Separation of Series Constraints for One-Machine Scheduling with - PowerPoint PPT Presentation

Constraints for Q Parallel Constraints Parallel Constraints Suppose that we schedule the jobs in order 1 , 2 , . . . , n with no idle time. Then it is easy to see that C j in this schedule is p 1 + p 2 + · · · + p j = P j . Therefore � � � p j C j = p j P j = p i p j . j ∈ J j ∈ J i ≤ j ∈ J Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 8 / 23

Constraints for Q Parallel Constraints Parallel Constraints Suppose that we schedule the jobs in order 1 , 2 , . . . , n with no idle time. Then it is easy to see that C j in this schedule is p 1 + p 2 + · · · + p j = P j . Therefore � � � p j C j = p j P j = p i p j . j ∈ J j ∈ J i ≤ j ∈ J More generally, for S ⊆ J define g ( S ) = � i ≤ j ∈ S p i p j . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 8 / 23

Constraints for Q Parallel Constraints Parallel Constraints Suppose that we schedule the jobs in order 1 , 2 , . . . , n with no idle time. Then it is easy to see that C j in this schedule is p 1 + p 2 + · · · + p j = P j . Therefore � � � p j C j = p j P j = p i p j . j ∈ J j ∈ J i ≤ j ∈ J More generally, for S ⊆ J define g ( S ) = � i ≤ j ∈ S p i p j . Then Queyranne ’93 showed that for any S ⊆ J , the parallel constraint � p j x j ≥ g ( S ) j ∈ S is valid for Q . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 8 / 23

Constraints for Q Parallel Constraints Separation of Parallel Constraints Separation can be solved via min S � j ∈ S p j ¯ x j − g ( S ) : Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 9 / 23

Constraints for Q Parallel Constraints Separation of Parallel Constraints Separation can be solved via min S � j ∈ S p j ¯ x j − g ( S ) : If the value of the min is non-negative, then ¯ x satisfies all parallel constraints. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 9 / 23

Constraints for Q Parallel Constraints Separation of Parallel Constraints Separation can be solved via min S � j ∈ S p j ¯ x j − g ( S ) : If the value of the min is non-negative, then ¯ x satisfies all parallel constraints. If the value of the min is negative, then a minimizing S gives a parallel constraint violated by ¯ x . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 9 / 23

Constraints for Q Parallel Constraints Separation of Parallel Constraints Separation can be solved via min S � j ∈ S p j ¯ x j − g ( S ) : If the value of the min is non-negative, then ¯ x satisfies all parallel constraints. If the value of the min is negative, then a minimizing S gives a parallel constraint violated by ¯ x . The term min S � j ∈ S p j ¯ x j is modular, and g ( S ) is supermodular, and so the separation problem is a special case of Submodular Function Minimization (SFM), and so can be solved in polynomial time (Mc ’06). Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 9 / 23

Constraints for Q Parallel Constraints Separation of Parallel Constraints Separation can be solved via min S � j ∈ S p j ¯ x j − g ( S ) : If the value of the min is non-negative, then ¯ x satisfies all parallel constraints. If the value of the min is negative, then a minimizing S gives a parallel constraint violated by ¯ x . The term min S � j ∈ S p j ¯ x j is modular, and g ( S ) is supermodular, and so the separation problem is a special case of Submodular Function Minimization (SFM), and so can be solved in polynomial time (Mc ’06). Even better, separation is a quadratic pseudo-boolean function with non-positive quadratic coefficients, and so can be solved via Min Cut in a max flow network (Picard & Queyranne ’80). Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 9 / 23

Constraints for Q Parallel Constraints Separation of Parallel Constraints Separation can be solved via min S � j ∈ S p j ¯ x j − g ( S ) : If the value of the min is non-negative, then ¯ x satisfies all parallel constraints. If the value of the min is negative, then a minimizing S gives a parallel constraint violated by ¯ x . The term min S � j ∈ S p j ¯ x j is modular, and g ( S ) is supermodular, and so the separation problem is a special case of Submodular Function Minimization (SFM), and so can be solved in polynomial time (Mc ’06). Even better, separation is a quadratic pseudo-boolean function with non-positive quadratic coefficients, and so can be solved via Min Cut in a max flow network (Picard & Queyranne ’80). Better yet, the “non-positive quadratic coefficients” have the form − p i p j , and so separation reduces to sorting the ¯ x and using an idea of Tseng, to get an O ( n log n ) separation algorithm. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 9 / 23

Constraints for Q Series Constraints What about Precedence? So far we’ve ignored the precedence constraints. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 10 / 23

Constraints for Q Series Constraints What about Precedence? So far we’ve ignored the precedence constraints. If S and T are disjoint subsets of J , then we write S → T to mean that i → j for all i ∈ S , all j ∈ T . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 10 / 23

Constraints for Q Series Constraints What about Precedence? So far we’ve ignored the precedence constraints. If S and T are disjoint subsets of J , then we write S → T to mean that i → j for all i ∈ S , all j ∈ T . If S → T , then in any feasible schedule there is some time t such that all jobs in S finish by t , and all jobs in T start after t . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 10 / 23

Constraints for Q Series Constraints What about Precedence? So far we’ve ignored the precedence constraints. If S and T are disjoint subsets of J , then we write S → T to mean that i → j for all i ∈ S , all j ∈ T . If S → T , then in any feasible schedule there is some time t such that all jobs in S finish by t , and all jobs in T start after t . t S T jobs in S finish by t jobs in T start after t Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 10 / 23

Constraints for Q Series Constraints What about Precedence? So far we’ve ignored the precedence constraints. If S and T are disjoint subsets of J , then we write S → T to mean that i → j for all i ∈ S , all j ∈ T . If S → T , then in any feasible schedule there is some time t such that all jobs in S finish by t , and all jobs in T start after t . t S T jobs in S finish by t jobs in T start after t Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 10 / 23

Constraints for Q Series Constraints Series Constraints “All jobs in T start after t ” translates to � p j ( C j − t ) ≥ g ( T ) . j ∈ T Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 11 / 23

Constraints for Q Series Constraints Series Constraints “All jobs in T start after t ” translates to � p j ( C j − t ) ≥ g ( T ) . j ∈ T “All jobs in S finish by t ” translates to � p j ( t − C j + p j ) ≥ g ( S ) . j ∈ S Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 11 / 23

Constraints for Q Series Constraints Series Constraints “All jobs in T start after t ” translates to � p j ( C j − t ) ≥ g ( T ) . j ∈ T “All jobs in S finish by t ” translates to � p j ( t − C j + p j ) ≥ g ( S ) . j ∈ S Then by taking the linear combination of these that eliminates t , Queyranne and Wang ’91 showed that for any S → T , this series constraint is valid for Q : � � p j C j ≥ p ( S ) g ( T ) + p ( T ) g ( S ) − p ( T ) p 2 ( S ) . p ( S ) p j C j − p ( T ) j ∈ T j ∈ S Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 11 / 23

Constraints for Q Series Constraints Series Constraints “All jobs in T start after t ” translates to � p j ( C j − t ) ≥ g ( T ) . j ∈ T “All jobs in S finish by t ” translates to � p j ( t − C j + p j ) ≥ g ( S ) . j ∈ S Then by taking the linear combination of these that eliminates t , Queyranne and Wang ’91 showed that for any S → T , this series constraint is valid for Q : � � p j C j ≥ p ( S ) g ( T ) + p ( T ) g ( S ) − p ( T ) p 2 ( S ) . p ( S ) p j C j − p ( T ) j ∈ T j ∈ S So, the big question is: Is there a polynomial separation algorithm for these constraints? Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 11 / 23

Separating Series Constraints The Parametric Subproblem Outline Polyhedral Scheduling 1 The Scheduling Problem Constraints for Q 2 Parallel Constraints Series Constraints Separating Series Constraints 3 The Parametric Subproblem Bad News, Confusing News Conclusions 4 Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 12 / 23

Separating Series Constraints The Parametric Subproblem How Tight a Schedule? Given a proposed job subset T that might be part of a violated series constraint S → T . . . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 13 / 23

Separating Series Constraints The Parametric Subproblem How Tight a Schedule? Given a proposed job subset T that might be part of a violated series constraint S → T . . . . . . it is useful to be able to compute t max ( T ) , which is the latest time t such that all parallel constraints are satisfied for { ¯ x j − t } j ∈ T . . . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 13 / 23

Separating Series Constraints The Parametric Subproblem How Tight a Schedule? Given a proposed job subset T that might be part of a violated series constraint S → T . . . . . . it is useful to be able to compute t max ( T ) , which is the latest time t such that all parallel constraints are satisfied for { ¯ x j − t } j ∈ T . . . . . . which solves � � � � � max t | min p j (¯ x j − t ) − g ( S ) ≥ 0 t S ⊆ T j ∈ S Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 13 / 23

Separating Series Constraints The Parametric Subproblem How Tight a Schedule? Given a proposed job subset T that might be part of a violated series constraint S → T . . . . . . it is useful to be able to compute t max ( T ) , which is the latest time t such that all parallel constraints are satisfied for { ¯ x j − t } j ∈ T . . . . . . which solves � � � � � max t | min p j (¯ x j − t ) − g ( S ) ≥ 0 t S ⊆ T j ∈ S This is a parametric optimization problem; could be solved via parametric SFM (Fleischer & Iwata) . . . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 13 / 23

Separating Series Constraints The Parametric Subproblem How Tight a Schedule? Given a proposed job subset T that might be part of a violated series constraint S → T . . . . . . it is useful to be able to compute t max ( T ) , which is the latest time t such that all parallel constraints are satisfied for { ¯ x j − t } j ∈ T . . . . . . which solves � � � � � max t | min p j (¯ x j − t ) − g ( S ) ≥ 0 t S ⊆ T j ∈ S This is a parametric optimization problem; could be solved via parametric SFM (Fleischer & Iwata) . . . . . . or by the GGT parametric Min Cut algorithm. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 13 / 23

Separating Series Constraints The Parametric Subproblem How Tight a Schedule? Given a proposed job subset T that might be part of a violated series constraint S → T . . . . . . it is useful to be able to compute t max ( T ) , which is the latest time t such that all parallel constraints are satisfied for { ¯ x j − t } j ∈ T . . . . . . which solves � � � � � max t | min p j (¯ x j − t ) − g ( S ) ≥ 0 t S ⊆ T j ∈ S This is a parametric optimization problem; could be solved via parametric SFM (Fleischer & Iwata) . . . . . . or by the GGT parametric Min Cut algorithm. Can we adapt the Tseng sorting algorithm to compute t max ( T ) ? Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 13 / 23

Separating Series Constraints The Parametric Subproblem Parametric Sorting Define T k as the subset of the k smallest ¯ x j in T . Then Tseng concludes that an optimal solution must be one of the T k . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 14 / 23

Separating Series Constraints The Parametric Subproblem Parametric Sorting Define T k as the subset of the k smallest ¯ x j in T . Then Tseng concludes that an optimal solution must be one of the T k . The order of the { ¯ x j − t } is the same as the { ¯ x j } = ⇒ can use same T k . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 14 / 23

Separating Series Constraints The Parametric Subproblem Parametric Sorting Define T k as the subset of the k smallest ¯ x j in T . Then Tseng concludes that an optimal solution must be one of the T k . The order of the { ¯ x j − t } is the same as the { ¯ x j } = ⇒ can use same T k . Bound on t max ( T ) coming from T k is � � � t k = p j ¯ x j − g ( T k ) /p ( T k ) . j ∈ T k Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 14 / 23

Separating Series Constraints The Parametric Subproblem Parametric Sorting Define T k as the subset of the k smallest ¯ x j in T . Then Tseng concludes that an optimal solution must be one of the T k . The order of the { ¯ x j − t } is the same as the { ¯ x j } = ⇒ can use same T k . Bound on t max ( T ) coming from T k is � � � t k = p j ¯ x j − g ( T k ) /p ( T k ) . j ∈ T k Then clearly t max ( T ) = min k t k . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 14 / 23

Separating Series Constraints The Parametric Subproblem Parametric Sorting Define T k as the subset of the k smallest ¯ x j in T . Then Tseng concludes that an optimal solution must be one of the T k . The order of the { ¯ x j − t } is the same as the { ¯ x j } = ⇒ can use same T k . Bound on t max ( T ) coming from T k is � � � t k = p j ¯ x j − g ( T k ) /p ( T k ) . j ∈ T k Then clearly t max ( T ) = min k t k . This is an O ( n log n ) algorithm for t max ( T ) which gives a subset achieving t max ( T ) . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 14 / 23

Separating Series Constraints The Parametric Subproblem Going from T to S → T Given a candidate T such that t max ( T ) is determined by T | T | = T , define S ( T ) = { i | i → j ∈ A ∀ j ∈ T } . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 15 / 23

Separating Series Constraints The Parametric Subproblem Going from T to S → T Given a candidate T such that t max ( T ) is determined by T | T | = T , define S ( T ) = { i | i → j ∈ A ∀ j ∈ T } . Use similar algorithm to compute t min ( S ( T )) , the smallest value of t such that { ¯ x i − t } i ∈ S ( T ) do not violate any (reversed) parallel constraints, with t min ( S ( T )) determined by S ∗ . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 15 / 23

Separating Series Constraints The Parametric Subproblem Going from T to S → T Given a candidate T such that t max ( T ) is determined by T | T | = T , define S ( T ) = { i | i → j ∈ A ∀ j ∈ T } . Use similar algorithm to compute t min ( S ( T )) , the smallest value of t such that { ¯ x i − t } i ∈ S ( T ) do not violate any (reversed) parallel constraints, with t min ( S ( T )) determined by S ∗ . If t max ( T ) < t min ( S ∗ ) , then clearly S ∗ → T is a violated series constraint; Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 15 / 23

Separating Series Constraints The Parametric Subproblem Going from T to S → T Given a candidate T such that t max ( T ) is determined by T | T | = T , define S ( T ) = { i | i → j ∈ A ∀ j ∈ T } . Use similar algorithm to compute t min ( S ( T )) , the smallest value of t such that { ¯ x i − t } i ∈ S ( T ) do not violate any (reversed) parallel constraints, with t min ( S ( T )) determined by S ∗ . If t max ( T ) < t min ( S ∗ ) , then clearly S ∗ → T is a violated series constraint; Conversely, if t max ( T ) ≥ t min ( S ∗ ) , then T cannot be part of a violated series constraint. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 15 / 23

Separating Series Constraints The Parametric Subproblem Going from T to S → T Given a candidate T such that t max ( T ) is determined by T | T | = T , define S ( T ) = { i | i → j ∈ A ∀ j ∈ T } . Use similar algorithm to compute t min ( S ( T )) , the smallest value of t such that { ¯ x i − t } i ∈ S ( T ) do not violate any (reversed) parallel constraints, with t min ( S ( T )) determined by S ∗ . If t max ( T ) < t min ( S ∗ ) , then clearly S ∗ → T is a violated series constraint; Conversely, if t max ( T ) ≥ t min ( S ∗ ) , then T cannot be part of a violated series constraint. So “all” we have to do is find a way to generate good candidate T ’s. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 15 / 23

Separating Series Constraints Bad News, Confusing News Bad News Series constraints are functions of S and T involving the submodular g ( S ) functions, and so maybe they are bisubmodular ? (Fujishige & Mc) But they are not bisubmodular. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 16 / 23

Separating Series Constraints Bad News, Confusing News Bad News Series constraints are functions of S and T involving the submodular g ( S ) functions, and so maybe they are bisubmodular ? (Fujishige & Mc) But they are not bisubmodular. We show instead that separating series constraints is NP Hard even when the precedence graph is bipartite ( L, R ) , even when Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 16 / 23

Separating Series Constraints Bad News, Confusing News Bad News Series constraints are functions of S and T involving the submodular g ( S ) functions, and so maybe they are bisubmodular ? (Fujishige & Mc) But they are not bisubmodular. We show instead that separating series constraints is NP Hard even when the precedence graph is bipartite ( L, R ) , even when p i is constant on L , Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 16 / 23

Separating Series Constraints Bad News, Confusing News Bad News Series constraints are functions of S and T involving the submodular g ( S ) functions, and so maybe they are bisubmodular ? (Fujishige & Mc) But they are not bisubmodular. We show instead that separating series constraints is NP Hard even when the precedence graph is bipartite ( L, R ) , even when p i is constant on L , p j is constant on R , Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 16 / 23

Separating Series Constraints Bad News, Confusing News Bad News Series constraints are functions of S and T involving the submodular g ( S ) functions, and so maybe they are bisubmodular ? (Fujishige & Mc) But they are not bisubmodular. We show instead that separating series constraints is NP Hard even when the precedence graph is bipartite ( L, R ) , even when p i is constant on L , p j is constant on R , C ≡ 0 on L , and Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 16 / 23

Separating Series Constraints Bad News, Confusing News Bad News Series constraints are functions of S and T involving the submodular g ( S ) functions, and so maybe they are bisubmodular ? (Fujishige & Mc) But they are not bisubmodular. We show instead that separating series constraints is NP Hard even when the precedence graph is bipartite ( L, R ) , even when p i is constant on L , p j is constant on R , C ≡ 0 on L , and C is constant at all but one node of R . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 16 / 23

Separating Series Constraints Bad News, Confusing News Bad News Series constraints are functions of S and T involving the submodular g ( S ) functions, and so maybe they are bisubmodular ? (Fujishige & Mc) But they are not bisubmodular. We show instead that separating series constraints is NP Hard even when the precedence graph is bipartite ( L, R ) , even when p i is constant on L , p j is constant on R , C ≡ 0 on L , and C is constant at all but one node of R . Recall that Min Node Cover is easy for bipartite graphs. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 16 / 23

Separating Series Constraints Bad News, Confusing News Bad News Series constraints are functions of S and T involving the submodular g ( S ) functions, and so maybe they are bisubmodular ? (Fujishige & Mc) But they are not bisubmodular. We show instead that separating series constraints is NP Hard even when the precedence graph is bipartite ( L, R ) , even when p i is constant on L , p j is constant on R , C ≡ 0 on L , and C is constant at all but one node of R . Recall that Min Node Cover is easy for bipartite graphs. However, Min Node Cover when we want a cover C with C ∩ R = l for some given l is NP Hard (Chen & Kanj); call this Min Node Cover with Cardinality Constraint (MNC-CC). Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 16 / 23

Separating Series Constraints Bad News, Confusing News Bad News Series constraints are functions of S and T involving the submodular g ( S ) functions, and so maybe they are bisubmodular ? (Fujishige & Mc) But they are not bisubmodular. We show instead that separating series constraints is NP Hard even when the precedence graph is bipartite ( L, R ) , even when p i is constant on L , p j is constant on R , C ≡ 0 on L , and C is constant at all but one node of R . Recall that Min Node Cover is easy for bipartite graphs. However, Min Node Cover when we want a cover C with C ∩ R = l for some given l is NP Hard (Chen & Kanj); call this Min Node Cover with Cardinality Constraint (MNC-CC). We reduce MNC-CC to our separation problem (technical). Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 16 / 23

Separating Series Constraints Bad News, Confusing News Confusing News: Wolsey’s Extended Formulation Define δ ij = 1 if job i precedes job j , and 0 otherwise. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 17 / 23

Separating Series Constraints Bad News, Confusing News Confusing News: Wolsey’s Extended Formulation Define δ ij = 1 if job i precedes job j , and 0 otherwise. Then constraints (1) δ ij + δ ji = 1 , δ ≥ 0 formulate these permutation constraints. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 17 / 23

Separating Series Constraints Bad News, Confusing News Confusing News: Wolsey’s Extended Formulation Define δ ij = 1 if job i precedes job j , and 0 otherwise. Then constraints (1) δ ij + δ ji = 1 , δ ≥ 0 formulate these permutation constraints. Then (2) C j ≥ p j + � i p i δ ij is valid, and it is easy to see that (1) and (2) imply the parallel constraints. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 17 / 23

Separating Series Constraints Bad News, Confusing News Confusing News: Wolsey’s Extended Formulation Define δ ij = 1 if job i precedes job j , and 0 otherwise. Then constraints (1) δ ij + δ ji = 1 , δ ≥ 0 formulate these permutation constraints. Then (2) C j ≥ p j + � i p i δ ij is valid, and it is easy to see that (1) and (2) imply the parallel constraints. For job i define S i = { j | i → j ∈ A } and P i = { j | j → i ∈ A } . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 17 / 23

Separating Series Constraints Bad News, Confusing News Confusing News: Wolsey’s Extended Formulation Define δ ij = 1 if job i precedes job j , and 0 otherwise. Then constraints (1) δ ij + δ ji = 1 , δ ≥ 0 formulate these permutation constraints. Then (2) C j ≥ p j + � i p i δ ij is valid, and it is easy to see that (1) and (2) imply the parallel constraints. For job i define S i = { j | i → j ∈ A } and P i = { j | j → i ∈ A } . Then (3) � � � C j − C i ≥ p i + p k + p k δ kj + p k δ ik k ∈ S i ∩ P j k ∈ S i − P j k ∈ P j − S i is valid, and (1) and (3) imply the series constraints (not so easy). Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 17 / 23

Separating Series Constraints Bad News, Confusing News Confusing News: Wolsey’s Extended Formulation Define δ ij = 1 if job i precedes job j , and 0 otherwise. Then constraints (1) δ ij + δ ji = 1 , δ ≥ 0 formulate these permutation constraints. Then (2) C j ≥ p j + � i p i δ ij is valid, and it is easy to see that (1) and (2) imply the parallel constraints. For job i define S i = { j | i → j ∈ A } and P i = { j | j → i ∈ A } . Then (3) � � � C j − C i ≥ p i + p k + p k δ kj + p k δ ik k ∈ S i ∩ P j k ∈ S i − P j k ∈ P j − S i is valid, and (1) and (3) imply the series constraints (not so easy). This extended formulation Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 17 / 23

Separating Series Constraints Bad News, Confusing News Confusing News: Wolsey’s Extended Formulation Define δ ij = 1 if job i precedes job j , and 0 otherwise. Then constraints (1) δ ij + δ ji = 1 , δ ≥ 0 formulate these permutation constraints. Then (2) C j ≥ p j + � i p i δ ij is valid, and it is easy to see that (1) and (2) imply the parallel constraints. For job i define S i = { j | i → j ∈ A } and P i = { j | j → i ∈ A } . Then (3) � � � C j − C i ≥ p i + p k + p k δ kj + p k δ ik k ∈ S i ∩ P j k ∈ S i − P j k ∈ P j − S i is valid, and (1) and (3) imply the series constraints (not so easy). This extended formulation ( + ) Is compact , so separation is trivial. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 17 / 23

Separating Series Constraints Bad News, Confusing News Confusing News: Wolsey’s Extended Formulation Define δ ij = 1 if job i precedes job j , and 0 otherwise. Then constraints (1) δ ij + δ ji = 1 , δ ≥ 0 formulate these permutation constraints. Then (2) C j ≥ p j + � i p i δ ij is valid, and it is easy to see that (1) and (2) imply the parallel constraints. For job i define S i = { j | i → j ∈ A } and P i = { j | j → i ∈ A } . Then (3) � � � C j − C i ≥ p i + p k + p k δ kj + p k δ ik k ∈ S i ∩ P j k ∈ S i − P j k ∈ P j − S i is valid, and (1) and (3) imply the series constraints (not so easy). This extended formulation ( + ) Is compact , so separation is trivial. ( + ) Is at least as tight as the QW formulation. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 17 / 23

Separating Series Constraints Bad News, Confusing News Confusing News: Wolsey’s Extended Formulation Define δ ij = 1 if job i precedes job j , and 0 otherwise. Then constraints (1) δ ij + δ ji = 1 , δ ≥ 0 formulate these permutation constraints. Then (2) C j ≥ p j + � i p i δ ij is valid, and it is easy to see that (1) and (2) imply the parallel constraints. For job i define S i = { j | i → j ∈ A } and P i = { j | j → i ∈ A } . Then (3) � � � C j − C i ≥ p i + p k + p k δ kj + p k δ ik k ∈ S i ∩ P j k ∈ S i − P j k ∈ P j − S i is valid, and (1) and (3) imply the series constraints (not so easy). This extended formulation ( + ) Is compact , so separation is trivial. ( + ) Is at least as tight as the QW formulation. ( − ) Has O ( n 2 ) variables. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 17 / 23

Separating Series Constraints Bad News, Confusing News A Paradox? Thus we have that QW ⊇ proj(Wolsey) , yet there is a class of constraints within QW that is NP Hard to separate, yet Wolsey is easy to separate ?!? Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 18 / 23

Separating Series Constraints Bad News, Confusing News A Paradox? Thus we have that QW ⊇ proj(Wolsey) , yet there is a class of constraints within QW that is NP Hard to separate, yet Wolsey is easy to separate ?!? According to Oriolo and Kaibel, this is not actually so surprising; other examples exist, though this one is perhaps more natural than most. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 18 / 23

Separating Series Constraints Bad News, Confusing News A Paradox? Thus we have that QW ⊇ proj(Wolsey) , yet there is a class of constraints within QW that is NP Hard to separate, yet Wolsey is easy to separate ?!? According to Oriolo and Kaibel, this is not actually so surprising; other examples exist, though this one is perhaps more natural than most. It implies something: QW showed that series plus parallel constraints are enough to characterize Q when A is series-parallel . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 18 / 23

Separating Series Constraints Bad News, Confusing News A Paradox? Thus we have that QW ⊇ proj(Wolsey) , yet there is a class of constraints within QW that is NP Hard to separate, yet Wolsey is easy to separate ?!? According to Oriolo and Kaibel, this is not actually so surprising; other examples exist, though this one is perhaps more natural than most. It implies something: QW showed that series plus parallel constraints are enough to characterize Q when A is series-parallel . Thus A series-parallel = ⇒ QW = proj(Wolsey) . Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 18 / 23

Separating Series Constraints Bad News, Confusing News A Paradox? Thus we have that QW ⊇ proj(Wolsey) , yet there is a class of constraints within QW that is NP Hard to separate, yet Wolsey is easy to separate ?!? According to Oriolo and Kaibel, this is not actually so surprising; other examples exist, though this one is perhaps more natural than most. It implies something: QW showed that series plus parallel constraints are enough to characterize Q when A is series-parallel . Thus A series-parallel = ⇒ QW = proj(Wolsey) . Is is known that solving the scheduling problem when A is series-parallel is polynomial. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 18 / 23

Separating Series Constraints Bad News, Confusing News A Paradox? Thus we have that QW ⊇ proj(Wolsey) , yet there is a class of constraints within QW that is NP Hard to separate, yet Wolsey is easy to separate ?!? According to Oriolo and Kaibel, this is not actually so surprising; other examples exist, though this one is perhaps more natural than most. It implies something: QW showed that series plus parallel constraints are enough to characterize Q when A is series-parallel . Thus A series-parallel = ⇒ QW = proj(Wolsey) . Is is known that solving the scheduling problem when A is series-parallel is polynomial. Therefore we must be able to separate series constraints in polynomial time when A is series-parallel. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 18 / 23

Separating Series Constraints Bad News, Confusing News Series Separation for Series-Parallel Precedence Recall that A is series-parallel iff it can be composed from trivial precedence relations via Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 19 / 23

Separating Series Constraints Bad News, Confusing News Series Separation for Series-Parallel Precedence Recall that A is series-parallel iff it can be composed from trivial precedence relations via Parallel composition: put precedence relations S and T in parallel (no new arcs), or Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 19 / 23

Separating Series Constraints Bad News, Confusing News Series Separation for Series-Parallel Precedence Recall that A is series-parallel iff it can be composed from trivial precedence relations via Parallel composition: put precedence relations S and T in parallel (no new arcs), or Series composition: put precedence relations S and T in series (an arc from every i ∈ S to every j ∈ T ). Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 19 / 23

Separating Series Constraints Bad News, Confusing News Series Separation for Series-Parallel Precedence Recall that A is series-parallel iff it can be composed from trivial precedence relations via Parallel composition: put precedence relations S and T in parallel (no new arcs), or Series composition: put precedence relations S and T in series (an arc from every i ∈ S to every j ∈ T ). In linear time we can either find such a decomposition, or prove that A is not series-parallel. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 19 / 23

Separating Series Constraints Bad News, Confusing News Series Separation for Series-Parallel Precedence Recall that A is series-parallel iff it can be composed from trivial precedence relations via Parallel composition: put precedence relations S and T in parallel (no new arcs), or Series composition: put precedence relations S and T in series (an arc from every i ∈ S to every j ∈ T ). In linear time we can either find such a decomposition, or prove that A is not series-parallel. Parallel composition is trivial to deal with. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 19 / 23

Separating Series Constraints Bad News, Confusing News Series Separation for Series-Parallel Precedence Recall that A is series-parallel iff it can be composed from trivial precedence relations via Parallel composition: put precedence relations S and T in parallel (no new arcs), or Series composition: put precedence relations S and T in series (an arc from every i ∈ S to every j ∈ T ). In linear time we can either find such a decomposition, or prove that A is not series-parallel. Parallel composition is trivial to deal with. Series composition induces a complete bipartite graph. Then we can use our parametric version of Tseng’s Algorithm to determine if there is a violated series constraint or not. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 19 / 23

Separating Series Constraints Bad News, Confusing News Series Separation for Series-Parallel Precedence Recall that A is series-parallel iff it can be composed from trivial precedence relations via Parallel composition: put precedence relations S and T in parallel (no new arcs), or Series composition: put precedence relations S and T in series (an arc from every i ∈ S to every j ∈ T ). In linear time we can either find such a decomposition, or prove that A is not series-parallel. Parallel composition is trivial to deal with. Series composition induces a complete bipartite graph. Then we can use our parametric version of Tseng’s Algorithm to determine if there is a violated series constraint or not. The end result is an O ( n log n ) separation algorithm. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 19 / 23

Conclusions Morals Outline Polyhedral Scheduling 1 The Scheduling Problem Constraints for Q 2 Parallel Constraints Series Constraints Separating Series Constraints 3 The Parametric Subproblem Bad News, Confusing News Conclusions 4 Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 20 / 23

Conclusions Morals Morals of this Story Sometimes extended formulations can be helpful for separation. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 21 / 23

Conclusions Morals Morals of this Story Sometimes extended formulations can be helpful for separation. Just because it is NP Hard to separate some class of constraints does not mean that there might not be an extended formulation where it becomes trivial. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 21 / 23

Conclusions Morals Morals of this Story Sometimes extended formulations can be helpful for separation. Just because it is NP Hard to separate some class of constraints does not mean that there might not be an extended formulation where it becomes trivial. The QW constraints are rich in submodularity, but are not easy to separate despite this — submodularity does not always lead to easy problems. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 21 / 23

Conclusions Morals Morals of this Story Sometimes extended formulations can be helpful for separation. Just because it is NP Hard to separate some class of constraints does not mean that there might not be an extended formulation where it becomes trivial. The QW constraints are rich in submodularity, but are not easy to separate despite this — submodularity does not always lead to easy problems. One can develop various greedy heuristic separation algorithms for series constraints in the “natural” QW formulation using parametric sorting. Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 21 / 23

Conclusions Morals Morals of this Story Sometimes extended formulations can be helpful for separation. Just because it is NP Hard to separate some class of constraints does not mean that there might not be an extended formulation where it becomes trivial. The QW constraints are rich in submodularity, but are not easy to separate despite this — submodularity does not always lead to easy problems. One can develop various greedy heuristic separation algorithms for series constraints in the “natural” QW formulation using parametric sorting. A more general polynomially solvable case: when A is 2-dimensional (Correa & Schulz; Amb¨ uhl & Mastrolilli). Can we characterize Q in this case? Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 21 / 23

Conclusions Morals Any questions? Questions? Comments? Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 22 / 23

Conclusions Morals Thank you On behalf of all attendees, we thank the organizers: Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 23 / 23

Conclusions Morals Thank you On behalf of all attendees, we thank the organizers: Fran¸ cois Margot Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 23 / 23