Combinatorial Algorithms to Solve Network Interdiction and - PowerPoint PPT Presentation

Network Interdiction Interdiction curves The overall interdiction curve We overlay the cut-wise interdiction curves to get the overall curve. cap c ( S ) cap c ( U ) S cap c ( T ) T cap c ( S ) � c 1 " residual capacity cap c ( S ) � c 1 � c 2 U 0 r 1 + r 2 r ( T ) r ( S ) r 1 r ( U ) B ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 6 / 32

Network Interdiction Interdiction curves The overall interdiction curve For a given value of B , we just select which S gives the minimum value at B , so the overall curve is the minimum of all the cut-wise curves. cap c ( S ) cap c ( U ) S cap c ( T ) T cap c ( S ) � c 1 " residual capacity cap c ( S ) � c 1 � c 2 U 0 r 1 + r 2 r ( T ) r ( S ) r 1 r ( U ) B ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 6 / 32

Network Interdiction Interdiction curves The overall interdiction curve Unfortunately, the minimum of a bunch of convex curves is not in general convex. cap c ( S ) cap c ( U ) S cap c ( T ) T cap c ( S ) � c 1 " residual capacity cap c ( S ) � c 1 � c 2 U 0 r 1 + r 2 r ( T ) r ( S ) r 1 r ( U ) B ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 6 / 32

Network Interdiction Interdiction curves The overall interdiction curve This is why Network Interdiction is NP Hard (Phillips ’93; Wood ’93). cap c ( S ) cap c ( U ) S cap c ( T ) T cap c ( S ) � c 1 " residual capacity cap c ( S ) � c 1 � c 2 U 0 r 1 + r 2 r ( T ) r ( S ) r 1 r ( U ) B ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 6 / 32

Network Interdiction Interdiction curves Linearizing the overall curve: the B -profile If we take the lower envelope, or convex hull, of the overall interdiction curve, we get something tractable, the B -profile. cap c ( S ) cap c ( U ) S cap c ( T ) T cap c ( S ) � c 1 infeasible values B " residual capacity feasible values B on U curve cap c ( S ) � c 1 � c 2 U 0 r 1 r 1 + r 2 r ( T ) r ( S ) r ( U ) B ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 7 / 32

Network Interdiction Interdiction curves Linearizing the overall curve: the B -profile Now budget B corresponds to a convex combination of points coming from the interdiction curves of (one or) two cuts, S 1 and S 2 . cap c ( S ) cap c ( U ) S cap c ( T ) T cap c ( S ) � c 1 S 1 " residual capacity S 2 cap c ( S ) � c 1 � c 2 U 0 r 1 B r 1 + r 2 r ( T ) r ( S ) r ( U ) B ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 7 / 32

Network Interdiction Interdiction curves Linearizing the overall curve: the B -profile Burch et al ’02 show that we can use S 1 and S 2 to get a pseudo-approximation algorithm for Network Interdiction: Given some ✏ > 0 , it will find either McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 8 / 32

Network Interdiction Interdiction curves Linearizing the overall curve: the B -profile Burch et al ’02 show that we can use S 1 and S 2 to get a pseudo-approximation algorithm for Network Interdiction: Given some ✏ > 0 , it will find either A feasible set of arcs to destroy whose residual capacity is within a factor of 1 + ✏ of optimal, or McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 8 / 32

Network Interdiction Interdiction curves Linearizing the overall curve: the B -profile Burch et al ’02 show that we can use S 1 and S 2 to get a pseudo-approximation algorithm for Network Interdiction: Given some ✏ > 0 , it will find either A feasible set of arcs to destroy whose residual capacity is within a factor of 1 + ✏ of optimal, or A set of arcs to destroy which costs at most (1 + ✏ ) B whose residual capacity is less than optimal. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 8 / 32

Network Interdiction Interdiction curves Linearizing the overall curve: the B -profile Burch et al ’02 show that we can use S 1 and S 2 to get a pseudo-approximation algorithm for Network Interdiction: Given some ✏ > 0 , it will find either A feasible set of arcs to destroy whose residual capacity is within a factor of 1 + ✏ of optimal, or A set of arcs to destroy which costs at most (1 + ✏ ) B whose residual capacity is less than optimal. The algorithmic question is then: Given B , how do we find S 1 and S 2 ? McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 8 / 32

Network Interdiction Interdiction curves Linearizing the overall curve: the B -profile Burch et al ’02 show that we can use S 1 and S 2 to get a pseudo-approximation algorithm for Network Interdiction: Given some ✏ > 0 , it will find either A feasible set of arcs to destroy whose residual capacity is within a factor of 1 + ✏ of optimal, or A set of arcs to destroy which costs at most (1 + ✏ ) B whose residual capacity is less than optimal. The algorithmic question is then: Given B , how do we find S 1 and S 2 ? Burch et al write a linear program that can do it, but here we want a combinatorial algorithm to do it. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 8 / 32

LP Duality Dual of interdiction Outline Network Interdiction 1 What is it? Interdiction curves LP Duality 2 Dual of interdiction Parametric Min Cut 3 Parametric curves The Breakpoint Subproblem 4 What is it? Algorithms Discrete Newton Multiple Parameters 5 What is it? Scheduling problem Multi-GGT McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 9 / 32

LP Duality Dual of interdiction The linear program and its dual The normal min cut dual LP is min P u ! v c uv y uv s.t. d u � d v + y uv 0 for u ! v 6 = t ! s , � d t � d s + y ts � 1 y uv 0 all u ! v . � McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 10 / 32

LP Duality Dual of interdiction The linear program and its dual The normal min cut dual LP is min P u ! v c uv y uv s.t. d u � d v + y uv 0 for u ! v 6 = t ! s , � d t � d s + y ts � 1 y uv 0 all u ! v . � To get an interdiction version, put a second dual variable z uv on each u ! v that represents what fraction of u ! v we are going to destroy. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 10 / 32

LP Duality Dual of interdiction The linear program and its dual The normal min cut dual LP is min P u ! v c uv y uv s.t. d u � d v + y uv 0 for u ! v 6 = t ! s , � d t � d s + y ts � 1 y uv 0 all u ! v . � To get an interdiction version, put a second dual variable z uv on each u ! v that represents what fraction of u ! v we are going to destroy. The new LP is then min P u ! v c uv y uv s.t. d u � d v + y uv + z uv 0 for u ! v 6 = t ! s , � d t � d s + y ts � 1 P u ! v r uv z uv B  y uv , z uv � 0 all u ! v . McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 10 / 32

LP Duality Dual of interdiction The linear program and its dual Repeat the new LP with dual variables: min P u ! v c uv y uv x uv : s.t. d u � d v + y uv + z uv � 0 for u ! v 6 = t ! s , x ts : : d t � d s + y ts � 1 � : P u ! v r uv z uv  B y uv , z uv 0 all u ! v . � McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 11 / 32

LP Duality Dual of interdiction The linear program and its dual Repeat the new LP with dual variables: min P u ! v c uv y uv x uv : s.t. d u � d v + y uv + z uv � 0 for u ! v 6 = t ! s , x ts : : d t � d s + y ts � 1 � : P u ! v r uv z uv  B y uv , z uv 0 all u ! v . � When we “primalize” this interdiction dual LP we get new primal variable � corresponding to the dual constraint P u ! v r uv z uv  B , and the z uv ’s give us a second set of capacities. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 11 / 32

LP Duality Dual of interdiction The linear program and its dual Repeat the new LP with dual variables: min P u ! v c uv y uv x uv : s.t. d u � d v + y uv + z uv � 0 for u ! v 6 = t ! s , x ts : : d t � d s + y ts � 1 � : P u ! v r uv z uv  B y uv , z uv 0 all u ! v . � When we “primalize” this interdiction dual LP we get new primal variable � corresponding to the dual constraint P u ! v r uv z uv  B , and the z uv ’s give us a second set of capacities. The primal interdiction LP is max x, λ ( x ts � B � ) d : s.t. conservation y uv : 0  x uv c uv  z uv : x uv � r uv �  0 . McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 11 / 32

LP Duality Dual of interdiction The linear program and its dual Repeat the primal interdiction LP and highlight the two capacities: max x, λ ( x ts � B � ) s.t. conservation 0  x uv c uv  x uv � r uv �  0 . McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 12 / 32

LP Duality Dual of interdiction The linear program and its dual Repeat the primal interdiction LP and highlight the two capacities: max x, λ ( x ts � B � ) s.t. conservation 0  x uv c uv  x uv � r uv �  0 . The two capacity constraints simplify into x uv  min( c uv , � r uv ) , a parametric capacity in the scalar parameter � . McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 12 / 32

LP Duality Dual of interdiction The linear program and its dual Repeat the primal interdiction LP and highlight the two capacities: max x, λ ( x ts � B � ) s.t. conservation 0  x uv c uv  x uv � r uv �  0 . The two capacity constraints simplify into x uv  min( c uv , � r uv ) , a parametric capacity in the scalar parameter � . So let’s investigate the behavior of this parametric min cut problem. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 12 / 32

Parametric Min Cut Parametric curves Outline Network Interdiction 1 What is it? Interdiction curves LP Duality 2 Dual of interdiction Parametric Min Cut 3 Parametric curves The Breakpoint Subproblem 4 What is it? Algorithms Discrete Newton Multiple Parameters 5 What is it? Scheduling problem Multi-GGT McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 13 / 32

Parametric Min Cut Parametric curves Parametric capacity of fixed cut S When � is small, cap( S, � ) = � cap r ( S ) . c 3 + ρ 3 ( r 1 + r 2 ) " parametric capacity slope r ( S ) = r 1 + r 2 + r 3 0 ρ 3 λ ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 14 / 32

Parametric Min Cut Parametric curves Parametric capacity of fixed cut S This continues as long as � r uv  c uv for all u ! v 2 � + ( S ) , or �  ⇢ uv . c 3 + ρ 3 ( r 1 + r 2 ) " parametric capacity slope r ( S ) = r 1 + r 2 + r 3 0 ρ 3 λ ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 14 / 32

Parametric Min Cut Parametric curves Parametric capacity of fixed cut S Thus the first breakpoint is when � hits min δ + ( S ) ⇢ uv . c 3 + ρ 3 ( r 1 + r 2 ) " parametric capacity slope r ( S ) = r 1 + r 2 + r 3 0 ρ 3 λ ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 14 / 32

Parametric Min Cut Parametric curves Parametric capacity of fixed cut S Thus the first breakpoint is when � hits min δ + ( S ) ⇢ uv . c 3 + c 2 + ρ 2 ( r 1 ) slope r 1 + r 2 c 3 + ρ 3 ( r 1 + r 2 ) " parametric capacity slope r ( S ) = r 1 + r 2 + r 3 0 ρ 3 ρ 2 λ ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 14 / 32

Parametric Min Cut Parametric curves Parametric capacity of fixed cut S Thus the first breakpoint is when � hits min δ + ( S ) ⇢ uv . cap c ( S ) slope r 1 c 3 + c 2 + ρ 2 ( r 1 ) slope r 1 + r 2 c 3 + ρ 3 ( r 1 + r 2 ) " parametric capacity slope r ( S ) = r 1 + r 2 + r 3 0 ρ 3 ρ 2 ρ 1 λ ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 14 / 32

Parametric Min Cut Parametric curves Parametric capacity of fixed cut S Thus the first breakpoint is when � hits min δ + ( S ) ⇢ uv . cap c ( S ) slope 0 slope r 1 c 3 + c 2 + ρ 2 ( r 1 ) slope r 1 + r 2 c 3 + ρ 3 ( r 1 + r 2 ) " parametric capacity slope r ( S ) = r 1 + r 2 + r 3 0 ρ 3 ρ 2 ρ 1 λ ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 14 / 32

Parametric Min Cut Parametric curves Parametric capacity of fixed cut S The parametric capacity curve for S is piecewise linear concave. cap c ( S ) slope 0 slope r 1 c 3 + c 2 + ρ 2 ( r 1 ) slope r 1 + r 2 c 3 + ρ 3 ( r 1 + r 2 ) " parametric capacity slope r ( S ) = r 1 + r 2 + r 3 0 ρ 3 ρ 2 ρ 1 λ ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 14 / 32

Parametric Min Cut Parametric curves Conjugate duality between interdiction and parametric capacity for S Both curves are piecewise linear; the interdiction curve (residual capacity curve) is convex, the parametric capacity curve is concave. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 15 / 32

Parametric Min Cut Parametric curves Conjugate duality between interdiction and parametric capacity for S Both curves are piecewise linear; the interdiction curve (residual capacity curve) is convex, the parametric capacity curve is concave. Index the arcs of � + ( S ) in descending order of ⇢ e . Then the breakpoints of S ’s interdiction curve are 0, r 1 , r 1 + r 2 , r 1 + r 2 + r 3 , . . . . The slopes of S ’s parametric capacity curve are . . . , r 1 + r 2 + r 3 , r 1 + r 2 , r 1 , 0. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 15 / 32

Parametric Min Cut Parametric curves Conjugate duality between interdiction and parametric capacity for S Both curves are piecewise linear; the interdiction curve (residual capacity curve) is convex, the parametric capacity curve is concave. Index the arcs of � + ( S ) in descending order of ⇢ e . Then the breakpoints of S ’s interdiction curve are 0, r 1 , r 1 + r 2 , r 1 + r 2 + r 3 , . . . . The slopes of S ’s parametric capacity curve are . . . , r 1 + r 2 + r 3 , r 1 + r 2 , r 1 , 0. The slopes of S ’s interdiction curve are � ⇢ 1 , � ⇢ 2 , . . . . The breakpoints of S ’s parametric capacity curve are . . . , ⇢ 3 , ⇢ 2 , ⇢ 1 . McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 15 / 32

Parametric Min Cut Parametric curves Conjugate duality between interdiction and parametric capacity for S Both curves are piecewise linear; the interdiction curve (residual capacity curve) is convex, the parametric capacity curve is concave. Index the arcs of � + ( S ) in descending order of ⇢ e . Then the breakpoints of S ’s interdiction curve are 0, r 1 , r 1 + r 2 , r 1 + r 2 + r 3 , . . . . The slopes of S ’s parametric capacity curve are . . . , r 1 + r 2 + r 3 , r 1 + r 2 , r 1 , 0. The slopes of S ’s interdiction curve are � ⇢ 1 , � ⇢ 2 , . . . . The breakpoints of S ’s parametric capacity curve are . . . , ⇢ 3 , ⇢ 2 , ⇢ 1 . Thus breakpoints and slopes are interchanged between S ’s interdiction curve and its parametric capacity curve, though in reverse order and modulo a minus sign. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 15 / 32

Parametric Min Cut Parametric curves Conjugate duality between interdiction and parametric capacity for S Both curves are piecewise linear; the interdiction curve (residual capacity curve) is convex, the parametric capacity curve is concave. Index the arcs of � + ( S ) in descending order of ⇢ e . Then the breakpoints of S ’s interdiction curve are 0, r 1 , r 1 + r 2 , r 1 + r 2 + r 3 , . . . . The slopes of S ’s parametric capacity curve are . . . , r 1 + r 2 + r 3 , r 1 + r 2 , r 1 , 0. The slopes of S ’s interdiction curve are � ⇢ 1 , � ⇢ 2 , . . . . The breakpoints of S ’s parametric capacity curve are . . . , ⇢ 3 , ⇢ 2 , ⇢ 1 . Thus breakpoints and slopes are interchanged between S ’s interdiction curve and its parametric capacity curve, though in reverse order and modulo a minus sign. In the language of conjugate duality, this is equivalent to saying that the parametric capacity curve cap( S, � ) is the negative of the conjugate dual of the interdiction curve for S , evaluated at � � . McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 15 / 32

Parametric Min Cut Parametric curves The overall parametric capacity curve: the λ -profile Now overlay the parametric capacity curves for all S . McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 16 / 32

Parametric Min Cut Parametric curves The overall parametric capacity curve: the λ -profile Now overlay the parametric capacity curves for all S . cap c ( S ) S slope 0 c 3 + c 2 + ρ 2 ( r 1 ) c 3 + ρ 3 ( r 1 + r 2 ) " parametric capacity slope r ( S ) 0 ρ 3 ρ 2 ρ 1 McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 16 / 32 λ !

Parametric Min Cut Parametric curves The overall parametric capacity curve: the λ -profile Now overlay the parametric capacity curves for all S . cap c ( S ) S slope 0 cap c ( T ) = cap ⇤ c c 3 + c 2 + ρ 2 ( r 1 ) T c 3 + ρ 3 ( r 1 + r 2 ) " parametric capacity slope r ( S ) 0 ρ 3 ρ 2 ρ 1 McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 16 / 32 λ !

Parametric Min Cut Parametric curves The overall parametric capacity curve: the λ -profile Now overlay the parametric capacity curves for all S . cap c ( S ) S slope 0 cap c ( U ) cap c ( T ) = cap ⇤ c c 3 + c 2 + ρ 2 ( r 1 ) T U c 3 + ρ 3 ( r 1 + r 2 ) " parametric capacity slope r ( S ) 0 ρ 3 ρ 2 ρ 1 McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 16 / 32 λ !

Parametric Min Cut Parametric curves The overall parametric capacity curve: the λ -profile For a fixed value of � , we want to find the S whose parametric capacity at � is minimum, so we just want the pointwise minimum of all these curves. cap c ( S ) S slope 0 cap c ( U ) cap c ( T ) = cap ⇤ c c 3 + c 2 + ρ 2 ( r 1 ) T U c 3 + ρ 3 ( r 1 + r 2 ) " parametric capacity slope r ( S ) 0 ρ 3 ρ 2 ρ 1 McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 16 / 32 λ !

Parametric Min Cut Parametric curves The overall parametric capacity curve: the λ -profile For a fixed value of � , we want to find the S whose parametric capacity at � is minimum, so we just want the pointwise minimum of all these curves. cap c ( S ) S slope 0 cap c ( U ) cap c ( T ) = cap ⇤ c c 3 + c 2 + ρ 2 ( r 1 ) T breakpoint is interesection of S and U ’s curves U c 3 + ρ 3 ( r 1 + r 2 ) breakpoint is a breakpoint of S ’s curve " parametric capacity slope r ( S ) 0 ρ 3 ρ 2 ρ 1 McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 16 / 32 λ !

Parametric Min Cut Parametric curves The overall parametric capacity curve: the λ -profile Since the minimum of a bunch of concave curves is again concave, this time we do not need to linearize. We call this overall parametric capacity curve the � -profile. cap c ( S ) S slope 0 cap c ( U ) cap c ( T ) = cap ⇤ c c 3 + c 2 + ρ 2 ( r 1 ) T breakpoint is interesection of S and U ’s curves U c 3 + ρ 3 ( r 1 + r 2 ) breakpoint is a breakpoint of S ’s curve " parametric capacity slope r ( S ) 0 ρ 3 ρ 2 ρ 1 McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 16 / 32 λ !

Parametric Min Cut Parametric curves The overall parametric capacity curve: the λ -profile We can compute things like cap ⇤ ( � ) easily using parametric min cut tech- nology. cap c ( S ) S slope 0 cap c ( U ) cap c ( T ) = cap ⇤ c c 3 + c 2 + ρ 2 ( r 1 ) T breakpoint is interesection of S and U ’s curves U c 3 + ρ 3 ( r 1 + r 2 ) breakpoint is a breakpoint of S ’s curve " parametric capacity slope r ( S ) 0 ρ 3 ρ 2 ρ 1 McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 16 / 32 λ !

Parametric Min Cut Parametric curves The overall parametric capacity curve: the λ -profile We can show that the conjugate duality between S ’s interdiction and para- metric capacity curves carries over to conjugate duality between the B - profile and the � -profile. cap c ( S ) S slope 0 cap c ( U ) cap c ( T ) = cap ⇤ c c 3 + c 2 + ρ 2 ( r 1 ) T breakpoint is interesection of S and U ’s curves U c 3 + ρ 3 ( r 1 + r 2 ) breakpoint is a breakpoint of S ’s curve " parametric capacity slope r ( S ) 0 ρ 3 ρ 2 ρ 1 McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 16 / 32 λ !

Parametric Min Cut Parametric curves The overall parametric capacity curve: the λ -profile Recall that to get our pseudo-approximation for a given B , we want to compute the two cuts S 1 and S 2 bracketing B on the B -profile. cap c ( S ) S slope 0 cap c ( U ) cap c ( T ) = cap ⇤ c c 3 + c 2 + ρ 2 ( r 1 ) T breakpoint is interesection of S and U ’s curves U c 3 + ρ 3 ( r 1 + r 2 ) breakpoint is a breakpoint of S ’s curve " parametric capacity slope r ( S ) 0 ρ 3 ρ 2 ρ 1 McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 16 / 32 λ !

Parametric Min Cut Parametric curves The overall parametric capacity curve: the λ -profile Conjugate duality implies that this is equivalent to finding a breakpoint on the � -profile whose adjacent slopes bracket B , here S and U . cap c ( S ) S slope 0 slope B cap c ( U ) cap c ( T ) = cap ⇤ c c 3 + c 2 + ρ 2 ( r 1 ) T breakpoint is interesection of S and U ’s curves U c 3 + ρ 3 ( r 1 + r 2 ) breakpoint is a breakpoint of S ’s curve " parametric capacity slope r ( S ) 0 ρ 3 ρ 2 ρ 1 McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 16 / 32 λ !

The Breakpoint Subproblem What is it? Outline Network Interdiction 1 What is it? Interdiction curves LP Duality 2 Dual of interdiction Parametric Min Cut 3 Parametric curves The Breakpoint Subproblem 4 What is it? Algorithms Discrete Newton Multiple Parameters 5 What is it? Scheduling problem Multi-GGT McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 17 / 32

The Breakpoint Subproblem What is it? The key breakpoint subproblem Notice that any breakpoint ˆ � of the � -profile is defined by the intersection of a segment to its left coming from cut S � (ˆ � ) with local slope sl � (ˆ � ) , and a segment to its right coming from cut S + (ˆ � ) with local slope sl + (ˆ � ) , with sl � (ˆ � ) > sl + (ˆ � ) by concavity. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 18 / 32

The Breakpoint Subproblem What is it? The key breakpoint subproblem Notice that any breakpoint ˆ � of the � -profile is defined by the intersection of a segment to its left coming from cut S � (ˆ � ) with local slope sl � (ˆ � ) , and a segment to its right coming from cut S + (ˆ � ) with local slope sl + (ˆ � ) , with sl � (ˆ � ) > sl + (ˆ � ) by concavity. The subproblem we now want to solve combinatorially: Given B , find breakpoint � B of the � -profile such that sl + ( � B )  B  sl � ( � B ) , along with the corresponding S � ( � B ) and S + ( � B ) . McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 18 / 32

The Breakpoint Subproblem What is it? The key breakpoint subproblem Notice that any breakpoint ˆ � of the � -profile is defined by the intersection of a segment to its left coming from cut S � (ˆ � ) with local slope sl � (ˆ � ) , and a segment to its right coming from cut S + (ˆ � ) with local slope sl + (ˆ � ) , with sl � (ˆ � ) > sl + (ˆ � ) by concavity. The subproblem we now want to solve combinatorially: Given B , find breakpoint � B of the � -profile such that sl + ( � B )  B  sl � ( � B ) , along with the corresponding S � ( � B ) and S + ( � B ) . A technical detail: Suppose I give you � B . Can you then use it to compute S � ( � B ) and S + ( � B ) ? McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 18 / 32

The Breakpoint Subproblem What is it? The key breakpoint subproblem Notice that any breakpoint ˆ � of the � -profile is defined by the intersection of a segment to its left coming from cut S � (ˆ � ) with local slope sl � (ˆ � ) , and a segment to its right coming from cut S + (ˆ � ) with local slope sl + (ˆ � ) , with sl � (ˆ � ) > sl + (ˆ � ) by concavity. The subproblem we now want to solve combinatorially: Given B , find breakpoint � B of the � -profile such that sl + ( � B )  B  sl � ( � B ) , along with the corresponding S � ( � B ) and S + ( � B ) . A technical detail: Suppose I give you � B . Can you then use it to compute S � ( � B ) and S + ( � B ) ? Yes: We can use a combination of Picard-Queyranne decomposition w.r.t. an optimal flow at � B , and min flow / max cut in the residual network to find them McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 18 / 32

The Breakpoint Subproblem What is it? The key breakpoint subproblem Notice that any breakpoint ˆ � of the � -profile is defined by the intersection of a segment to its left coming from cut S � (ˆ � ) with local slope sl � (ˆ � ) , and a segment to its right coming from cut S + (ˆ � ) with local slope sl + (ˆ � ) , with sl � (ˆ � ) > sl + (ˆ � ) by concavity. The subproblem we now want to solve combinatorially: Given B , find breakpoint � B of the � -profile such that sl + ( � B )  B  sl � ( � B ) , along with the corresponding S � ( � B ) and S + ( � B ) . A technical detail: Suppose I give you � B . Can you then use it to compute S � ( � B ) and S + ( � B ) ? Yes: We can use a combination of Picard-Queyranne decomposition w.r.t. an optimal flow at � B , and min flow / max cut in the residual network to find them So let’s just concentrate on finding � B . McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 18 / 32

The Breakpoint Subproblem Algorithms Binary search solves it 1 Set � L = 0 and � R = cap ⇤ r ; then all interesting values of � are in [ � L , � R ] . McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 19 / 32

The Breakpoint Subproblem Algorithms Binary search solves it 1 Set � L = 0 and � R = cap ⇤ r ; then all interesting values of � are in [ � L , � R ] . 2 Compute ˆ � = ( � L + � R ) / 2 , a max flow w.r.t. ˆ � , and sl � (ˆ � ) and sl + (ˆ � ) . McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 19 / 32

The Breakpoint Subproblem Algorithms Binary search solves it 1 Set � L = 0 and � R = cap ⇤ r ; then all interesting values of � are in [ � L , � R ] . 2 Compute ˆ � = ( � L + � R ) / 2 , a max flow w.r.t. ˆ � , and sl � (ˆ � ) and sl + (ˆ � ) . 3 If B 2 [sl + (ˆ � ) , sl � (ˆ � )] , then � B = ˆ � and we can stop. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 19 / 32

The Breakpoint Subproblem Algorithms Binary search solves it 1 Set � L = 0 and � R = cap ⇤ r ; then all interesting values of � are in [ � L , � R ] . 2 Compute ˆ � = ( � L + � R ) / 2 , a max flow w.r.t. ˆ � , and sl � (ˆ � ) and sl + (ˆ � ) . 3 If B 2 [sl + (ˆ � ) , sl � (ˆ � )] , then � B = ˆ � and we can stop. 4 Otherwise, if B < sl + (ˆ � ) then replace � L by ˆ � ; else ( B > sl � (ˆ � ) ) replace � R by ˆ � and go to 2. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 19 / 32

The Breakpoint Subproblem Algorithms Binary search solves it 1 Set � L = 0 and � R = cap ⇤ r ; then all interesting values of � are in [ � L , � R ] . 2 Compute ˆ � = ( � L + � R ) / 2 , a max flow w.r.t. ˆ � , and sl � (ˆ � ) and sl + (ˆ � ) . 3 If B 2 [sl + (ˆ � ) , sl � (ˆ � )] , then � B = ˆ � and we can stop. 4 Otherwise, if B < sl + (ˆ � ) then replace � L by ˆ � ; else ( B > sl � (ˆ � ) ) replace � R by ˆ � and go to 2. This runs in something like Θ (log( nD )) time, where D is the size of the data. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 19 / 32

The Breakpoint Subproblem Algorithms Binary search solves it 1 Set � L = 0 and � R = cap ⇤ r ; then all interesting values of � are in [ � L , � R ] . 2 Compute ˆ � = ( � L + � R ) / 2 , a max flow w.r.t. ˆ � , and sl � (ˆ � ) and sl + (ˆ � ) . 3 If B 2 [sl + (ˆ � ) , sl � (ˆ � )] , then � B = ˆ � and we can stop. 4 Otherwise, if B < sl + (ˆ � ) then replace � L by ˆ � ; else ( B > sl � (ˆ � ) ) replace � R by ˆ � and go to 2. This runs in something like Θ (log( nD )) time, where D is the size of the data. Can we do better? McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 19 / 32

The Breakpoint Subproblem Discrete Newton Discrete Newton gives a better algorithm r as before. Denote sl + ( � L ) by sl + Set � L = 0 and � R = cap ⇤ L and sl � ( � R ) by sl � R . S + 1 L a 1 L λ + b 1 L cap ⇤ ( λ ) cap λ 1 L ( S + 1 L ) " cap λ ( S ) λ 1 L λ 1 R λ ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 20 / 32

The Breakpoint Subproblem Discrete Newton Discrete Newton gives a better algorithm r as before. Denote sl + ( � L ) by sl + Set � L = 0 and � R = cap ⇤ L and sl � ( � R ) by sl � R . S + 1 L a 1 L λ + b 1 L cap ⇤ ( λ ) cap λ 1 L ( S + 1 L ) " cap λ ( S ) λ 1 L λ 1 R λ ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 20 / 32

The Breakpoint Subproblem Discrete Newton Discrete Newton gives a better algorithm r as before. Denote sl + ( � L ) by sl + Set � L = 0 and � R = cap ⇤ L and sl � ( � R ) by sl � R . S + 1 L a 1 L λ + b 1 L S � 1 R cap λ 1 R ( S � 1 R ) = cap λ 1 R ( S + 1 R ) a 1 R λ + b 1 R cap ⇤ ( λ ) cap λ 1 L ( S + 1 L ) " cap λ ( S ) λ 1 L λ 1 R λ ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 20 / 32

The Breakpoint Subproblem Discrete Newton Discrete Newton gives a better algorithm ˆ sl + Compute as the intersection of the line of slope through � L ( � L , cap ⇤ ( � L )) , and the line of slope sl � R through ( � R , cap ⇤ ( � R )) , a max flow w.r.t. ˆ � , and sl � (ˆ � ) and sl + (ˆ � ) . S + 1 L a 1 L λ + b 1 L S � 1 R cap λ 1 R ( S � 1 R ) = cap λ 1 R ( S + 1 R ) a 1 R λ + b 1 R cap ⇤ ( λ ) " cap λ ( S ) λ 1 L λ 2 = λ 2 L λ 1 R λ ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 20 / 32

The Breakpoint Subproblem Discrete Newton Discrete Newton gives a better algorithm If B 2 [sl + (ˆ � ) , sl � (ˆ � )] , then � B = ˆ � and we can stop. S + 1 L a 1 L λ + b 1 L S � 1 R cap λ 1 R ( S � 1 R ) = cap λ 1 R ( S + 1 R ) a 1 R λ + b 1 R cap ⇤ ( λ ) " cap λ ( S ) λ 1 L λ 2 = λ 2 L λ 1 R λ ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 20 / 32

The Breakpoint Subproblem Discrete Newton Discrete Newton gives a better algorithm Otherwise, if B < sl + (ˆ � ) then replace � L by ˆ � ; else ( B > sl � (ˆ � ) ) replace � R by ˆ � and go to 2. S + 1 L a 1 L λ + b 1 L S � 1 R = S � 2 R cap λ 1 R ( S � 1 R ) = cap λ 1 R ( S + 1 R ) = a 1 R λ + b 1 R = cap λ 2 ( S + 1 L ) = cap λ 2 ( S � 1 R ) cap ⇤ ( λ ) cap λ 2 L ( S + 2 L ) λ B S + 2 L a 2 L λ + b 2 L cap λ 1 L ( S + 2 L ) cap λ 1 L ( S + 1 L ) " cap λ ( S ) λ 1 L λ 2 = λ 2 L λ 3 λ 1 R = λ 2 R λ ! McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 20 / 32

The Breakpoint Subproblem Discrete Newton Defining gaps How can we analyze the running time of this Newton- B algorithm? McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 21 / 32

The Breakpoint Subproblem Discrete Newton Defining gaps How can we analyze the running time of this Newton- B algorithm? Let’s think in terms of lines of slope B . Let L ⇤ denote the line of slope B through the (as-yet unknown) point ( � B , cap ⇤ ( � B )) . This L ⇤ is the highest possible line of slope B through any point of the � -profile. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 21 / 32

The Breakpoint Subproblem Discrete Newton Defining gaps How can we analyze the running time of this Newton- B algorithm? Let’s think in terms of lines of slope B . Let L ⇤ denote the line of slope B through the (as-yet unknown) point ( � B , cap ⇤ ( � B )) . This L ⇤ is the highest possible line of slope B through any point of the � -profile. Thus the line of slope B through, e.g., ( � L , cap ⇤ ( � L )) lies below L ⇤ . McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 21 / 32

The Breakpoint Subproblem Discrete Newton Defining gaps How can we analyze the running time of this Newton- B algorithm? Let’s think in terms of lines of slope B . Let L ⇤ denote the line of slope B through the (as-yet unknown) point ( � B , cap ⇤ ( � B )) . This L ⇤ is the highest possible line of slope B through any point of the � -profile. Thus the line of slope B through, e.g., ( � L , cap ⇤ ( � L )) lies below L ⇤ . Since the lines defining ˆ � are tangents to the � -profile, their intersection must lie above L ⇤ , and so the line of slope B through this intersection point lies above L ⇤ . McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 21 / 32

The Breakpoint Subproblem Discrete Newton Defining gaps How can we analyze the running time of this Newton- B algorithm? Let’s think in terms of lines of slope B . Let L ⇤ denote the line of slope B through the (as-yet unknown) point ( � B , cap ⇤ ( � B )) . This L ⇤ is the highest possible line of slope B through any point of the � -profile. Thus the line of slope B through, e.g., ( � L , cap ⇤ ( � L )) lies below L ⇤ . Since the lines defining ˆ � are tangents to the � -profile, their intersection must lie above L ⇤ , and so the line of slope B through this intersection point lies above L ⇤ . Define vgap L to be the vertical distance between the line of slope B through the intersection point, and the line of slope B through ( � L , cap ⇤ ( � L )) , and similarly for vgap R . McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 21 / 32

The Breakpoint Subproblem Discrete Newton Defining gaps How can we analyze the running time of this Newton- B algorithm? Let’s think in terms of lines of slope B . Let L ⇤ denote the line of slope B through the (as-yet unknown) point ( � B , cap ⇤ ( � B )) . This L ⇤ is the highest possible line of slope B through any point of the � -profile. Thus the line of slope B through, e.g., ( � L , cap ⇤ ( � L )) lies below L ⇤ . Since the lines defining ˆ � are tangents to the � -profile, their intersection must lie above L ⇤ , and so the line of slope B through this intersection point lies above L ⇤ . Define vgap L to be the vertical distance between the line of slope B through the intersection point, and the line of slope B through ( � L , cap ⇤ ( � L )) , and similarly for vgap R . Also define slgap L to be sl + L � B and slgap R to be B � sl � R . McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 21 / 32

The Breakpoint Subproblem Discrete Newton vgap illustrated S + 1 L a 1 L λ + b 1 L S � 1 R = S � 1 R ) = cap λ 1 R ( S + 2 R vgap 1 R cap λ 1 R ( S � 1 R ) = a 1 R λ + b 1 R = cap λ 2 ( S + 1 L ) = cap λ 2 ( S � 1 R ) vgap 2 R vgap 2 L cap ⇤ ( λ ) B ( λ 1 L � λ 2 ) + cap λ 2 ( S + 1 L ) cap λ 2 L ( S + 2 L ) λ B B ( λ 1 L � λ 3 ) + cap λ 3 ( S + 2 L ) S + 2 L a 2 L λ + b 2 L vgap 1 L cap λ 1 L ( S + 2 L ) cap λ 1 L ( S + 1 L ) " cap λ ( S ) λ 2 = λ 2 L λ 1 R = λ 2 R λ 1 L λ 3 McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 22 / 32 λ !

The Breakpoint Subproblem Discrete Newton The key inequality We use primes to denote new values. When ˆ � becomes the new � L then the key inequality is + slgap 0 vgap 0 L L < 1 (1) vgap L slgap L McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 23 / 32

The Breakpoint Subproblem Discrete Newton The key inequality We use primes to denote new values. When ˆ � becomes the new � L then the key inequality is + slgap 0 vgap 0 L L < 1 (1) vgap L slgap L This immediately implies that at each iteration, one of vgap L , vgap R , slgap L , or slgap R is cut down by a factor of at least 2. Thus Newton- B is never worse than Binary Search. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 23 / 32

The Breakpoint Subproblem Discrete Newton The key inequality We use primes to denote new values. When ˆ � becomes the new � L then the key inequality is + slgap 0 vgap 0 L L < 1 (1) vgap L slgap L This immediately implies that at each iteration, one of vgap L , vgap R , slgap L , or slgap R is cut down by a factor of at least 2. Thus Newton- B is never worse than Binary Search. (1) was originally proved in Mc+Ervolina ’94. Then Rote, and Radzik ’92–’98 showed that Newton- B is sometimes faster than Binary Search, and has a strongly polynomial bound. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 23 / 32

The Breakpoint Subproblem Discrete Newton Some implications We didn’t use much network structure in this analysis. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 24 / 32

The Breakpoint Subproblem Discrete Newton Some implications We didn’t use much network structure in this analysis. Thus we could define an interdiction version of any capacitated problem; primalizing would give a conjugate dual parametric problem that we could then solve via Newton- B . The Burch et al pseudo-approximation framework carries through also. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 24 / 32

The Breakpoint Subproblem Discrete Newton Some implications We didn’t use much network structure in this analysis. Thus we could define an interdiction version of any capacitated problem; primalizing would give a conjugate dual parametric problem that we could then solve via Newton- B . The Burch et al pseudo-approximation framework carries through also. Indeed, this Newton- B algorithm and its analysis works for any concave (or convex) function, even continuous ones. McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 24 / 32

Multiple Parameters What is it? Outline Network Interdiction 1 What is it? Interdiction curves LP Duality 2 Dual of interdiction Parametric Min Cut 3 Parametric curves The Breakpoint Subproblem 4 What is it? Algorithms Discrete Newton Multiple Parameters 5 What is it? Scheduling problem Multi-GGT McCormick et al (UBC-Rome-Berlin) Parametric Interdiction CanaDAM June 2013 25 / 32