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Periodic Event Scheduling and its Industrial Application Tobias Hofmann Table of Contents I Precedence Relationships II Periodic Event Scheduling III Appropriate Cycle Bases Tobias Hofmann Algorithmic and Discrete Mathematics 1 | 14

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Periodic Event Scheduling and its Industrial Application Tobias Hofmann

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Table of Contents

I Precedence Relationships II Periodic Event Scheduling III Appropriate Cycle Bases

Tobias Hofmann Algorithmic and Discrete Mathematics 1 | 14

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Precedence Relationships

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Precedence Relationships The Scenario

Tobias Hofmann Algorithmic and Discrete Mathematics 2 | 14

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Precedence Relationships The Scenario

◾ Fixed environment

Tobias Hofmann Algorithmic and Discrete Mathematics 2 | 14

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Precedence Relationships The Scenario

◾ Fixed environment ◾ Fixed number of robots

Tobias Hofmann Algorithmic and Discrete Mathematics 2 | 14

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Precedence Relationships The Scenario

◾ Fixed environment ◾ Fixed number of robots ◾ Fixed motions between points

Tobias Hofmann Algorithmic and Discrete Mathematics 2 | 14

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Precedence Relationships The Scenario

◾ Fixed environment ◾ Fixed number of robots ◾ Fixed motions between points ◾ Adjustable robot schedules

Tobias Hofmann Algorithmic and Discrete Mathematics 2 | 14

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Precedence Relationships The Scenario

Tobias Hofmann Algorithmic and Discrete Mathematics 3 | 14

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Precedence Relationships The Scenario

Tobias Hofmann Algorithmic and Discrete Mathematics 3 | 14

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Precedence Relationships The Scenario

Tobias Hofmann Algorithmic and Discrete Mathematics 3 | 14

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Precedence Relationships The Scenario

Tobias Hofmann Algorithmic and Discrete Mathematics 3 | 14

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Precedence Relationships The Scenario

Tobias Hofmann Algorithmic and Discrete Mathematics 3 | 14

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Precedence Relationships The Scenario

Tobias Hofmann Algorithmic and Discrete Mathematics 3 | 14

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Precedence Relationships The Scenario

We consider a digraph D = (V , A) with l, u ∈ QA.

Tobias Hofmann Algorithmic and Discrete Mathematics 3 | 14

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Precedence Relationships Conflicts

Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Conflicts

Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Conflicts

Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Conflicts

Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Conflicts

Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Conflicts

Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Conflicts

Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Conflicts

Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Conflicts

Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Conflicts

Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Conflicts

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Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Conflicts

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Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Conflicts

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Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Conflicts

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Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Conflicts

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Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Conflicts

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Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Conflicts

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Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Conflicts

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Tobias Hofmann Algorithmic and Discrete Mathematics 4 | 14

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Precedence Relationships Subproblem: Set Cover (Karp, 1972)

n m

Tobias Hofmann Algorithmic and Discrete Mathematics 5 | 14

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Precedence Relationships Subproblem: Set Cover (Karp, 1972)

Given a finite set U and T ⊂ 2U n m

Tobias Hofmann Algorithmic and Discrete Mathematics 5 | 14

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Precedence Relationships Subproblem: Set Cover (Karp, 1972)

Given a finite set U and T ⊂ 2U, a cover is a subset C ⊂ T whose union is U. n m

Tobias Hofmann Algorithmic and Discrete Mathematics 5 | 14

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Precedence Relationships Subproblem: Set Cover (Karp, 1972)

Given a finite set U and T ⊂ 2U, a cover is a subset C ⊂ T whose union is U. n m Minimize

  • S∈T

xS

Tobias Hofmann Algorithmic and Discrete Mathematics 5 | 14

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Precedence Relationships Subproblem: Set Cover (Karp, 1972)

Given a finite set U and T ⊂ 2U, a cover is a subset C ⊂ T whose union is U. n m Minimize

  • S∈T

xS

  • s. t.
  • S:e∈S

xS ≥ 1 ∀ e ∈ U,

Tobias Hofmann Algorithmic and Discrete Mathematics 5 | 14

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Precedence Relationships Subproblem: Set Cover (Karp, 1972)

Given a finite set U and T ⊂ 2U, a cover is a subset C ⊂ T whose union is U. n m Minimize

  • S∈T

xS

  • s. t.
  • S:e∈S

xS ≥ 1 ∀ e ∈ U, xS ∈ {0, 1} ∀ S ∈ T .

Tobias Hofmann Algorithmic and Discrete Mathematics 5 | 14

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Precedence Relationships Subproblem: Set Cover (Karp, 1972)

Given a finite set U and T ⊂ 2U, a cover is a subset C ⊂ T whose union is U. n m Minimize

  • S∈T

xS

  • s. t.
  • S:e∈S

xS ≥ 1 ∀ e ∈ U, xS ∈ {0, 1} ∀ S ∈ T .

Tobias Hofmann Algorithmic and Discrete Mathematics 5 | 14

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Precedence Relationships Subproblem: Set Cover (Karp, 1972)

Given a finite set U and T ⊂ 2U, a cover is a subset C ⊂ T whose union is U. n m Minimize

  • S∈T

xS

  • s. t.
  • S:e∈S

xS ≥ 1 ∀ e ∈ U, xS ∈ {0, 1} ∀ S ∈ T .

Tobias Hofmann Algorithmic and Discrete Mathematics 5 | 14

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Precedence Relationships Subproblem: Set Cover (Karp, 1972)

Given a finite set U and T ⊂ 2U, a cover is a subset C ⊂ T whose union is U. n m Minimize

  • S∈T

xS

  • s. t.
  • S:e∈S

xS ≥ 1 ∀ e ∈ U, xS ∈ {0, 1} ∀ S ∈ T .

Tobias Hofmann Algorithmic and Discrete Mathematics 5 | 14

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Precedence Relationships Subproblem: Set Cover (Karp, 1972)

Given a finite set U and T ⊂ 2U, a cover is a subset C ⊂ T whose union is U. n m Minimize

  • S∈T

xS

  • s. t.
  • S:e∈S

xS ≥ 1 ∀ e ∈ U, xS ∈ {0, 1} ∀ S ∈ T .

Tobias Hofmann Algorithmic and Discrete Mathematics 5 | 14

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Precedence Relationships Subproblem: Set Cover (Karp, 1972)

Given a finite set U and T ⊂ 2U, a cover is a subset C ⊂ T whose union is U. n m Minimize

  • S∈T

xS

  • s. t.
  • S:e∈S

xS ≥ 1 ∀ e ∈ U, xS ∈ {0, 1} ∀ S ∈ T .

Tobias Hofmann Algorithmic and Discrete Mathematics 5 | 14

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Precedence Relationships Subproblem: Set Cover (Karp, 1972)

Given a finite set U and T ⊂ 2U, a cover is a subset C ⊂ T whose union is U. n m Minimize

  • S∈T

xS

  • s. t.
  • S:e∈S

xS ≥ 1 ∀ e ∈ U, xS ∈ {0, 1} ∀ S ∈ T . Theorem The algorithms takes O(min{m, n}). Proof.

  • Tobias Hofmann

Algorithmic and Discrete Mathematics 5 | 14

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Precedence Relationships Subproblem: Set Cover (Karp, 1972)

Given a finite set U and T ⊂ 2U, a cover is a subset C ⊂ T whose union is U. n m Minimize

  • S∈T

xS

  • s. t.
  • S:e∈S

xS ≥ 1 ∀ e ∈ U, xS ∈ {0, 1} ∀ S ∈ T . Theorem The algorithms takes O(min{m, n}). Proof.

  • Tobias Hofmann

Algorithmic and Discrete Mathematics 5 | 14

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Precedence Relationships Subproblem: Set Cover (Karp, 1972)

Given a finite set U and T ⊂ 2U, a cover is a subset C ⊂ T whose union is U. n m Minimize

  • S∈T

xS

  • s. t.
  • S:e∈S

xS ≥ 1 ∀ e ∈ U, xS ∈ {0, 1} ∀ S ∈ T . Theorem The algorithms takes O(min{m, n}). Proof.

  • Tobias Hofmann

Algorithmic and Discrete Mathematics 5 | 14

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Precedence Relationships Subproblem: Set Cover (Karp, 1972)

Given a finite set U and T ⊂ 2U, a cover is a subset C ⊂ T whose union is U. n m Minimize

  • S∈T

xS

  • s. t.
  • S:e∈S

xS ≥ 1 ∀ e ∈ U, xS ∈ {0, 1} ∀ S ∈ T . Theorem The algorithms takes O(min{m, n}). Proof.

  • Tobias Hofmann

Algorithmic and Discrete Mathematics 5 | 14

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Precedence Relationships Conflicts

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Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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Precedence Relationships Conflicts

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Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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Precedence Relationships Conflicts

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Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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Precedence Relationships Conflicts

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Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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Periodic Event Scheduling

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Periodic Event Scheduling

Periodic Event Scheduling Problem [PESP] (Serafini and Ukovich, 1989) Given a digraph D = (V , A) as well as l, u ∈ QA, find x ∈ QV such that ∀ a = (i, j) ∈ A : 0 ≤ (xj − xi − la) mod T ≤ ua − la

Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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Periodic Event Scheduling

Periodic Event Scheduling Problem [PESP] (Serafini and Ukovich, 1989) Given a digraph D = (V , A) as well as l, u ∈ QA, find x ∈ QV such that ∀ a = (i, j) ∈ A : 0 ≤ (xj − xi − la) mod T ≤ ua − la

Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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Periodic Event Scheduling

Periodic Event Scheduling Problem [PESP] (Serafini and Ukovich, 1989) Given a digraph D = (V , A) as well as l, u ∈ QA, find x ∈ QV such that ∀ a = (i, j) ∈ A : 0 ≤ (xj − xi − la) mod T ≤ ua − la ⇔ ∀ a = (i, j) ∈ A : 0 ≤ xj − xi − la − T max{z ∈ Z | xj − xi − la − z T ≥ 0}

Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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Periodic Event Scheduling

Periodic Event Scheduling Problem [PESP] (Serafini and Ukovich, 1989) Given a digraph D = (V , A) as well as l, u ∈ QA, find x ∈ QV such that ∀ a = (i, j) ∈ A : 0 ≤ (xj − xi − la) mod T ≤ ua − la ⇔ ∀ a = (i, j) ∈ A : 0 ≤ xj − xi − la − T max{z ∈ Z | xj − xi − la − z T ≥ 0}

Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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SLIDE 63

Periodic Event Scheduling

Periodic Event Scheduling Problem [PESP] (Serafini and Ukovich, 1989) Given a digraph D = (V , A) as well as l, u ∈ QA, find x ∈ QV and p ∈ ZA such that ∀ a = (i, j) ∈ A : 0 ≤ (xj − xi − la) mod T ≤ ua − la ⇔ ∀ a = (i, j) ∈ A : 0 ≤ xj − xi − la − T max{z ∈ Z | xj − xi − la − z T ≥ 0} ⇔ ∀ a = (i, j) ∈ A : la ≤ xj − xi + T pa ≤ ua.

Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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Periodic Event Scheduling

Periodic Event Scheduling Problem [PESP] (Serafini and Ukovich, 1989) Given a digraph D = (V , A) as well as l, u ∈ QA, find x ∈ QV and p ∈ ZA such that ∀ a = (i, j) ∈ A : 0 ≤ (xj − xi − la) mod T ≤ ua − la ⇔ ∀ a = (i, j) ∈ A : 0 ≤ xj − xi − la − T max{z ∈ Z | xj − xi − la − z T ≥ 0} ⇔ ∀ a = (i, j) ∈ A : la ≤ xj − xi + T pa ≤ ua. Theorem (Odijk, 1994) PESP is NP-complete.

Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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Periodic Event Scheduling

Periodic Event Scheduling Problem [PESP] (Serafini and Ukovich, 1989) Given a digraph D = (V , A) as well as l, u ∈ QA, find x ∈ QV and p ∈ ZA such that ∀ a = (i, j) ∈ A : 0 ≤ (xj − xi − la) mod T ≤ ua − la ⇔ ∀ a = (i, j) ∈ A : 0 ≤ xj − xi − la − T max{z ∈ Z | xj − xi − la − z T ≥ 0} ⇔ ∀ a = (i, j) ∈ A : la ≤ xj − xi + T pa ≤ ua. Theorem (Odijk, 1994) PESP is NP-complete. Theorem (e. g. Liebchen, 2006) For any fixed p ∈ ZA PESP can be solved in O(|V ||A|).

Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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Periodic Event Scheduling

Periodic Event Scheduling Problem [PESP] (Serafini and Ukovich, 1989) Given a digraph D = (V , A) as well as l, u ∈ QA, find x ∈ QV and p ∈ ZA such that ∀ a = (i, j) ∈ A : 0 ≤ (xj − xi − la) mod T ≤ ua − la ⇔ ∀ a = (i, j) ∈ A : 0 ≤ xj − xi − la − T max{z ∈ Z | xj − xi − la − z T ≥ 0} ⇔ ∀ a = (i, j) ∈ A : la ≤ xj − xi + T pa ≤ ua. Theorem (Odijk, 1994) PESP is NP-complete. Theorem (e. g. Liebchen, 2006) For any fixed p ∈ ZA PESP can be solved in O(|V ||A|). Annotation (Liebchen and Möhring, 2007) The PESP comprises rich modeling capabilities in the context of railway timetable design.

Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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Periodic Event Scheduling

Additional Structure Given a set of robots R, we consider digraphs D = (V , A) with

Tobias Hofmann Algorithmic and Discrete Mathematics 7 | 14

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Periodic Event Scheduling

Additional Structure Given a set of robots R, we consider digraphs D = (V , A) with ◾ a decomposition V = ·

r∈R

Vr of V into subsets Vr each inducing a chordless directed cycle C(Vr) and for C(V1) C(V2)

Tobias Hofmann Algorithmic and Discrete Mathematics 7 | 14

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Periodic Event Scheduling

Additional Structure Given a set of robots R, we consider digraphs D = (V , A) with ◾ a decomposition V = ·

r∈R

Vr of V into subsets Vr each inducing a chordless directed cycle C(Vr) and for ◾ a set of conflicts S = {{(u1, u2), (v1, v2)} | ∃ r1, r2 ∈ R : u1, v2 ∈ Vr1 ∧ u2, v1 ∈ Vr2} C(V1) C(V2)

Tobias Hofmann Algorithmic and Discrete Mathematics 7 | 14

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Periodic Event Scheduling

Additional Structure Given a set of robots R, we consider digraphs D = (V , A) with ◾ a decomposition V = ·

r∈R

Vr of V into subsets Vr each inducing a chordless directed cycle C(Vr) and for ◾ a set of conflicts S = {{(u1, u2), (v1, v2)} | ∃ r1, r2 ∈ R : u1, v2 ∈ Vr1 ∧ u2, v1 ∈ Vr2} a decomposition A = ·

r∈R

C(Vr) ∪ · {a | ∃ s ∈ S : a ∈ s}. C(V1) C(V2)

Tobias Hofmann Algorithmic and Discrete Mathematics 7 | 14

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Periodic Event Scheduling

Additional Structure Given a set of robots R, we consider digraphs D = (V , A) with ◾ a decomposition V = ·

r∈R

Vr of V into subsets Vr each inducing a chordless directed cycle C(Vr) and for ◾ a set of conflicts S = {{(u1, u2), (v1, v2)} | ∃ r1, r2 ∈ R : u1, v2 ∈ Vr1 ∧ u2, v1 ∈ Vr2} a decomposition A = ·

r∈R

C(Vr) ∪ · {a | ∃ s ∈ S : a ∈ s}. Observation For every conflict s ∈ S there is a unique directed cycle D(s) using the arcs of s, but no arcs

  • f any other conflict.

Tobias Hofmann Algorithmic and Discrete Mathematics 7 | 14

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Periodic Event Scheduling

Additional Structure Given a set of robots R, we consider digraphs D = (V , A) with ◾ a decomposition V = ·

r∈R

Vr of V into subsets Vr each inducing a chordless directed cycle C(Vr) and for ◾ a set of conflicts S = {{(u1, u2), (v1, v2)} | ∃ r1, r2 ∈ R : u1, v2 ∈ Vr1 ∧ u2, v1 ∈ Vr2} a decomposition A = ·

r∈R

C(Vr) ∪ · {a | ∃ s ∈ S : a ∈ s}. Observation For every conflict s ∈ S there is a unique directed cycle D(s) using the arcs of s, but no arcs

  • f any other conflict. Proof.

Tobias Hofmann Algorithmic and Discrete Mathematics 7 | 14

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SLIDE 73

Periodic Event Scheduling

Additional Structure Given a set of robots R, we consider digraphs D = (V , A) with ◾ a decomposition V = ·

r∈R

Vr of V into subsets Vr each inducing a chordless directed cycle C(Vr) and for ◾ a set of conflicts S = {{(u1, u2), (v1, v2)} | ∃ r1, r2 ∈ R : u1, v2 ∈ Vr1 ∧ u2, v1 ∈ Vr2} a decomposition A = ·

r∈R

C(Vr) ∪ · {a | ∃ s ∈ S : a ∈ s}. Observation For every conflict s ∈ S there is a unique directed cycle D(s) using the arcs of s, but no arcs

  • f any other conflict. Proof.
  • Tobias Hofmann

Algorithmic and Discrete Mathematics 7 | 14

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Periodic Event Scheduling

ISO-PESP (Node Potential Formulation) Given a set of robots R and a set of conflicts S as well as the arising digraph D = (V = ·

r∈R

Vr, A = ·

r∈R

C(Vr) ∪ · {a | ∃ s ∈ S : a ∈ s}) with l, u ∈ QA,

Tobias Hofmann Algorithmic and Discrete Mathematics 8 | 14

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Periodic Event Scheduling

ISO-PESP (Node Potential Formulation) Given a set of robots R and a set of conflicts S as well as the arising digraph D = (V = ·

r∈R

Vr, A = ·

r∈R

C(Vr) ∪ · {a | ∃ s ∈ S : a ∈ s}) with l, u ∈ QA, minimize T

Tobias Hofmann Algorithmic and Discrete Mathematics 8 | 14

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SLIDE 76

Periodic Event Scheduling

ISO-PESP (Node Potential Formulation) Given a set of robots R and a set of conflicts S as well as the arising digraph D = (V = ·

r∈R

Vr, A = ·

r∈R

C(Vr) ∪ · {a | ∃ s ∈ S : a ∈ s}) with l, u ∈ QA, minimize T

  • s. t.

la ≤ xj − xi + T pa ≤ ua ∀ a = (i, j) ∈ A,

Tobias Hofmann Algorithmic and Discrete Mathematics 8 | 14

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SLIDE 77

Periodic Event Scheduling

ISO-PESP (Node Potential Formulation) Given a set of robots R and a set of conflicts S as well as the arising digraph D = (V = ·

r∈R

Vr, A = ·

r∈R

C(Vr) ∪ · {a | ∃ s ∈ S : a ∈ s}) with l, u ∈ QA, minimize T

  • s. t.

la ≤ xj − xi + T pa ≤ ua ∀ a = (i, j) ∈ A,

  • a∈C(Vr )

pa = 1 ∀ r ∈ R,

Tobias Hofmann Algorithmic and Discrete Mathematics 8 | 14

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SLIDE 78

Periodic Event Scheduling

ISO-PESP (Node Potential Formulation) Given a set of robots R and a set of conflicts S as well as the arising digraph D = (V = ·

r∈R

Vr, A = ·

r∈R

C(Vr) ∪ · {a | ∃ s ∈ S : a ∈ s}) with l, u ∈ QA, minimize T

  • s. t.

la ≤ xj − xi + T pa ≤ ua ∀ a = (i, j) ∈ A,

  • a∈C(Vr )

pa = 1 ∀ r ∈ R,

  • a∈D(s)

pa = 1 ∀ s ∈ S,

Tobias Hofmann Algorithmic and Discrete Mathematics 8 | 14

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SLIDE 79

Periodic Event Scheduling

ISO-PESP (Node Potential Formulation) Given a set of robots R and a set of conflicts S as well as the arising digraph D = (V = ·

r∈R

Vr, A = ·

r∈R

C(Vr) ∪ · {a | ∃ s ∈ S : a ∈ s}) with l, u ∈ QA, minimize T

  • s. t.

la ≤ xj − xi + T pa ≤ ua ∀ a = (i, j) ∈ A,

  • a∈C(Vr )

pa = 1 ∀ r ∈ R,

  • a∈D(s)

pa = 1 ∀ s ∈ S, (x, p) ∈ QV × ZA.

Tobias Hofmann Algorithmic and Discrete Mathematics 8 | 14

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Periodic Event Scheduling

Definition. Given a PESP instance D = (V , A) and a node potential x ∈ QV , then y ∈ QA is called periodic tension, if ∃ p ∈ ZA : ∀ a = (i, j) ∈ A : ya = xj − xi + T pa.

Tobias Hofmann Algorithmic and Discrete Mathematics 9 | 14

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SLIDE 81

Periodic Event Scheduling

Definition. Given a PESP instance D = (V , A) and a node potential x ∈ QV , then y ∈ QA is called periodic tension, if ∃ p ∈ ZA : ∀ a = (i, j) ∈ A : ya = xj − xi + T pa. Lemma (Cycle Periodicity Property, e. g. Liebchen, 2006). Given a PESP instance D = (V , A) as well as a periodic tension y ∈ QA, then ∀ cycles C ∈ D : ∃ pC ∈ Z :

  • a∈C+

ya −

a∈C−

ya = T pC.

Tobias Hofmann Algorithmic and Discrete Mathematics 9 | 14

slide-82
SLIDE 82

Periodic Event Scheduling

Definition. Given a PESP instance D = (V , A) and a node potential x ∈ QV , then y ∈ QA is called periodic tension, if ∃ p ∈ ZA : ∀ a = (i, j) ∈ A : ya = xj − xi + T pa. Lemma (Cycle Periodicity Property, e. g. Liebchen, 2006). Given a PESP instance D = (V , A) as well as a periodic tension y ∈ QA, then ∀ cycles C ∈ D : ∃ pC ∈ Z :

  • a∈C+

ya −

a∈C−

ya = T pC.

  • Proof. Given a cycle C ∈ D,
  • a∈C+

ya −

a∈C−

ya

Tobias Hofmann Algorithmic and Discrete Mathematics 9 | 14

slide-83
SLIDE 83

Periodic Event Scheduling

Definition. Given a PESP instance D = (V , A) and a node potential x ∈ QV , then y ∈ QA is called periodic tension, if ∃ p ∈ ZA : ∀ a = (i, j) ∈ A : ya = xj − xi + T pa. Lemma (Cycle Periodicity Property, e. g. Liebchen, 2006). Given a PESP instance D = (V , A) as well as a periodic tension y ∈ QA, then ∀ cycles C ∈ D : ∃ pC ∈ Z :

  • a∈C+

ya −

a∈C−

ya = T pC.

  • Proof. Given a cycle C ∈ D,
  • a∈C+

ya −

a∈C−

ya =

a=(i,j)∈C+

(xj − xi + T pa) −

a=(i,j)∈C−

(xj − xi + T pa)

Tobias Hofmann Algorithmic and Discrete Mathematics 9 | 14

slide-84
SLIDE 84

Periodic Event Scheduling

Definition. Given a PESP instance D = (V , A) and a node potential x ∈ QV , then y ∈ QA is called periodic tension, if ∃ p ∈ ZA : ∀ a = (i, j) ∈ A : ya = xj − xi + T pa. Lemma (Cycle Periodicity Property, e. g. Liebchen, 2006). Given a PESP instance D = (V , A) as well as a periodic tension y ∈ QA, then ∀ cycles C ∈ D : ∃ pC ∈ Z :

  • a∈C+

ya −

a∈C−

ya = T pC.

  • Proof. Given a cycle C ∈ D,
  • a∈C+

ya −

a∈C−

ya =

a=(i,j)∈C+

(xj − xi + T pa) −

a=(i,j)∈C−

(xj − xi + T pa) = T

  • a∈C+

pa −

a∈C−

pa

  • Tobias Hofmann

Algorithmic and Discrete Mathematics 9 | 14

slide-85
SLIDE 85

Periodic Event Scheduling

Definition. Given a PESP instance D = (V , A) and a node potential x ∈ QV , then y ∈ QA is called periodic tension, if ∃ p ∈ ZA : ∀ a = (i, j) ∈ A : ya = xj − xi + T pa. Lemma (Cycle Periodicity Property, e. g. Liebchen, 2006). Given a PESP instance D = (V , A) as well as a periodic tension y ∈ QA, then ∀ cycles C ∈ D : ∃ pC ∈ Z :

  • a∈C+

ya −

a∈C−

ya = T pC.

  • Proof. Given a cycle C ∈ D,
  • a∈C+

ya −

a∈C−

ya =

a=(i,j)∈C+

(xj − xi + T pa) −

a=(i,j)∈C−

(xj − xi + T pa) = T

  • a∈C+

pa −

a∈C−

pa

  • = T pC.
  • Tobias Hofmann

Algorithmic and Discrete Mathematics 9 | 14

slide-86
SLIDE 86

Periodic Event Scheduling

Corollary. Given an instance of ISO-PESP as well as a periodic tension y, then (i) ∀ r ∈ R :

  • a∈C(Vr )

ya = T. (ii) ∀ s ∈ S :

  • a∈C(s)

ya = T.

  • Proof. (i) [(ii) analogous] Given an r ∈ R, we write C = C(Vr).

Tobias Hofmann Algorithmic and Discrete Mathematics 10 | 14

slide-87
SLIDE 87

Periodic Event Scheduling

Corollary. Given an instance of ISO-PESP as well as a periodic tension y, then (i) ∀ r ∈ R :

  • a∈C(Vr )

ya = T. (ii) ∀ s ∈ S :

  • a∈C(s)

ya = T.

  • Proof. (i) [(ii) analogous] Given an r ∈ R, we write C = C(Vr).
  • a∈C

ya

Tobias Hofmann Algorithmic and Discrete Mathematics 10 | 14

slide-88
SLIDE 88

Periodic Event Scheduling

Corollary. Given an instance of ISO-PESP as well as a periodic tension y, then (i) ∀ r ∈ R :

  • a∈C(Vr )

ya = T. (ii) ∀ s ∈ S :

  • a∈C(s)

ya = T.

  • Proof. (i) [(ii) analogous] Given an r ∈ R, we write C = C(Vr).
  • a∈C

ya =

a∈C+

ya −

a∈C−

ya

= 0 Tobias Hofmann Algorithmic and Discrete Mathematics 10 | 14

slide-89
SLIDE 89

Periodic Event Scheduling

Corollary. Given an instance of ISO-PESP as well as a periodic tension y, then (i) ∀ r ∈ R :

  • a∈C(Vr )

ya = T. (ii) ∀ s ∈ S :

  • a∈C(s)

ya = T.

  • Proof. (i) [(ii) analogous] Given an r ∈ R, we write C = C(Vr).
  • a∈C

ya =

a∈C+

ya −

a∈C−

ya

= 0

= T

  

  • a∈C+

pa −

a∈C−

pa

  

Tobias Hofmann Algorithmic and Discrete Mathematics 10 | 14

slide-90
SLIDE 90

Periodic Event Scheduling

Corollary. Given an instance of ISO-PESP as well as a periodic tension y, then (i) ∀ r ∈ R :

  • a∈C(Vr )

ya = T. (ii) ∀ s ∈ S :

  • a∈C(s)

ya = T.

  • Proof. (i) [(ii) analogous] Given an r ∈ R, we write C = C(Vr).
  • a∈C

ya =

a∈C+

ya −

a∈C−

ya

= 0

= T

  

  • a∈C+

pa

= 1

−

a∈C−

pa

= 0

   = T.

  • Tobias Hofmann

Algorithmic and Discrete Mathematics 10 | 14

slide-91
SLIDE 91

Periodic Event Scheduling

Corollary. Given an instance of ISO-PESP as well as a periodic tension y, then (i) ∀ r ∈ R :

  • a∈C(Vr )

ya = T. (ii) ∀ s ∈ S :

  • a∈C(s)

ya = T.

  • Proof. (i) [(ii) analogous] Given an r ∈ R, we write C = C(Vr).
  • a∈C

ya =

a∈C+

ya −

a∈C−

ya

= 0

= T

  

  • a∈C+

pa

= 1

−

a∈C−

pa

= 0

   = T.

  • Theorem (Nachtigall, 1994)

To describe a connected PESP instance it suffices to require the cycle periodicity property for ν = |A| − |V | + 1 fundamental cycles.

Tobias Hofmann Algorithmic and Discrete Mathematics 10 | 14

slide-92
SLIDE 92

Periodic Event Scheduling

ISO-PESP (Cycle Periodicity Formulation) Given a set of robots R and a set of conflicts S as well as the arising digraph D = (V = ·

r∈R

Vr, A = ·

r∈R

C(Vr) ∪ · {a | ∃ s ∈ S : a ∈ s}) with l, u ∈ QA,

Tobias Hofmann Algorithmic and Discrete Mathematics 11 | 14

slide-93
SLIDE 93

Periodic Event Scheduling

ISO-PESP (Cycle Periodicity Formulation) Given a set of robots R and a set of conflicts S as well as the arising digraph D = (V = ·

r∈R

Vr, A = ·

r∈R

C(Vr) ∪ · {a | ∃ s ∈ S : a ∈ s}) with l, u ∈ QA, minimize T

Tobias Hofmann Algorithmic and Discrete Mathematics 11 | 14

slide-94
SLIDE 94

Periodic Event Scheduling

ISO-PESP (Cycle Periodicity Formulation) Given a set of robots R and a set of conflicts S as well as the arising digraph D = (V = ·

r∈R

Vr, A = ·

r∈R

C(Vr) ∪ · {a | ∃ s ∈ S : a ∈ s}) with l, u ∈ QA, minimize T

  • s. t.
  • a∈C(Vr )

ya = T ∀ r ∈ R,

Tobias Hofmann Algorithmic and Discrete Mathematics 11 | 14

slide-95
SLIDE 95

Periodic Event Scheduling

ISO-PESP (Cycle Periodicity Formulation) Given a set of robots R and a set of conflicts S as well as the arising digraph D = (V = ·

r∈R

Vr, A = ·

r∈R

C(Vr) ∪ · {a | ∃ s ∈ S : a ∈ s}) with l, u ∈ QA, minimize T

  • s. t.
  • a∈C(Vr )

ya = T ∀ r ∈ R,

  • a∈C(s)

ya = T ∀ s ∈ S,

Tobias Hofmann Algorithmic and Discrete Mathematics 11 | 14

slide-96
SLIDE 96

Periodic Event Scheduling

ISO-PESP (Cycle Periodicity Formulation) Given a set of robots R and a set of conflicts S as well as the arising digraph D = (V = ·

r∈R

Vr, A = ·

r∈R

C(Vr) ∪ · {a | ∃ s ∈ S : a ∈ s}) with l, u ∈ QA, minimize T

  • s. t.
  • a∈C(Vr )

ya = T ∀ r ∈ R,

  • a∈C(s)

ya = T ∀ s ∈ S,

  • a∈C+

ya −

a∈C−

ya = T pC, ∀ ν − (|R| + |S|) fundamental cycles C ∈ B, where B is an appropriately chosen subset of a cycle basis.

Tobias Hofmann Algorithmic and Discrete Mathematics 11 | 14

slide-97
SLIDE 97

Periodic Event Scheduling

ISO-PESP (Cycle Periodicity Formulation) Given a set of robots R and a set of conflicts S as well as the arising digraph D = (V = ·

r∈R

Vr, A = ·

r∈R

C(Vr) ∪ · {a | ∃ s ∈ S : a ∈ s}) with l, u ∈ QA, minimize T

  • s. t.
  • a∈C(Vr )

ya = T ∀ r ∈ R,

  • a∈C(s)

ya = T ∀ s ∈ S,

  • a∈C+

ya −

a∈C−

ya = T pC, ∀ ν − (|R| + |S|) fundamental cycles C ∈ B, (y, p) ∈ QV × Zν−(|R|+|S|), where B is an appropriately chosen subset of a cycle basis.

Tobias Hofmann Algorithmic and Discrete Mathematics 11 | 14

slide-98
SLIDE 98

Appropriate Cycle Bases

slide-99
SLIDE 99

Appropriate Cycle Bases How many further cycles do we have to choose?

Tobias Hofmann Algorithmic and Discrete Mathematics 12 | 14

slide-100
SLIDE 100

Appropriate Cycle Bases How many further cycles do we have to choose?

Theorem Given a connected ISO-PESP instance, ν = |A| − |V | + 1 ≥ |R| + |S|.

Tobias Hofmann Algorithmic and Discrete Mathematics 12 | 14

slide-101
SLIDE 101

Appropriate Cycle Bases How many further cycles do we have to choose?

Theorem Given a connected ISO-PESP instance, ν = |A| − |V | + 1 ≥ |R| + |S|. Proof. |R| + |S|

Tobias Hofmann Algorithmic and Discrete Mathematics 12 | 14

slide-102
SLIDE 102

Appropriate Cycle Bases How many further cycles do we have to choose?

Theorem Given a connected ISO-PESP instance, ν = |A| − |V | + 1 ≥ |R| + |S|. Proof. |R| + |S| ≤ 2|S| + 1

Tobias Hofmann Algorithmic and Discrete Mathematics 12 | 14

slide-103
SLIDE 103

Appropriate Cycle Bases How many further cycles do we have to choose?

Theorem Given a connected ISO-PESP instance, ν = |A| − |V | + 1 ≥ |R| + |S|. Proof. |R| + |S| ≤ 2|S| + 1 = |{a | ∃ s ∈ S : a ∈ s}| +

  • ·

r∈R

Vr

  • −
  • ·

r∈R

C(Vr)

  • = 0

+ 1

Tobias Hofmann Algorithmic and Discrete Mathematics 12 | 14

slide-104
SLIDE 104

Appropriate Cycle Bases How many further cycles do we have to choose?

Theorem Given a connected ISO-PESP instance, ν = |A| − |V | + 1 ≥ |R| + |S|. Proof. |R| + |S| ≤ 2|S| + 1 = |{a | ∃ s ∈ S : a ∈ s}| +

  • ·

r∈R

Vr

  • −
  • ·

r∈R

C(Vr)

  • = 0

+ 1 = |A| − |V | + 1.

  • Tobias Hofmann

Algorithmic and Discrete Mathematics 12 | 14

slide-105
SLIDE 105

Appropriate Cycle Bases Which cycles should we choose?

◾ The cycles of a cycle basis have to be linearly independent.

Tobias Hofmann Algorithmic and Discrete Mathematics 13 | 14

slide-106
SLIDE 106

Appropriate Cycle Bases Which cycles should we choose?

◾ The cycles of a cycle basis have to be linearly independent. ◾ We should choose cycles C for which we have good bounds on pC.

Tobias Hofmann Algorithmic and Discrete Mathematics 13 | 14

slide-107
SLIDE 107

Appropriate Cycle Bases Which cycles should we choose?

◾ The cycles of a cycle basis have to be linearly independent. ◾ We should choose cycles C for which we have good bounds on pC. C(V1) C(V2) D(s1) D(s2)

Tobias Hofmann Algorithmic and Discrete Mathematics 13 | 14

slide-108
SLIDE 108

Appropriate Cycle Bases Which cycles should we choose?

◾ The cycles of a cycle basis have to be linearly independent. ◾ We should choose cycles C for which we have good bounds on pC. C(V1) C(V2) D(s1) D(s2) C ⇒ C linearly independent and pC ∈ {−1, 0, 1}

Tobias Hofmann Algorithmic and Discrete Mathematics 13 | 14

slide-109
SLIDE 109

Appropriate Cycle Bases Which cycles should we choose?

◾ The cycles of a cycle basis have to be linearly independent. ◾ We should choose cycles C for which we have good bounds on pC. C(V1) C(V2) D(s1) D(s2) C ⇒ C linearly independent and pC ∈ {−1, 0, 1} ◾ If pC is bounded, we can use big M methods to replace the nonlinear term T pC in the cycle periodicity constraints.

Tobias Hofmann Algorithmic and Discrete Mathematics 13 | 14

slide-110
SLIDE 110

In a Nutshell

slide-111
SLIDE 111

In a Nutshell

✓ Precedence Relationships

Tobias Hofmann Algorithmic and Discrete Mathematics 14 | 14

slide-112
SLIDE 112

In a Nutshell

✓ Precedence Relationships ✓ ISO-PESP Formulation

Tobias Hofmann Algorithmic and Discrete Mathematics 14 | 14

slide-113
SLIDE 113

In a Nutshell

✓ Precedence Relationships ✓ ISO-PESP Formulation ✓ Solved Practical Data Sets

Tobias Hofmann Algorithmic and Discrete Mathematics 14 | 14

slide-114
SLIDE 114

In a Nutshell

✓ Precedence Relationships ✓ ISO-PESP Formulation ✓ Solved Practical Data Sets ➟ Instance Generator

Tobias Hofmann Algorithmic and Discrete Mathematics 14 | 14

slide-115
SLIDE 115

In a Nutshell

✓ Precedence Relationships ✓ ISO-PESP Formulation ✓ Solved Practical Data Sets ➟ Instance Generator ➟ Stochastic Variants

Tobias Hofmann Algorithmic and Discrete Mathematics 14 | 14

slide-116
SLIDE 116

In a Nutshell

✓ Precedence Relationships ✓ ISO-PESP Formulation ✓ Solved Practical Data Sets ➟ Instance Generator ➟ Stochastic Variants ➟ Max-Plus Algebra Formulations

Tobias Hofmann Algorithmic and Discrete Mathematics 14 | 14

slide-117
SLIDE 117

Our viRAL Project Partners

slide-118
SLIDE 118

Literature

[1] Karp, Richard M. Reducibility among Combinatorial Problems. Complexity of Computer Computations, pages 85–103, 1972. [2] Liebchen, Christian. Periodic Timetable Optimization in Public Transport. dissertation.de, 2006. [3] Liebchen, Christian and Möhring, Rolf H. The Modeling Power of the Periodic Event Scheduling Problem: Railway Timetables and Beyond. Algorithmic Methods for Railway Optimization, pages 3–40, 2007. [4] Nachtigall, Karl. A Branch-And-Cut Approach for Periodic Network Programming. Hildesheimer Informatik Berichte 29, 1994. [5] Odijk, Michiel A. Construction of Periodic Timetables, Part 1: A Cutting Plane Algorithm. Technical Report 94-61, TU Delft, 1994. [6] Serafini, Paul and Ukovich Walter. A Mathematical Model for Periodic Scheduling Problems. SIAM Journal on Discrete Mathematics, 2(4):550–581, 1989.