Separation of Series Constraints for One-Machine Scheduling with - - PowerPoint PPT Presentation
Separation of Series Constraints for One-Machine Scheduling with Precedence Y. Kobayashi, A. Ridha Mahjoub, S. Thomas McCormick U. Tokyo/Paris-Dauphine/Sauder School of Business, UBC Aussois January 2012 Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC)
Aussois January 2012
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 1 / 23
Aussois January 2012
Sauder School of Business University of British Columbia
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 1 / 23
Aussois January 2012
Sauder School of Business The best research b-school in Canada! University of British Columbia
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 1 / 23
1
Polyhedral Scheduling The Scheduling Problem
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 2 / 23
1
Polyhedral Scheduling The Scheduling Problem
2
Constraints for Q Parallel Constraints Series Constraints
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 2 / 23
1
Polyhedral Scheduling The Scheduling Problem
2
Constraints for Q Parallel Constraints Series Constraints
3
Separating Series Constraints The Parametric Subproblem Bad News, Confusing News
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 2 / 23
1
Polyhedral Scheduling The Scheduling Problem
2
Constraints for Q Parallel Constraints Series Constraints
3
Separating Series Constraints The Parametric Subproblem Bad News, Confusing News
4
Conclusions
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 2 / 23
Polyhedral Scheduling The Scheduling Problem
1
Polyhedral Scheduling The Scheduling Problem
2
Constraints for Q Parallel Constraints Series Constraints
3
Separating Series Constraints The Parametric Subproblem Bad News, Confusing News
4
Conclusions
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 3 / 23
Polyhedral Scheduling The Scheduling Problem
We have set J = {1, 2, . . . , n} of jobs on a single machine.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 4 / 23
Polyhedral Scheduling The Scheduling Problem
We have set J = {1, 2, . . . , n} of jobs on a single machine. Job j has processing time pj (no release times or due dates), and weight wj.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 4 / 23
Polyhedral Scheduling The Scheduling Problem
We have set J = {1, 2, . . . , n} of jobs on a single machine. Job j has processing time pj (no release times or due dates), and weight wj. We have a precedence graph (J, A) where i → j ∈ A means that i must be scheduled before j.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 4 / 23
Polyhedral Scheduling The Scheduling Problem
We have set J = {1, 2, . . . , n} of jobs on a single machine. Job j has processing time pj (no release times or due dates), and weight wj. We have a precedence graph (J, A) where i → j ∈ A means that i must be scheduled before j. The machine can process only one job at a time, and there is no preemption.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 4 / 23
Polyhedral Scheduling The Scheduling Problem
We have set J = {1, 2, . . . , n} of jobs on a single machine. Job j has processing time pj (no release times or due dates), and weight wj. We have a precedence graph (J, A) where i → j ∈ A means that i must be scheduled before j. The machine can process only one job at a time, and there is no preemption. The objective is to minimize the weighted sum of completion times, minj∈J wjCj.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 4 / 23
Polyhedral Scheduling The Scheduling Problem
We have set J = {1, 2, . . . , n} of jobs on a single machine. Job j has processing time pj (no release times or due dates), and weight wj. We have a precedence graph (J, A) where i → j ∈ A means that i must be scheduled before j. The machine can process only one job at a time, and there is no preemption. The objective is to minimize the weighted sum of completion times, minj∈J wjCj. Problem is 1 | pj, prec | wjCj in classic scheduling notation.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 4 / 23
Polyhedral Scheduling The Scheduling Problem
This problem is known to be NP Complete.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 5 / 23
Polyhedral Scheduling The Scheduling Problem
This problem is known to be NP Complete. It is a basic problem that arises as a subproblem in more general scheduling problems, so we need some way to attack it.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 5 / 23
Polyhedral Scheduling The Scheduling Problem
This problem is known to be NP Complete. It is a basic problem that arises as a subproblem in more general scheduling problems, so we need some way to attack it. Pioneering work by Balas, Queyranne, and Queyranne & Wang developed a polyhedral approach to the problem:
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 5 / 23
Polyhedral Scheduling The Scheduling Problem
This problem is known to be NP Complete. It is a basic problem that arises as a subproblem in more general scheduling problems, so we need some way to attack it. Pioneering work by Balas, Queyranne, and Queyranne & Wang developed a polyhedral approach to the problem:
Consider Q = hull{Cj | Cj is a set of feasible completion times}.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 5 / 23
Polyhedral Scheduling The Scheduling Problem
This problem is known to be NP Complete. It is a basic problem that arises as a subproblem in more general scheduling problems, so we need some way to attack it. Pioneering work by Balas, Queyranne, and Queyranne & Wang developed a polyhedral approach to the problem:
Consider Q = hull{Cj | Cj is a set of feasible completion times}.
In order for this to be computationally useful, we need to characterize the facets of Q, and give separation algorithms for these facets.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 5 / 23
Polyhedral Scheduling The Scheduling Problem
Suppose that Q is a subproblem of a bigger problem that we’re solving using an LP relaxation such as Branch and Cut.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 6 / 23
Polyhedral Scheduling The Scheduling Problem
Suppose that Q is a subproblem of a bigger problem that we’re solving using an LP relaxation such as Branch and Cut. Suppose that the LP relaxation has a current point ¯ x = (¯ xj).
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 6 / 23
Polyhedral Scheduling The Scheduling Problem
Suppose that Q is a subproblem of a bigger problem that we’re solving using an LP relaxation such as Branch and Cut. Suppose that the LP relaxation has a current point ¯ x = (¯ xj). We want to know whether ¯ x ∈ Q or not, and if not, find some constraint of Q separating ¯ x from Q that we can add to the relaxation.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 6 / 23
Polyhedral Scheduling The Scheduling Problem
Suppose that Q is a subproblem of a bigger problem that we’re solving using an LP relaxation such as Branch and Cut. Suppose that the LP relaxation has a current point ¯ x = (¯ xj). We want to know whether ¯ x ∈ Q or not, and if not, find some constraint of Q separating ¯ x from Q that we can add to the relaxation. Thus Separation: Given ¯ x ∈ RJ, either prove that ¯ x ∈ Q, or find some constraint satisfied by all points of Q, but violated by ¯ x.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 6 / 23
Constraints for Q Parallel Constraints
1
Polyhedral Scheduling The Scheduling Problem
2
Constraints for Q Parallel Constraints Series Constraints
3
Separating Series Constraints The Parametric Subproblem Bad News, Confusing News
4
Conclusions
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 7 / 23
Constraints for Q Parallel Constraints
Suppose that we schedule the jobs in order 1, 2, . . . , n with no idle time.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 8 / 23
Constraints for Q Parallel Constraints
Suppose that we schedule the jobs in order 1, 2, . . . , n with no idle time. Then it is easy to see that Cj in this schedule is p1 + p2 + · · · + pj = Pj. Therefore
pjCj =
pjPj =
pipj.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 8 / 23
Constraints for Q Parallel Constraints
Suppose that we schedule the jobs in order 1, 2, . . . , n with no idle time. Then it is easy to see that Cj in this schedule is p1 + p2 + · · · + pj = Pj. Therefore
pjCj =
pjPj =
pipj. More generally, for S ⊆ J define g(S) =
i≤j∈S pipj.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 8 / 23
Constraints for Q Parallel Constraints
Suppose that we schedule the jobs in order 1, 2, . . . , n with no idle time. Then it is easy to see that Cj in this schedule is p1 + p2 + · · · + pj = Pj. Therefore
pjCj =
pjPj =
pipj. More generally, for S ⊆ J define g(S) =
i≤j∈S pipj.
Then Queyranne ’93 showed that for any S ⊆ J, the parallel constraint
pjxj ≥ g(S) is valid for Q.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 8 / 23
Constraints for Q Parallel Constraints
Separation can be solved via minS
xj − g(S):
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 9 / 23
Constraints for Q Parallel Constraints
Separation can be solved via minS
xj − 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 can be solved via minS
xj − 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 can be solved via minS
xj − 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 minS
xj 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 can be solved via minS
xj − 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 minS
xj 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 can be solved via minS
xj − 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 minS
xj 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 −pipj, and so separation reduces to sorting the ¯ x and using an idea
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 9 / 23
Constraints for Q Series Constraints
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
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
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
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.
S T
jobs in S finish by t jobs in T start after t t Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 10 / 23
Constraints for Q Series Constraints
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
“All jobs in T start after t” translates to
pj(Cj − t) ≥ g(T).
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 11 / 23
Constraints for Q Series Constraints
“All jobs in T start after t” translates to
pj(Cj − t) ≥ g(T). “All jobs in S finish by t” translates to
pj(t − Cj + pj) ≥ g(S).
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 11 / 23
Constraints for Q Series Constraints
“All jobs in T start after t” translates to
pj(Cj − t) ≥ g(T). “All jobs in S finish by t” translates to
pj(t − Cj + pj) ≥ g(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(S)
pjCj − p(T)
pjCj ≥ p(S)g(T) + p(T)g(S) − p(T)p2(S).
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 11 / 23
Constraints for Q Series Constraints
“All jobs in T start after t” translates to
pj(Cj − t) ≥ g(T). “All jobs in S finish by t” translates to
pj(t − Cj + pj) ≥ g(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(S)
pjCj − p(T)
pjCj ≥ p(S)g(T) + p(T)g(S) − p(T)p2(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
1
Polyhedral Scheduling The Scheduling Problem
2
Constraints for Q Parallel Constraints Series Constraints
3
Separating Series Constraints The Parametric Subproblem Bad News, Confusing News
4
Conclusions
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 12 / 23
Separating Series Constraints The Parametric Subproblem
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
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 tmax(T), which is the latest time t such that all parallel constraints are satisfied for {¯ xj − t}j∈T . . .
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 13 / 23
Separating Series Constraints The Parametric Subproblem
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 tmax(T), which is the latest time t such that all parallel constraints are satisfied for {¯ xj − t}j∈T . . . . . . which solves max
t
S⊆T j∈S
pj(¯ xj − t) − g(S)
Series Separation Aussois January 2012 13 / 23
Separating Series Constraints The Parametric Subproblem
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 tmax(T), which is the latest time t such that all parallel constraints are satisfied for {¯ xj − t}j∈T . . . . . . which solves max
t
S⊆T j∈S
pj(¯ xj − t) − g(S)
parametric SFM (Fleischer & Iwata) . . .
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 13 / 23
Separating Series Constraints The Parametric Subproblem
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 tmax(T), which is the latest time t such that all parallel constraints are satisfied for {¯ xj − t}j∈T . . . . . . which solves max
t
S⊆T j∈S
pj(¯ xj − t) − g(S)
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
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 tmax(T), which is the latest time t such that all parallel constraints are satisfied for {¯ xj − t}j∈T . . . . . . which solves max
t
S⊆T j∈S
pj(¯ xj − t) − g(S)
parametric SFM (Fleischer & Iwata) . . . . . . or by the GGT parametric Min Cut algorithm. Can we adapt the Tseng sorting algorithm to compute tmax(T)?
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 13 / 23
Separating Series Constraints The Parametric Subproblem
Define Tk as the subset of the k smallest ¯ xj in T. Then Tseng concludes that an optimal solution must be one of the Tk.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 14 / 23
Separating Series Constraints The Parametric Subproblem
Define Tk as the subset of the k smallest ¯ xj in T. Then Tseng concludes that an optimal solution must be one of the Tk. The order of the {¯ xj − t} is the same as the {¯ xj} = ⇒ can use same Tk.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 14 / 23
Separating Series Constraints The Parametric Subproblem
Define Tk as the subset of the k smallest ¯ xj in T. Then Tseng concludes that an optimal solution must be one of the Tk. The order of the {¯ xj − t} is the same as the {¯ xj} = ⇒ can use same Tk. Bound on tmax(T) coming from Tk is tk =
j∈Tk
pj ¯ xj − g(Tk)
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 14 / 23
Separating Series Constraints The Parametric Subproblem
Define Tk as the subset of the k smallest ¯ xj in T. Then Tseng concludes that an optimal solution must be one of the Tk. The order of the {¯ xj − t} is the same as the {¯ xj} = ⇒ can use same Tk. Bound on tmax(T) coming from Tk is tk =
j∈Tk
pj ¯ xj − g(Tk)
Then clearly tmax(T) = mink tk.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 14 / 23
Separating Series Constraints The Parametric Subproblem
Define Tk as the subset of the k smallest ¯ xj in T. Then Tseng concludes that an optimal solution must be one of the Tk. The order of the {¯ xj − t} is the same as the {¯ xj} = ⇒ can use same Tk. Bound on tmax(T) coming from Tk is tk =
j∈Tk
pj ¯ xj − g(Tk)
Then clearly tmax(T) = mink tk. This is an O(n log n) algorithm for tmax(T) which gives a subset achieving tmax(T).
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 14 / 23
Separating Series Constraints The Parametric Subproblem
Given a candidate T such that tmax(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
Given a candidate T such that tmax(T) is determined by T|T| = T, define S(T) = {i | i → j ∈ A ∀j ∈ T}. Use similar algorithm to compute tmin(S(T)), the smallest value of t such that {¯ xi − t}i∈S(T) do not violate any (reversed) parallel constraints, with tmin(S(T)) determined by S∗.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 15 / 23
Separating Series Constraints The Parametric Subproblem
Given a candidate T such that tmax(T) is determined by T|T| = T, define S(T) = {i | i → j ∈ A ∀j ∈ T}. Use similar algorithm to compute tmin(S(T)), the smallest value of t such that {¯ xi − t}i∈S(T) do not violate any (reversed) parallel constraints, with tmin(S(T)) determined by S∗. If tmax(T) < tmin(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
Given a candidate T such that tmax(T) is determined by T|T| = T, define S(T) = {i | i → j ∈ A ∀j ∈ T}. Use similar algorithm to compute tmin(S(T)), the smallest value of t such that {¯ xi − t}i∈S(T) do not violate any (reversed) parallel constraints, with tmin(S(T)) determined by S∗. If tmax(T) < tmin(S∗), then clearly S∗ → T is a violated series constraint; Conversely, if tmax(T) ≥ tmin(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
Given a candidate T such that tmax(T) is determined by T|T| = T, define S(T) = {i | i → j ∈ A ∀j ∈ T}. Use similar algorithm to compute tmin(S(T)), the smallest value of t such that {¯ xi − t}i∈S(T) do not violate any (reversed) parallel constraints, with tmin(S(T)) determined by S∗. If tmax(T) < tmin(S∗), then clearly S∗ → T is a violated series constraint; Conversely, if tmax(T) ≥ tmin(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
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
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
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
pi is constant on L,
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 16 / 23
Separating Series Constraints Bad News, Confusing 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
pi is constant on L, pj is constant on R,
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 16 / 23
Separating Series Constraints Bad News, Confusing 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
pi is constant on L, pj 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
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
pi is constant on L, pj 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
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
pi is constant on L, pj 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
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
pi is constant on L, pj 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
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
pi is constant on L, pj 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
Define δij = 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
Define δij = 1 if job i precedes job j, and 0 otherwise. Then constraints (1) δij + δji = 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
Define δij = 1 if job i precedes job j, and 0 otherwise. Then constraints (1) δij + δji = 1, δ ≥ 0 formulate these permutation constraints. Then (2) Cj ≥ pj +
i piδij 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
Define δij = 1 if job i precedes job j, and 0 otherwise. Then constraints (1) δij + δji = 1, δ ≥ 0 formulate these permutation constraints. Then (2) Cj ≥ pj +
i piδij is valid, and it is easy to see that (1)
and (2) imply the parallel constraints. For job i define Si = {j | i → j ∈ A} and Pi = {j | j → i ∈ A}.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 17 / 23
Separating Series Constraints Bad News, Confusing News
Define δij = 1 if job i precedes job j, and 0 otherwise. Then constraints (1) δij + δji = 1, δ ≥ 0 formulate these permutation constraints. Then (2) Cj ≥ pj +
i piδij is valid, and it is easy to see that (1)
and (2) imply the parallel constraints. For job i define Si = {j | i → j ∈ A} and Pi = {j | j → i ∈ A}. Then (3) Cj − Ci ≥ pi +
pk +
pkδkj +
pkδik 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
Define δij = 1 if job i precedes job j, and 0 otherwise. Then constraints (1) δij + δji = 1, δ ≥ 0 formulate these permutation constraints. Then (2) Cj ≥ pj +
i piδij is valid, and it is easy to see that (1)
and (2) imply the parallel constraints. For job i define Si = {j | i → j ∈ A} and Pi = {j | j → i ∈ A}. Then (3) Cj − Ci ≥ pi +
pk +
pkδkj +
pkδik 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
Define δij = 1 if job i precedes job j, and 0 otherwise. Then constraints (1) δij + δji = 1, δ ≥ 0 formulate these permutation constraints. Then (2) Cj ≥ pj +
i piδij is valid, and it is easy to see that (1)
and (2) imply the parallel constraints. For job i define Si = {j | i → j ∈ A} and Pi = {j | j → i ∈ A}. Then (3) Cj − Ci ≥ pi +
pk +
pkδkj +
pkδik 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
Define δij = 1 if job i precedes job j, and 0 otherwise. Then constraints (1) δij + δji = 1, δ ≥ 0 formulate these permutation constraints. Then (2) Cj ≥ pj +
i piδij is valid, and it is easy to see that (1)
and (2) imply the parallel constraints. For job i define Si = {j | i → j ∈ A} and Pi = {j | j → i ∈ A}. Then (3) Cj − Ci ≥ pi +
pk +
pkδkj +
pkδik 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
Define δij = 1 if job i precedes job j, and 0 otherwise. Then constraints (1) δij + δji = 1, δ ≥ 0 formulate these permutation constraints. Then (2) Cj ≥ pj +
i piδij is valid, and it is easy to see that (1)
and (2) imply the parallel constraints. For job i define Si = {j | i → j ∈ A} and Pi = {j | j → i ∈ A}. Then (3) Cj − Ci ≥ pi +
pk +
pkδkj +
pkδik 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(n2) variables.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 17 / 23
Separating Series Constraints Bad News, Confusing News
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
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
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
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
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
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
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
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
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
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
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
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
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
1
Polyhedral Scheduling The Scheduling Problem
2
Constraints for Q Parallel Constraints Series Constraints
3
Separating Series Constraints The Parametric Subproblem Bad News, Confusing News
4
Conclusions
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 20 / 23
Conclusions Morals
Sometimes extended formulations can be helpful for separation.
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 21 / 23
Conclusions Morals
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
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
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
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
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 22 / 23
Conclusions Morals
On behalf of all attendees, we thank the organizers:
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 23 / 23
Conclusions Morals
On behalf of all attendees, we thank the organizers:
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 23 / 23
Conclusions Morals
On behalf of all attendees, we thank the organizers:
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 23 / 23
Conclusions Morals
On behalf of all attendees, we thank the organizers:
and especially the originator of Aussois
Kobayashi-Mahjoub-Mc (Tokyo-Paris-UBC) Series Separation Aussois January 2012 23 / 23