(

weight processing time

pj wj

i ≺g j i ≺ℓ j j ≺g i j ≺ℓ i

Constraint programming approach:

(joint work with Jens Schulz & Daniela Luft)

start time based formulations with disjunctive constraint and domain propagation (SCIP 2.1.1) again major numerical problems (for t2, t3, t4)

  • W. H¨
  • hn and T. Jacobs
slide-84
SLIDE 84

Conclusions

Single machine scheduling with weighted convex/concave cost:

  • W. H¨
  • hn and T. Jacobs
slide-85
SLIDE 85

Conclusions

Single machine scheduling with weighted convex/concave cost: Approximation algorithms: tight (parametrized) analysis of Smith’s rule

  • W. H¨
  • hn and T. Jacobs
slide-86
SLIDE 86

Conclusions

Single machine scheduling with weighted convex/concave cost: Approximation algorithms: tight (parametrized) analysis of Smith’s rule asymptotic approximation factor k

k−1 k+1 for cost tk

  • W. H¨
  • hn and T. Jacobs
slide-87
SLIDE 87

Conclusions

Single machine scheduling with weighted convex/concave cost: Approximation algorithms: tight (parametrized) analysis of Smith’s rule asymptotic approximation factor k

k−1 k+1 for cost tk

Exact algorithms for monomial cost: generic solvers have major numerical problems while problem-specific enumeration schemes don’t

  • W. H¨
  • hn and T. Jacobs
slide-88
SLIDE 88

Conclusions

Single machine scheduling with weighted convex/concave cost: Approximation algorithms: tight (parametrized) analysis of Smith’s rule asymptotic approximation factor k

k−1 k+1 for cost tk

Exact algorithms for monomial cost: generic solvers have major numerical problems while problem-specific enumeration schemes don’t complexity (almost) completely open

  • W. H¨
  • hn and T. Jacobs
slide-89
SLIDE 89

Conclusions

Single machine scheduling with weighted convex/concave cost: Approximation algorithms: tight (parametrized) analysis of Smith’s rule asymptotic approximation factor k

k−1 k+1 for cost tk

Exact algorithms for monomial cost: generic solvers have major numerical problems while problem-specific enumeration schemes don’t complexity (almost) completely open Thank you!

  • W. H¨
  • hn and T. Jacobs