Risk Averse Shortest Path Interdiction Yongjia Song 1 and Siqian Shen 2 1: Virginia Commonwealth University 2: University of Michigan ICS 2015, Richmond, VA Song and Shen Risk Averse Shortest Path Interdiction 1/28
Network interdiction: a two-player game Stackelberg game (two player; sequential moves) played on a network. (a) Leader: Interdictor (b) Follower: Operator ◮ Goal: maximally restrict a follower’s utility gained in the network by damaging arcs or nodes. Song and Shen Risk Averse Shortest Path Interdiction 2/28
Applications ◮ Smugglers (followers) evade authorities (leaders) who lead the game by placing checkpoints. ◮ Emergency service providers (leaders) allocate resources and fortify arcs/nodes against malicious attacks (followers). ◮ . . . Song and Shen Risk Averse Shortest Path Interdiction 3/28
Deterministic Network Interdiction A shortest path network interdiction on Graph G ( V , A ) : ◮ x a ∈ { 0 , 1 } , ∀ a ∈ A : whether or not interdict arc a ◮ y a ∈ { 0 , 1 } , ∀ a ∈ A : whether or not arc a is on the path chosen by the follower Assume d a = 3 , ∀ a , and we � max x ∈ X min ( c a + d a x a ) y a y can interdict up to two a ∈ A arcs 1 if i = s � � s.t. y a − y a = , ∀ i ∈ V − 1 if i = t a ∈ δ + ( i ) a ∈ δ − ( i ) 0 o.w. y a ≥ 0 , ∀ a ∈ A where X = { x ∈ { 0 , 1 } | A | | � a ∈ A r a x a ≤ R } Song and Shen Risk Averse Shortest Path Interdiction 4/28
Deterministic Network Interdiction A shortest path network interdiction on Graph G ( V , A ) : ◮ x a ∈ { 0 , 1 } , ∀ a ∈ A : whether or not interdict arc a ◮ y a ∈ { 0 , 1 } , ∀ a ∈ A : whether or not arc a is on the path chosen by the follower Assume d a = 3 , ∀ a , and we � max x ∈ X min ( c a + d a x a ) y a can interdict up to two y a ∈ A arcs 1 if i = s � � s.t. y a − y a = , ∀ i ∈ V − 1 if i = t a ∈ δ + ( i ) a ∈ δ − ( i ) 0 o.w. y a ≥ 0 , ∀ a ∈ A where X = { x ∈ { 0 , 1 } | A | | � a ∈ A r a x a ≤ R } Song and Shen Risk Averse Shortest Path Interdiction 4/28
Deterministic Network Interdiction A shortest path network interdiction on Graph G ( V , A ) : ◮ x a ∈ { 0 , 1 } , ∀ a ∈ A : whether or not interdict arc a ◮ y a ∈ { 0 , 1 } , ∀ a ∈ A : whether or not arc a is on the path chosen by the follower Assume d a = 3 , ∀ a , and we � max x ∈ X min ( c a + d a x a ) y a y can interdict up to two a ∈ A arcs 1 if i = s � � s.t. y a − y a = , ∀ i ∈ V − 1 if i = t a ∈ δ + ( i ) a ∈ δ − ( i ) 0 o.w. y a ≥ 0 , ∀ a ∈ A where X = { x ∈ { 0 , 1 } | A | | � a ∈ A r a x a ≤ R } Song and Shen Risk Averse Shortest Path Interdiction 4/28
Solution Approaches (Morton 2011) Given a relaxed interdiction ^ x , the follower chooses a shortest path using c a + d a ^ x a as the length for each arc a : ◮ Extended formulation: take the dual of the inner shortest path LP x ∈ X ,π π t max s.t. π j − π i ≤ c a + d a x a , ∀ a = ( i , j ) ∈ A π s = 0 ◮ Benders formulation: � max min ( c a + d a x a ) x ∈ X P ∈P a ∈ P Song and Shen Risk Averse Shortest Path Interdiction 5/28
Solution Approaches (Morton 2011) Given a relaxed interdiction ^ x , the follower chooses a shortest path using c a + d a ^ x a as the length for each arc a : ◮ Extended formulation: take the dual of the inner shortest path LP x ∈ X ,π π t max s.t. π j − π i ≤ c a + d a x a , ∀ a = ( i , j ) ∈ A π s = 0 ◮ Benders formulation: � max x ∈ X { θ | θ ≤ ( c a + d a x a ) , ∀ P ∈ P } a ∈ P Song and Shen Risk Averse Shortest Path Interdiction 5/28
Stochastic Network Interdiction c a and interdiction effects ˜ Assume the arc lengths ˜ d a are uncertain, and the uncertainty can be characterized by a finite set of scenarios { ( c k a , d k a ) } k ∈ N � � ( c k a + x a d k a ) y k max p k min a x ∈ X y k k ∈ N a ∈ A 1 if i = s � � y k y k s.t. a − a = − 1 if i = t , ∀ i ∈ V , ∀ k a ∈ δ + ( i ) a ∈ δ − ( i ) 0 o.w. y k a ≥ 0 , ∀ a ∈ A , ∀ k ∈ N Song and Shen Risk Averse Shortest Path Interdiction 6/28
Stochastic Network Interdiction c a and interdiction effects ˜ Assume the arc lengths ˜ d a are uncertain, and the uncertainty can be characterized by a finite set of scenarios { ( c k a , d k a ) } k ∈ N � p k θ k max x ∈ X k ∈ N � s.t. θ k ≤ ( c k a + d k a x a ) , ∀ P ∈ P a ∈ P ◮ Benders formulation is preferred, since it enables scenario decomposition ◮ Could be strengthened by additional valid inequalities, e.g., the step inequalities (Pan and Morton 2008) Song and Shen Risk Averse Shortest Path Interdiction 6/28
Stochastic Network Interdiction c a and interdiction effects ˜ Assume the arc lengths ˜ d a are uncertain, and the uncertainty can be characterized by a finite set of scenarios { ( c k a , d k a ) } k ∈ N � p k θ k max x ∈ X k ∈ N � s.t. θ k ≤ ( c k a + d k a x a ) , ∀ P ∈ P a ∈ P ◮ Benders formulation is preferred, since it enables scenario decomposition ◮ Could be strengthened by additional valid inequalities, e.g., the step inequalities (Pan and Morton 2008) Limit: the risk aversion of the players are not considered Song and Shen Risk Averse Shortest Path Interdiction 6/28
Risk Averse Shortest Path Interdiction (RASPI) Model risk aversion by chance constraint: risk averse interdictor (leader) targets on high probability of enforcing a long distance for the traveler Two settings: ◮ Wait-and-see follower: make optimal response after observing the random outcome ◮ We do not need the follower’s risk attitude in this case ◮ Traditional stochastic shortest path interdiction problem assumes a risk neutral leader ◮ Here-and-now follower: must make a decision before the observation of the random outcome ◮ We assume the follower is risk neutral in the here-and-now setting: choose a path that has the shortest expected distance Song and Shen Risk Averse Shortest Path Interdiction 7/28
Outline Chance-constrained RASPI with Wait-and-see Follower Chance-constrained RASPI with Here-and-now Follower Song and Shen Risk Averse Shortest Path Interdiction 8/28
Outline Chance-constrained RASPI with Wait-and-see Follower Chance-constrained RASPI with Here-and-now Follower Song and Shen Risk Averse Shortest Path Interdiction 9/28
Risk averse interdictor with wait-and-see follower: a chance-constrained model Idea: Ensure that the follower’s shortest possible traveling distance from s to t exceeds a given length φ with high probability x , z r ⊤ x min � ( c k a + d k a x a ) y k s.t. a ( x ) ≥ φ z k , ∀ k ∈ N , a ∈ A � p k z k ≥ 1 − ǫ, k ∈ N z k ∈ { 0 , 1 } , ∀ k ∈ N , x a ∈ { 0 , 1 } , ∀ a ∈ A � where y k ( x ) ∈ arg min ( c k a + d k a x a ) y a , Y : flow balance equations y ∈ Y a ∈ A ◮ z k ∈ { 0 , 1 } : whether or not scenario k is satisfied Song and Shen Risk Averse Shortest Path Interdiction 10/28
Risk averse interdictor with wait-and-see follower: a chance-constrained model Idea: Ensure that the follower’s shortest possible traveling distance from s to t exceeds a given length φ with high probability x , z r ⊤ x min � ( c k a + d k s.t. a x a ) ≥ φ z k , ∀ P ∈ P , ∀ k ∈ N , a ∈ P � p k z k ≥ 1 − ǫ, k ∈ N z k ∈ { 0 , 1 } , ∀ k ∈ N , x a ∈ { 0 , 1 } , ∀ a ∈ A ◮ z k ∈ { 0 , 1 } : whether or not scenario k is satisfied Song and Shen Risk Averse Shortest Path Interdiction 10/28
Standard Benders decomposition Given a relaxation solution ^ x , ^ z of the master problem (with a subset of paths) ◮ Solve a shortest path problem for each scenario k using c k a + d k x a as the arc length, and get the shortest path P k a ^ ◮ Check if inequality � a ∈ P k ( c k a + d k a ^ x a ) ≥ φ ^ z k is violated, and add a Benders cut if so ◮ Could be applied for both integer and fractional solutions Song and Shen Risk Averse Shortest Path Interdiction 11/28
Implicit covering structure Scenario-based path inequality: � ( c k a + d k a x a ) ≥ φ z k , ∀ P ∈ P a ∈ P Song and Shen Risk Averse Shortest Path Interdiction 12/28
Implicit covering structure Scenario-based path inequality: � d k a x a ≥ ( φ − l k P ) z k , ∀ P ∈ P a ∈ P where l k P is the length of path P using c k a as the arc length Song and Shen Risk Averse Shortest Path Interdiction 12/28
Implicit covering structure Scenario-based path inequality: � d k a x a ≥ ( φ − l k P ) z k , ∀ P ∈ P a ∈ P where l k P is the length of path P using c k a as the arc length Structure: exponentially many covering constraints ◮ Related to Song and Luedtke (2013), � a ∈ C k x a ≥ z k , “scenario-based graph cut inequalities” ◮ Related to Song, Luedtke, and Kücükyavuz (2014), multi-dimensional binary packing problems with a small (non-exponential) number of constraints Song and Shen Risk Averse Shortest Path Interdiction 12/28
Pack-based formulation: Motivation Fix a scenario k , given a set of arcs C , if none is interdicted in C , we cannot achieve the target ⇒ C is a pack in that scenario k ! � � c k ( c k a + d k ∃ P ∈ P : a + a ) < φ a ∈ P ∩ C a ∈ P \ C Song and Shen Risk Averse Shortest Path Interdiction 13/28
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