On the Computational Complexity of Periodic Scheduling PhD defense - - PowerPoint PPT Presentation

On the Computational Complexity

• f Periodic Scheduling

PhD defense Thomas Rothvoß

Real-time Scheduling

Given: (synchronous) tasks τ1, . . . , τn with τi = (c(τi), d(τi), p(τi))

Real-time Scheduling

Given: (synchronous) tasks τ1, . . . , τn with τi = (c(τi), d(τi), p(τi)) running time

Real-time Scheduling

Given: (synchronous) tasks τ1, . . . , τn with τi = (c(τi), d(τi), p(τi)) running time (relative) deadline

Real-time Scheduling

Given: (synchronous) tasks τ1, . . . , τn with τi = (c(τi), d(τi), p(τi)) running time (relative) deadline period

Real-time Scheduling

Given: (synchronous) tasks τ1, . . . , τn with τi = (c(τi), d(τi), p(τi)) running time (relative) deadline period W.l.o.g.: Task τi releases job of length c(τi) at z · p(τi) and absolute deadline z · p(τi) + d(τi) (z ∈ N0)

Real-time Scheduling

Given: (synchronous) tasks τ1, . . . , τn with τi = (c(τi), d(τi), p(τi)) running time (relative) deadline period W.l.o.g.: Task τi releases job of length c(τi) at z · p(τi) and absolute deadline z · p(τi) + d(τi) (z ∈ N0) Utilization: u(τi) = c(τi)

p(τi)

Real-time Scheduling

Given: (synchronous) tasks τ1, . . . , τn with τi = (c(τi), d(τi), p(τi)) running time (relative) deadline period W.l.o.g.: Task τi releases job of length c(τi) at z · p(τi) and absolute deadline z · p(τi) + d(τi) (z ∈ N0) Utilization: u(τi) = c(τi)

p(τi)

Settings:

◮ static priorities ↔ dynamic priorities

Real-time Scheduling

Given: (synchronous) tasks τ1, . . . , τn with τi = (c(τi), d(τi), p(τi)) running time (relative) deadline period W.l.o.g.: Task τi releases job of length c(τi) at z · p(τi) and absolute deadline z · p(τi) + d(τi) (z ∈ N0) Utilization: u(τi) = c(τi)

p(τi)

Settings:

◮ static priorities ↔ dynamic priorities ◮ single-processor ↔ multi-processor

Real-time Scheduling

Given: (synchronous) tasks τ1, . . . , τn with τi = (c(τi), d(τi), p(τi)) running time (relative) deadline period W.l.o.g.: Task τi releases job of length c(τi) at z · p(τi) and absolute deadline z · p(τi) + d(τi) (z ∈ N0) Utilization: u(τi) = c(τi)

p(τi)

Settings:

◮ static priorities ↔ dynamic priorities ◮ single-processor ↔ multi-processor ◮ preemptive scheduling ↔ non-preemptive

Real-time Scheduling

Given: (synchronous) tasks τ1, . . . , τn with τi = (c(τi), d(τi), p(τi)) running time (relative) deadline period W.l.o.g.: Task τi releases job of length c(τi) at z · p(τi) and absolute deadline z · p(τi) + d(τi) (z ∈ N0) Utilization: u(τi) = c(τi)

p(τi)

Settings:

◮ static priorities ↔ dynamic priorities ◮ single-processor ↔ multi-processor ◮ preemptive scheduling ↔ non-preemptive

Implicit deadlines: d(τi) = p(τi) Constrained deadlines: d(τi) ≤ p(τi)

Example: Implicit deadlines & static priorities

c(τ1) = 1 d(τ1) = p(τ1) = 2 c(τ2) = 2 d(τ2) = p(τ2) = 5

b b b

time 0 1 2 3 4 5 6 7 8 9 10

Example: Implicit deadlines & static priorities

Theorem (Liu & Layland ’73)

Optimal priorities:

1 p(τi) for τi (Rate-monotonic schedule)

c(τ1) = 1 d(τ1) = p(τ1) = 2 c(τ2) = 2 d(τ2) = p(τ2) = 5

b b b

time 0 1 2 3 4 5 6 7 8 9 10

Example: Implicit deadlines & static priorities

Theorem (Liu & Layland ’73)

Optimal priorities:

1 p(τi) for τi (Rate-monotonic schedule)

c(τ1) = 1 d(τ1) = p(τ1) = 2 c(τ2) = 2 d(τ2) = p(τ2) = 5

b b b

time 0 1 2 3 4 5 6 7 8 9 10

Example: Implicit deadlines & static priorities

Theorem (Liu & Layland ’73)

Optimal priorities:

1 p(τi) for τi (Rate-monotonic schedule)

c(τ1) = 1 d(τ1) = p(τ1) = 2 c(τ2) = 2 d(τ2) = p(τ2) = 5

b b b

time 0 1 2 3 4 5 6 7 8 9 10

Feasibility test for implicit-deadline tasks

Theorem (Lehoczky et al. ’89)

If p(τ1) ≤ . . . ≤ p(τn) then the response time r(τi) in a Rate-monotonic schedule is the smallest non-negative value s.t. c(τi) +

• j<i

r(τi) p(τj)

• c(τj) ≤ r(τi)

1 machine suffices ⇔ ∀i : r(τi) ≤ p(τi).

Simultaneous Diophantine Approximation (SDA)

Given:

◮ α1, . . . , αn ∈ Q ◮ bound N ∈ N ◮ error bound ε > 0

Decide: ∃Q ∈ {1, . . . , N} : max

i=1,...,n

• αi − Z

Q

• ≤ ε

Q

Simultaneous Diophantine Approximation (SDA)

Given:

◮ α1, . . . , αn ∈ Q ◮ bound N ∈ N ◮ error bound ε > 0

Decide: ∃Q ∈ {1, . . . , N} : max

i=1,...,n

• αi − Z

Q

• ≤ ε

Q ⇔ ∃Q ∈ {1, . . . , N} : max

i=1,...,n |⌈Qαi⌋ − Qαi| ≤ ε

Simultaneous Diophantine Approximation (SDA)

Given:

◮ α1, . . . , αn ∈ Q ◮ bound N ∈ N ◮ error bound ε > 0

Decide: ∃Q ∈ {1, . . . , N} : max

i=1,...,n

• αi − Z

Q

• ≤ ε

Q ⇔ ∃Q ∈ {1, . . . , N} : max

i=1,...,n |⌈Qαi⌋ − Qαi| ≤ ε ◮ NP-hard [Lagarias ’85]

Simultaneous Diophantine Approximation (SDA)

Given:

◮ α1, . . . , αn ∈ Q ◮ bound N ∈ N ◮ error bound ε > 0

Decide: ∃Q ∈ {1, . . . , N} : max

i=1,...,n

• αi − Z

Q

• ≤ ε

Q ⇔ ∃Q ∈ {1, . . . , N} : max

i=1,...,n |⌈Qαi⌋ − Qαi| ≤ ε ◮ NP-hard [Lagarias ’85] ◮ Gap version NP-hard [R¨

• ssner & Seifert ’96,

Chen & Meng ’07]

Simultaneous Diophantine Approximation (2)

Theorem (R¨

• ssner & Seifert ’96, Chen & Meng ’07)

Given α1, . . . , αn, N, ε > 0 it is NP-hard to distinguish

◮ ∃Q ∈ {1, . . . , N} : max i=1,...,n |⌈Qαi⌋ − Qαi| ≤ ε ◮ ∄Q ∈ {1, . . . , n

O(1) log log n N} : max

i=1,...,n |⌈Qαi⌋ − Qαi| ≤ n

O(1) log log n ε

Simultaneous Diophantine Approximation (2)

Theorem (R¨

• ssner & Seifert ’96, Chen & Meng ’07)

Given α1, . . . , αn, N, ε > 0 it is NP-hard to distinguish

◮ ∃Q ∈ {1, . . . , N} : max i=1,...,n |⌈Qαi⌋ − Qαi| ≤ ε ◮ ∄Q ∈ {1, . . . , n

O(1) log log n N} : max

i=1,...,n |⌈Qαi⌋ − Qαi| ≤ n

O(1) log log n ε

Theorem (Eisenbrand & R. - APPROX’09)

Given α1, . . . , αn, N, ε > 0 it is NP-hard to distinguish

◮ ∃Q ∈ {N/2, . . . , N} : max i=1,...,n |⌈Qαi⌋ − Qαi| ≤ ε ◮ ∄Q ∈ {1, . . . , 2nO(1) · N} : max i=1,...,n |⌈Qαi⌋ − Qαi| ≤ n

O(1) log log n · ε

even if ε ≤ (1

2)nO(1).

Directed Diophantine Approximation (DDA)

Theorem (Eisenbrand & R. - APPROX’09)

Given α1, . . . , αn, N, ε > 0 it is NP-hard to distinguish

◮ ∃Q ∈ {N/2, . . . , N} : max i=1,...,n |⌈Qαi⌉ − Qαi| ≤ ε ◮ ∄Q ∈ {1, . . . , n

O(1) log log n · N} : max

i=1,...,n |⌈Qαi⌉ − Qαi| ≤ 2nO(1) · ε

even if ε ≤ (1

2)nO(1).

Directed Diophantine Approximation (DDA)

Theorem (Eisenbrand & R. - APPROX’09)

Given α1, . . . , αn, N, ε > 0 it is NP-hard to distinguish

◮ ∃Q ∈ {N/2, . . . , N} : max i=1,...,n |⌈Qαi⌉ − Qαi| ≤ ε ◮ ∄Q ∈ {1, . . . , n

O(1) log log n · N} : max

i=1,...,n |⌈Qαi⌉ − Qαi| ≤ 2nO(1) · ε

even if ε ≤ (1

2)nO(1).

Theorem (Eisenbrand & R. - SODA’10)

Given α1, . . . , αn, w1, . . . , wn ≥ 0, N, ε > 0 it is NP-hard to distinguish

◮ ∃Q ∈ [N/2, N] : n i=1 wi (Qαi − ⌊Qαi⌋) ≤ ε ◮ ∄Q ∈ [1, n

O(1) log log n · N] : n

i=1 wi (Qαi − ⌊Qαi⌋) ≤ 2nO(1) · ε

even if ε ≤ (1

2)nO(1).

Hardness of Response Time Computation

Theorem (Eisenbrand & R. - RTSS’08)

Computing response times for implicit-deadline tasks w.r.t. to a Rate-monotonic schedule, i.e. solving min

• r ≥ 0 | c(τn) +

n−1

• i=1
• r

p(τi)

• c(τi) ≤ r
• (p(τ1) ≤ . . . ≤ p(τn)) is NP-hard (even to approximate within a

factor of n

O(1) log log n ).

◮ Reduction from Directed Diophantine Approximation

Mixing Set

min css + cT y s + aiyi ≥ bi ∀i = 1, . . . , n s ∈ R≥0 y ∈ Zn

Mixing Set

min css + cT y s + aiyi ≥ bi ∀i = 1, . . . , n s ∈ R≥0 y ∈ Zn

◮ Complexity status? [Conforti, Di Summa & Wolsey ’08]

Mixing Set

min css + cT y s + aiyi ≥ bi ∀i = 1, . . . , n s ∈ R≥0 y ∈ Zn

◮ Complexity status? [Conforti, Di Summa & Wolsey ’08]

Theorem (Eisenbrand & R. - APPROX’09)

Solving Mixing Set is NP-hard.

• 1. Model Directed Diophantine Approximation (almost) as

Mixing Set

• 2. Simulate missing constraint with Lagrangian relaxation

Testing EDF-schedulability

Setting: Constrained deadline tasks, i.e. d(τi) ≤ p(τi)

Theorem (Baruah, Mok & Rosier ’90)

Q − d(τi) p(τi)

• + 1
• · c(τi)
• =: DBF(τi, Q)

d(τi) d(τi) + 1 · p(τi) d(τi) + 2 · p(τi) c(τi) 2c(τi) 3c(τi) Q DBF(τi, Q)

Testing EDF-schedulability

Setting: Constrained deadline tasks, i.e. d(τi) ≤ p(τi)

Theorem (Baruah, Mok & Rosier ’90)

Earliest-Deadline-First schedule is feasible iff ∀Q ≥ 0 :

n

• i=1

Q − d(τi) p(τi)

• + 1
• · c(τi)
• =: DBF(τi, Q)

≤ Q

Testing EDF-schedulability

Setting: Constrained deadline tasks, i.e. d(τi) ≤ p(τi)

Theorem (Baruah, Mok & Rosier ’90)

Earliest-Deadline-First schedule is feasible iff ∀Q ≥ 0 :

n

• i=1

Q − d(τi) p(τi)

• + 1
• · c(τi)
• =: DBF(τi, Q)

≤ Q

◮ Complexity status is “Problem 2” in Open Problems in

Real-Time Scheduling [Baruah & Pruhs ’09]

Testing EDF-schedulability

Setting: Constrained deadline tasks, i.e. d(τi) ≤ p(τi)

Theorem (Baruah, Mok & Rosier ’90)

Earliest-Deadline-First schedule is feasible iff ∀Q ≥ 0 :

n

• i=1

Q − d(τi) p(τi)

• + 1
• · c(τi)
• =: DBF(τi, Q)

≤ Q

◮ Complexity status is “Problem 2” in Open Problems in

Real-Time Scheduling [Baruah & Pruhs ’09]

Theorem (Eisenbrand & R. - SODA’10)

Testing EDF-schedulability is coNP-hard.

Reduction (1)

Given: DDA instance α1, . . . , αn, w1, . . . , wn, N ∈ N, ε > 0

Reduction (1)

Given: DDA instance α1, . . . , αn, w1, . . . , wn, N ∈ N, ε > 0 Define: Task set S = {τ0, . . . , τn} s.t.

◮ Yes: ∃Q ∈ [N/2, N] : n i=1 wi(Qαi − ⌊Qαi⌋) ≤ ε

⇒ S not EDF-schedulable (∃Q ≥ 0 : DBF(S, Q) > β · Q)

◮ No: ∄Q ∈ [N/2, 3N] : n i=1 wi(Qαi − ⌊Qαi⌋) ≤ 3ε

⇒ S EDF-schedulable (∀Q ≥ 0 : DBF(S, Q) ≤ β · Q)

Reduction (1)

Given: DDA instance α1, . . . , αn, w1, . . . , wn, N ∈ N, ε > 0 Define: Task set S = {τ0, . . . , τn} s.t.

◮ Yes: ∃Q ∈ [N/2, N] : n i=1 wi(Qαi − ⌊Qαi⌋) ≤ ε

⇒ S not EDF-schedulable (∃Q ≥ 0 : DBF(S, Q) > β · Q)

◮ No: ∄Q ∈ [N/2, 3N] : n i=1 wi(Qαi − ⌊Qαi⌋) ≤ 3ε

⇒ S EDF-schedulable (∀Q ≥ 0 : DBF(S, Q) ≤ β · Q) Task system: τi = (c(τi), d(τi), p(τi)) :=

• wi, 1

αi , 1 αi

• ∀i = 1, . . . , n

U :=

n

• i=1

c(τi) p(τi)

Reduction (2)

Q · U −

n

• i=1

Q − d(τi) p(τi)

• + 1
• c(τi)

Reduction (2)

Q · U −

n

• i=1

Q − d(τi) p(τi)

• + 1
• c(τi)

=

n

• i=1

Q p(τi) − Q p(τi)

• c(τi)

Reduction (2)

Q · U −

n

• i=1

Q − d(τi) p(τi)

• + 1
• c(τi)

=

n

• i=1

Q p(τi) − Q p(τi)

• c(τi)

=

n

• i=1

wi(Qαi − ⌊Qαi⌋)

Reduction (2)

Q · U −

n

• i=1

Q − d(τi) p(τi)

• + 1
• c(τi)

=

n

• i=1

Q p(τi) − Q p(τi)

• c(τi)

=

n

• i=1

wi(Qαi − ⌊Qαi⌋) ⇒ DBF({τ1, . . . , τn}, Q) Q ≈ U ⇔

• approx. error small

Reduction (2)

Q · U −

n

• i=1

Q − d(τi) p(τi)

• + 1
• c(τi)

=

n

• i=1

Q p(τi) − Q p(τi)

• c(τi)

=

n

• i=1

wi(Qαi − ⌊Qαi⌋) ⇒ DBF({τ1, . . . , τn}, Q) Q ≈ U ⇔

• approx. error small

DBF({τ1, . . . , τn}, Q) Q Q

Reduction (2)

Q · U −

n

• i=1

Q − d(τi) p(τi)

• + 1
• c(τi)

=

n

• i=1

Q p(τi) − Q p(τi)

• c(τi)

=

n

• i=1

wi(Qαi − ⌊Qαi⌋) ⇒ DBF({τ1, . . . , τn}, Q) Q ≈ U ⇔

• approx. error small

DBF({τ1, . . . , τn}, Q) Q Q scm(p(τ1), . . . , p(τn)) U

b

Reduction (2)

Q · U −

n

• i=1

Q − d(τi) p(τi)

• + 1
• c(τi)

=

n

• i=1

Q p(τi) − Q p(τi)

• c(τi)

=

n

• i=1

wi(Qαi − ⌊Qαi⌋) ⇒ DBF({τ1, . . . , τn}, Q) Q ≈ U ⇔

• approx. error small

DBF({τ1, . . . , τn}, Q) Q Q scm(p(τ1), . . . , p(τn)) U

b

N/2 N

Reduction (2)

Q · U −

n

• i=1

Q − d(τi) p(τi)

• + 1
• c(τi)

=

n

• i=1

Q p(τi) − Q p(τi)

• c(τi)

=

n

• i=1

wi(Qαi − ⌊Qαi⌋) ⇒ DBF({τ1, . . . , τn}, Q) Q ≈ U ⇔

• approx. error small

DBF({τ1, . . . , τn}, Q) Q Q scm(p(τ1), . . . , p(τn)) U

b

N/2 N

b

Q∗

Reduction (3)

Add a special task τ0 = (c(τ0), d(τ0), p(τ0)) := (3ε, N/2, ∞) N/2 N DBF(τ0, Q) Q Q

Reduction (4)

Q N/2 N U

Reduction (4)

DBF(S, Q) Q Q N/2 N U

Reduction (4)

DBF(S, Q) Q Q N/2 N U β := U + ε

N

U − ε

N

Algorithms for Multi-processor Scheduling

Setting: Implicit deadlines (d(τi) = p(τi)), multi-processor

Algorithms for Multi-processor Scheduling

Setting: Implicit deadlines (d(τi) = p(τi)), multi-processor Known results:

◮ NP-hard (generalization of Bin Packing)

Algorithms for Multi-processor Scheduling

Setting: Implicit deadlines (d(τi) = p(τi)), multi-processor Known results:

◮ NP-hard (generalization of Bin Packing) ◮ asymp. 1.75-apx [Burchard et al. ’95]

Algorithms for Multi-processor Scheduling

Setting: Implicit deadlines (d(τi) = p(τi)), multi-processor Known results:

◮ NP-hard (generalization of Bin Packing) ◮ asymp. 1.75-apx [Burchard et al. ’95]

Theorem (Eisenbrand & R. - ICALP’08)

In time Oε(1) · n(1/ε)O(1) one can find an assignment to APX ≤ (1 + ε) · OPT + O(1) machines such that the RM-schedule is feasible if the machines are speed up by a factor of 1 + ε (resource augmentation).

• 1. Rounding, clustering & merging small tasks
• 2. Use relaxed feasibility notion
• 3. Dynamic programming

Algorithms for Multi-processor Scheduling (2)

Theorem

Let k ∈ N be an arbitrary parameter. In time O(n3) one can find an assignment of implicit-deadline tasks to APX ≤ 3 2 + 1 k

• OPT + 9k

machines (→ asymptotic 3

2-apx).

• 1. Create graph G = (S, E) with tasks as nodes and edges

(τ1, τ2) ∈ E :⇔ {τ1, τ2} RM-schedulable (on 1 machine)

• 2. Define suitable edge weights
• 3. Mincost matching ⇒ good assignment

Algorithms for Multi-processor Scheduling (3)

P = {x ∈ {0, 1}n | {τi | xi = 1} RM-schedulable}

Algorithms for Multi-processor Scheduling (3)

P = {x ∈ {0, 1}n | {τi | xi = 1} RM-schedulable} λx =

• 1

pack machine with {τi | xi = 1}

• therwise

Algorithms for Multi-processor Scheduling (3)

P = {x ∈ {0, 1}n | {τi | xi = 1} RM-schedulable} λx =

• 1

pack machine with {τi | xi = 1}

• therwise

Column-based LP (or configuration LP): min 1T λ

• x∈P

λxx ≥ 1 λ ≥

Algorithms for Multi-processor Scheduling (3)

P = {x ∈ {0, 1}n | {τi | xi = 1} RM-schedulable} λx =

• 1

pack machine with {τi | xi = 1}

• therwise

Column-based LP (or configuration LP): min 1T λ

• x∈P

λxx ≥ 1 λ ≥

◮ For Bin Packing: OPT − OPTf ≤ O(log2 n)

[Karmarkar & Karp ’82]

Algorithms for Multi-processor Scheduling (3)

P = {x ∈ {0, 1}n | {τi | xi = 1} RM-schedulable} λx =

• 1

pack machine with {τi | xi = 1}

• therwise

Column-based LP (or configuration LP): min 1T λ

• x∈P

λxx ≥ 1 λ ≥

◮ For Bin Packing: OPT − OPTf ≤ O(log2 n)

[Karmarkar & Karp ’82]

Theorem

1.33 ≈ 4 3 ≤ sup

instances

OPT OPTf

• ≤ 1 + ln(2) ≈ 1.69

Algorithms for Multi-processor Scheduling (4)

Theorem (Eisenbrand & R. - ICALP’08)

For all ε > 0, there is no polynomial time algorithm with APX ≤ OPT + n1−ε unless P = NP (⇒ no AFPTAS).

◮ Cloning of 3-Partition instances

Algorithms for Multi-processor Scheduling (5)

◮ FFMP is a simple First Fit like heuristic with running time

O(n log n)

Algorithms for Multi-processor Scheduling (5)

◮ FFMP is a simple First Fit like heuristic with running time

O(n log n)

Theorem (Karrenbauer & R. - ESA’09)

For n tasks S = {τ1, . . . , τn} with u(τi) ∈ [0, 1] uniformly at random E[FFMP(S)] ≤ E[u(S)] + O(n3/4(log n)3/8) (average utilization → 100%).

◮ Reduce to known results from the average case analysis of

Bin Packing

Dynamic vs. static priorities

◮ Setting: constrained-deadline tasks, i.e. d(τi) ≤ p(τi)

Dynamic vs. static priorities

◮ Setting: constrained-deadline tasks, i.e. d(τi) ≤ p(τi) ◮ Worst-case speedup factor

f := sup

instances

• proc. speed for static priority schedule
• proc. speed for arbitrary schedule
• ≥ 1

Dynamic vs. static priorities

◮ Setting: constrained-deadline tasks, i.e. d(τi) ≤ p(τi) ◮ Worst-case speedup factor

f := sup

instances

• proc. speed for static priority schedule
• proc. speed for arbitrary schedule
• ≥ 1

◮ Known: 3 2 ≤ f ≤ 2 [Baruah & Burns ’08]

Dynamic vs. static priorities

◮ Setting: constrained-deadline tasks, i.e. d(τi) ≤ p(τi) ◮ Worst-case speedup factor

f := sup

instances

• proc. speed for static priority schedule
• proc. speed for arbitrary schedule
• ≥ 1

◮ Known: 3 2 ≤ f ≤ 2 [Baruah & Burns ’08]

Theorem (Davis, R., Baruah & Burns - RT Systems ’09)

f = 1 Ω ≈ 1.76 where Ω ≈ 0.567 is the unique positive real root of x = ln(1/x).

The end Thanks for your attention