Periodic Event Scheduling and its Industrial Application Tobias - PowerPoint PPT Presentation

Precedence Relationships Conflicts 7 4 6 8 3 5 9 2 1 10 1 2 3 4 5 6 7 8 9 10 Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

Precedence Relationships Conflicts 7 4 6 8 3 5 9 2 1 10 1 2 3 4 5 6 7 8 9 10 Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

Precedence Relationships Conflicts 7 4 6 8 3 5 9 2 1 10 1 2 3 4 5 6 7 8 9 10 Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

Precedence Relationships Conflicts 7 4 6 8 3 5 9 2 1 10 1 2 3 4 5 6 7 8 9 10 Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

Precedence Relationships Conflicts 7 4 6 8 3 5 9 2 1 10 1 2 3 4 5 6 7 8 9 10 Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

Precedence Relationships Conflicts 7 4 6 8 3 5 9 2 1 10 1 2 3 4 5 6 7 8 9 10 Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

Periodic Event Scheduling

Periodic Event Scheduling Periodic Event Scheduling Problem [PESP] (Serafini and Ukovich, 1989) Given a digraph D = ( V , A ) as well as l , u ∈ Q A , find x ∈ Q V such that ∀ a = ( i , j ) ∈ A : 0 ≤ ( x j − x i − l a ) mod T ≤ u a − l a Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

Periodic Event Scheduling Periodic Event Scheduling Problem [PESP] (Serafini and Ukovich, 1989) Given a digraph D = ( V , A ) as well as l , u ∈ Q A , find x ∈ Q V such that ∀ a = ( i , j ) ∈ A : 0 ≤ ( x j − x i − l a ) mod T ≤ u a − l a Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

Periodic Event Scheduling Periodic Event Scheduling Problem [PESP] (Serafini and Ukovich, 1989) Given a digraph D = ( V , A ) as well as l , u ∈ Q A , find x ∈ Q V such that ∀ a = ( i , j ) ∈ A : 0 ≤ ( x j − x i − l a ) mod T ≤ u a − l a ⇔ ∀ a = ( i , j ) ∈ A : 0 ≤ x j − x i − l a − T max { z ∈ Z | x j − x i − l a − z T ≥ 0 } Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

Periodic Event Scheduling Periodic Event Scheduling Problem [PESP] (Serafini and Ukovich, 1989) Given a digraph D = ( V , A ) as well as l , u ∈ Q A , find x ∈ Q V such that ∀ a = ( i , j ) ∈ A : 0 ≤ ( x j − x i − l a ) mod T ≤ u a − l a ⇔ ∀ a = ( i , j ) ∈ A : 0 ≤ x j − x i − l a − T max { z ∈ Z | x j − x i − l a − z T ≥ 0 } Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

Periodic Event Scheduling Periodic Event Scheduling Problem [PESP] (Serafini and Ukovich, 1989) Given a digraph D = ( V , A ) as well as l , u ∈ Q A , find x ∈ Q V and p ∈ Z A such that ∀ a = ( i , j ) ∈ A : 0 ≤ ( x j − x i − l a ) mod T ≤ u a − l a ⇔ ∀ a = ( i , j ) ∈ A : 0 ≤ x j − x i − l a − T max { z ∈ Z | x j − x i − l a − z T ≥ 0 } ⇔ ∀ a = ( i , j ) ∈ A : l a ≤ x j − x i + T p a ≤ u a . Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

Periodic Event Scheduling Periodic Event Scheduling Problem [PESP] (Serafini and Ukovich, 1989) Given a digraph D = ( V , A ) as well as l , u ∈ Q A , find x ∈ Q V and p ∈ Z A such that ∀ a = ( i , j ) ∈ A : 0 ≤ ( x j − x i − l a ) mod T ≤ u a − l a ⇔ ∀ a = ( i , j ) ∈ A : 0 ≤ x j − x i − l a − T max { z ∈ Z | x j − x i − l a − z T ≥ 0 } ⇔ ∀ a = ( i , j ) ∈ A : l a ≤ x j − x i + T p a ≤ u a . Theorem (Odijk, 1994) PESP is NP-complete. Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

Periodic Event Scheduling Periodic Event Scheduling Problem [PESP] (Serafini and Ukovich, 1989) Given a digraph D = ( V , A ) as well as l , u ∈ Q A , find x ∈ Q V and p ∈ Z A such that ∀ a = ( i , j ) ∈ A : 0 ≤ ( x j − x i − l a ) mod T ≤ u a − l a ⇔ ∀ a = ( i , j ) ∈ A : 0 ≤ x j − x i − l a − T max { z ∈ Z | x j − x i − l a − z T ≥ 0 } ⇔ ∀ a = ( i , j ) ∈ A : l a ≤ x j − x i + T p a ≤ u a . Theorem (Odijk, 1994) PESP is NP-complete. Theorem (e. g. Liebchen, 2006) For any fixed p ∈ Z A PESP can be solved in O ( | V || A | ). Tobias Hofmann Algorithmic and Discrete Mathematics 6 | 14

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

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

Periodic Event Scheduling Additional Structure Given a set of robots R , we consider digraphs D = ( V , A ) with a decomposition V = � V r of V into subsets V r each inducing a chordless directed ◾ · r ∈ R cycle C ( V r ) and for C ( V 1 ) C ( V 2 ) Tobias Hofmann Algorithmic and Discrete Mathematics 7 | 14

Periodic Event Scheduling Additional Structure Given a set of robots R , we consider digraphs D = ( V , A ) with a decomposition V = � V r of V into subsets V r each inducing a chordless directed ◾ · r ∈ R cycle C ( V r ) and for a set of conflicts S = {{ ( u 1 , u 2 ) , ( v 1 , v 2 ) } | ∃ r 1 , r 2 ∈ R : u 1 , v 2 ∈ V r 1 ∧ u 2 , v 1 ∈ V r 2 } ◾ C ( V 1 ) C ( V 2 ) Tobias Hofmann Algorithmic and Discrete Mathematics 7 | 14

Periodic Event Scheduling Additional Structure Given a set of robots R , we consider digraphs D = ( V , A ) with a decomposition V = � V r of V into subsets V r each inducing a chordless directed ◾ · r ∈ R cycle C ( V r ) and for a set of conflicts S = {{ ( u 1 , u 2 ) , ( v 1 , v 2 ) } | ∃ r 1 , r 2 ∈ R : u 1 , v 2 ∈ V r 1 ∧ u 2 , v 1 ∈ V r 2 } ◾ a decomposition A = � C ( V r ) ∪ · { a | ∃ s ∈ S : a ∈ s } . · r ∈ R C ( V 1 ) C ( V 2 ) Tobias Hofmann Algorithmic and Discrete Mathematics 7 | 14

Periodic Event Scheduling Additional Structure Given a set of robots R , we consider digraphs D = ( V , A ) with a decomposition V = � V r of V into subsets V r each inducing a chordless directed ◾ · r ∈ R cycle C ( V r ) and for a set of conflicts S = {{ ( u 1 , u 2 ) , ( v 1 , v 2 ) } | ∃ r 1 , r 2 ∈ R : u 1 , v 2 ∈ V r 1 ∧ u 2 , v 1 ∈ V r 2 } ◾ a decomposition A = � C ( V r ) ∪ · { a | ∃ s ∈ S : a ∈ s } . · r ∈ R Observation For every conflict s ∈ S there is a unique directed cycle D ( s ) using the arcs of s , but no arcs of any other conflict. Tobias Hofmann Algorithmic and Discrete Mathematics 7 | 14

Periodic Event Scheduling Additional Structure Given a set of robots R , we consider digraphs D = ( V , A ) with a decomposition V = � V r of V into subsets V r each inducing a chordless directed ◾ · r ∈ R cycle C ( V r ) and for a set of conflicts S = {{ ( u 1 , u 2 ) , ( v 1 , v 2 ) } | ∃ r 1 , r 2 ∈ R : u 1 , v 2 ∈ V r 1 ∧ u 2 , v 1 ∈ V r 2 } ◾ a decomposition A = � C ( V r ) ∪ · { a | ∃ s ∈ S : a ∈ s } . · r ∈ R Observation For every conflict s ∈ S there is a unique directed cycle D ( s ) using the arcs of s , but no arcs of any other conflict. Proof. Tobias Hofmann Algorithmic and Discrete Mathematics 7 | 14

Periodic Event Scheduling Additional Structure Given a set of robots R , we consider digraphs D = ( V , A ) with a decomposition V = � V r of V into subsets V r each inducing a chordless directed ◾ · r ∈ R cycle C ( V r ) and for a set of conflicts S = {{ ( u 1 , u 2 ) , ( v 1 , v 2 ) } | ∃ r 1 , r 2 ∈ R : u 1 , v 2 ∈ V r 1 ∧ u 2 , v 1 ∈ V r 2 } ◾ a decomposition A = � C ( V r ) ∪ · { a | ∃ s ∈ S : a ∈ s } . · r ∈ R Observation For every conflict s ∈ S there is a unique directed cycle D ( s ) using the arcs of s , but no arcs of any other conflict. Proof. � Tobias Hofmann Algorithmic and Discrete Mathematics 7 | 14

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 = � V r , A = � · { a | ∃ s ∈ S : a ∈ s } ) with l , u ∈ Q A , C ( V r ) ∪ · · r ∈ R r ∈ R Tobias Hofmann Algorithmic and Discrete Mathematics 8 | 14

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 = � V r , A = � · { a | ∃ s ∈ S : a ∈ s } ) with l , u ∈ Q A , C ( V r ) ∪ · · r ∈ R r ∈ R minimize T Tobias Hofmann Algorithmic and Discrete Mathematics 8 | 14

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 = � V r , A = � · { a | ∃ s ∈ S : a ∈ s } ) with l , u ∈ Q A , C ( V r ) ∪ · · r ∈ R r ∈ R minimize T s. t. l a ≤ x j − x i + T p a ≤ u a ∀ a = ( i , j ) ∈ A , Tobias Hofmann Algorithmic and Discrete Mathematics 8 | 14

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 = � V r , A = � · { a | ∃ s ∈ S : a ∈ s } ) with l , u ∈ Q A , C ( V r ) ∪ · · r ∈ R r ∈ R minimize T s. t. l a ≤ x j − x i + T p a ≤ u a ∀ a = ( i , j ) ∈ A , � p a = 1 ∀ r ∈ R , a ∈ C ( V r ) Tobias Hofmann Algorithmic and Discrete Mathematics 8 | 14

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 = � V r , A = � · { a | ∃ s ∈ S : a ∈ s } ) with l , u ∈ Q A , C ( V r ) ∪ · · r ∈ R r ∈ R minimize T s. t. l a ≤ x j − x i + T p a ≤ u a ∀ a = ( i , j ) ∈ A , � p a = 1 ∀ r ∈ R , a ∈ C ( V r ) � p a = 1 ∀ s ∈ S , a ∈ D ( s ) Tobias Hofmann Algorithmic and Discrete Mathematics 8 | 14

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 = � V r , A = � · { a | ∃ s ∈ S : a ∈ s } ) with l , u ∈ Q A , C ( V r ) ∪ · · r ∈ R r ∈ R minimize T s. t. l a ≤ x j − x i + T p a ≤ u a ∀ a = ( i , j ) ∈ A , � p a = 1 ∀ r ∈ R , a ∈ C ( V r ) � p a = 1 ∀ s ∈ S , a ∈ D ( s ) ( x , p ) ∈ Q V × Z A . Tobias Hofmann Algorithmic and Discrete Mathematics 8 | 14

Periodic Event Scheduling Definition. Given a PESP instance D = ( V , A ) and a node potential x ∈ Q V , then y ∈ Q A is called periodic tension , if ∃ p ∈ Z A : ∀ a = ( i , j ) ∈ A : y a = x j − x i + T p a . Tobias Hofmann Algorithmic and Discrete Mathematics 9 | 14

Periodic Event Scheduling Definition. Given a PESP instance D = ( V , A ) and a node potential x ∈ Q V , then y ∈ Q A is called periodic tension , if ∃ p ∈ Z A : ∀ a = ( i , j ) ∈ A : y a = x j − x i + T p a . Lemma (Cycle Periodicity Property, e. g. Liebchen, 2006). Given a PESP instance D = ( V , A ) as well as a periodic tension y ∈ Q A , then � y a − � ∀ cycles C ∈ D : ∃ p C ∈ Z : y a = T p C . a ∈ C + a ∈ C − Tobias Hofmann Algorithmic and Discrete Mathematics 9 | 14

Periodic Event Scheduling Definition. Given a PESP instance D = ( V , A ) and a node potential x ∈ Q V , then y ∈ Q A is called periodic tension , if ∃ p ∈ Z A : ∀ a = ( i , j ) ∈ A : y a = x j − x i + T p a . Lemma (Cycle Periodicity Property, e. g. Liebchen, 2006). Given a PESP instance D = ( V , A ) as well as a periodic tension y ∈ Q A , then � y a − � ∀ cycles C ∈ D : ∃ p C ∈ Z : y a = T p C . a ∈ C + a ∈ C − Proof. Given a cycle C ∈ D , � y a − � y a a ∈ C + a ∈ C − Tobias Hofmann Algorithmic and Discrete Mathematics 9 | 14

Periodic Event Scheduling Definition. Given a PESP instance D = ( V , A ) and a node potential x ∈ Q V , then y ∈ Q A is called periodic tension , if ∃ p ∈ Z A : ∀ a = ( i , j ) ∈ A : y a = x j − x i + T p a . Lemma (Cycle Periodicity Property, e. g. Liebchen, 2006). Given a PESP instance D = ( V , A ) as well as a periodic tension y ∈ Q A , then � y a − � ∀ cycles C ∈ D : ∃ p C ∈ Z : y a = T p C . a ∈ C + a ∈ C − Proof. Given a cycle C ∈ D , � y a − � y a = � ( x j − x i + T p a ) − � ( x j − x i + T p a ) a ∈ C + a =( i , j ) ∈ C + a ∈ C − a =( i , j ) ∈ C − Tobias Hofmann Algorithmic and Discrete Mathematics 9 | 14

Periodic Event Scheduling Definition. Given a PESP instance D = ( V , A ) and a node potential x ∈ Q V , then y ∈ Q A is called periodic tension , if ∃ p ∈ Z A : ∀ a = ( i , j ) ∈ A : y a = x j − x i + T p a . Lemma (Cycle Periodicity Property, e. g. Liebchen, 2006). Given a PESP instance D = ( V , A ) as well as a periodic tension y ∈ Q A , then � y a − � ∀ cycles C ∈ D : ∃ p C ∈ Z : y a = T p C . a ∈ C + a ∈ C − Proof. Given a cycle C ∈ D , � y a − � y a = � ( x j − x i + T p a ) − � ( x j − x i + T p a ) a ∈ C + a =( i , j ) ∈ C + a ∈ C − a =( i , j ) ∈ C − � � � p a − � = T p a a ∈ C + a ∈ C − Tobias Hofmann Algorithmic and Discrete Mathematics 9 | 14

Periodic Event Scheduling Definition. Given a PESP instance D = ( V , A ) and a node potential x ∈ Q V , then y ∈ Q A is called periodic tension , if ∃ p ∈ Z A : ∀ a = ( i , j ) ∈ A : y a = x j − x i + T p a . Lemma (Cycle Periodicity Property, e. g. Liebchen, 2006). Given a PESP instance D = ( V , A ) as well as a periodic tension y ∈ Q A , then � y a − � ∀ cycles C ∈ D : ∃ p C ∈ Z : y a = T p C . a ∈ C + a ∈ C − Proof. Given a cycle C ∈ D , � y a − � y a = � ( x j − x i + T p a ) − � ( x j − x i + T p a ) a ∈ C + a =( i , j ) ∈ C + a ∈ C − a =( i , j ) ∈ C − � � � p a − � = T p a = T p C . � a ∈ C + a ∈ C − Tobias Hofmann Algorithmic and Discrete Mathematics 9 | 14

Periodic Event Scheduling Corollary. Given an instance of ISO-PESP as well as a periodic tension y , then � (i) ∀ r ∈ R : y a = T . a ∈ C ( V r ) � (ii) ∀ s ∈ S : y a = T . a ∈ C ( s ) Proof. (i) [(ii) analogous] Given an r ∈ R , we write C = C ( V r ). Tobias Hofmann Algorithmic and Discrete Mathematics 10 | 14

Periodic Event Scheduling Corollary. Given an instance of ISO-PESP as well as a periodic tension y , then � (i) ∀ r ∈ R : y a = T . a ∈ C ( V r ) � (ii) ∀ s ∈ S : y a = T . a ∈ C ( s ) Proof. (i) [(ii) analogous] Given an r ∈ R , we write C = C ( V r ). � y a a ∈ C Tobias Hofmann Algorithmic and Discrete Mathematics 10 | 14

Periodic Event Scheduling Corollary. Given an instance of ISO-PESP as well as a periodic tension y , then � (i) ∀ r ∈ R : y a = T . a ∈ C ( V r ) � (ii) ∀ s ∈ S : y a = T . a ∈ C ( s ) Proof. (i) [(ii) analogous] Given an r ∈ R , we write C = C ( V r ). � y a = � y a − � y a a ∈ C + a ∈ C a ∈ C − = 0 Tobias Hofmann Algorithmic and Discrete Mathematics 10 | 14

Periodic Event Scheduling Corollary. Given an instance of ISO-PESP as well as a periodic tension y , then � (i) ∀ r ∈ R : y a = T . a ∈ C ( V r ) � (ii) ∀ s ∈ S : y a = T . a ∈ C ( s ) Proof. (i) [(ii) analogous] Given an r ∈ R , we write C = C ( V r ).   � y a = � y a − � � p a − �   = T y a p a   a ∈ C + a ∈ C + a ∈ C a ∈ C − a ∈ C − = 0 Tobias Hofmann Algorithmic and Discrete Mathematics 10 | 14

Periodic Event Scheduling Corollary. Given an instance of ISO-PESP as well as a periodic tension y , then � (i) ∀ r ∈ R : y a = T . a ∈ C ( V r ) � (ii) ∀ s ∈ S : y a = T . a ∈ C ( s ) Proof. (i) [(ii) analogous] Given an r ∈ R , we write C = C ( V r ).   � y a = � y a − � � − �   = T  = T . y a p a p a �  a ∈ C + a ∈ C + a ∈ C a ∈ C − a ∈ C − = 0 = 1 = 0 Tobias Hofmann Algorithmic and Discrete Mathematics 10 | 14

Periodic Event Scheduling Corollary. Given an instance of ISO-PESP as well as a periodic tension y , then � (i) ∀ r ∈ R : y a = T . a ∈ C ( V r ) � (ii) ∀ s ∈ S : y a = T . a ∈ C ( s ) Proof. (i) [(ii) analogous] Given an r ∈ R , we write C = C ( V r ).   � y a = � y a − � � − �   = T  = T . y a p a p a �  a ∈ C + a ∈ C + a ∈ C a ∈ C − a ∈ C − = 0 = 1 = 0 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

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 = � V r , A = � · { a | ∃ s ∈ S : a ∈ s } ) with l , u ∈ Q A , C ( V r ) ∪ · · r ∈ R r ∈ R Tobias Hofmann Algorithmic and Discrete Mathematics 11 | 14

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 = � V r , A = � · { a | ∃ s ∈ S : a ∈ s } ) with l , u ∈ Q A , C ( V r ) ∪ · · r ∈ R r ∈ R minimize T Tobias Hofmann Algorithmic and Discrete Mathematics 11 | 14

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 = � V r , A = � · { a | ∃ s ∈ S : a ∈ s } ) with l , u ∈ Q A , C ( V r ) ∪ · · r ∈ R r ∈ R minimize T � s. t. y a = T ∀ r ∈ R , a ∈ C ( V r ) Tobias Hofmann Algorithmic and Discrete Mathematics 11 | 14

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 = � V r , A = � · { a | ∃ s ∈ S : a ∈ s } ) with l , u ∈ Q A , C ( V r ) ∪ · · r ∈ R r ∈ R minimize T � s. t. y a = T ∀ r ∈ R , a ∈ C ( V r ) � y a = T ∀ s ∈ S , a ∈ C ( s ) Tobias Hofmann Algorithmic and Discrete Mathematics 11 | 14

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 = � V r , A = � · { a | ∃ s ∈ S : a ∈ s } ) with l , u ∈ Q A , C ( V r ) ∪ · · r ∈ R r ∈ R minimize T � s. t. y a = T ∀ r ∈ R , a ∈ C ( V r ) � y a = T ∀ s ∈ S , a ∈ C ( s ) � y a − � y a = T p C , ∀ ν − ( | R | + | S | ) fundamental cycles C ∈ B , a ∈ C + a ∈ C − where B is an appropriately chosen subset of a cycle basis. Tobias Hofmann Algorithmic and Discrete Mathematics 11 | 14

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 = � V r , A = � · { a | ∃ s ∈ S : a ∈ s } ) with l , u ∈ Q A , C ( V r ) ∪ · · r ∈ R r ∈ R minimize T � s. t. y a = T ∀ r ∈ R , a ∈ C ( V r ) � y a = T ∀ s ∈ S , a ∈ C ( s ) � y a − � y a = T p C , ∀ ν − ( | R | + | S | ) fundamental cycles C ∈ B , a ∈ C + a ∈ C − ( y , p ) ∈ Q V × Z ν − ( | R | + | S | ) , where B is an appropriately chosen subset of a cycle basis. Tobias Hofmann Algorithmic and Discrete Mathematics 11 | 14

Appropriate Cycle Bases

Appropriate Cycle Bases How many further cycles do we have to choose? Tobias Hofmann Algorithmic and Discrete Mathematics 12 | 14

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