On Integer and Bilevel Formulations for the k -Vertex Cut Problem Ivana Ljubi´ c • , joint work with Fabio Furini ◦ , Enrico Malaguti ∗ and Paolo Paronuzzi ∗ ∗ DEI “Guglielmo Marconi”, University of Bologna ◦ Paris Dauphine University • ESSEC Business School, Paris INOC 2019, June 12-14 2019, Avignon On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 1
Problem setting and motivation On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 2
Problem setting A k-vertex cut is a subset of vertices whose removal disconnects the graph in at least k (not-empty) components. On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 3
Problem setting Example of a 3-vertex cut: On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 4
The k -Vertex Cut Problem Definition Given an undirected graph G = ( V , E ) with vertex weights w v , v ∈ V , and a integer k ≥ 2, find a subset of vertices of minimum weight whose removal disconnects G in at least k (not-empty) components. On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 5
Motivation Family of Critical Node Detection Problems (M. Lalou, M. A. Tahraoui, and H. Kheddouci. The critical node detection problem in networks: A survey. Computer Science Review, 2018); Analysis of networks (D. Kempe, J. Kleinberg, and ´ E. Tardos. Influential nodes in a diffusion model for social networks. Automata, Languages and Programming, 2005.); Decomposition method for linear equation systems . On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 6
Compact Model and Representative Formulation On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 7
Compact formulation We associate a binary variable y i v to all vertices v ∈ V and for all integers i ∈ K , such that: � 1 if vertex v belongs to component i y i i ∈ K , v ∈ V . v = 0 otherwise On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 8
Compact Model Compact ILP formulation for k-Vertex-Cut Problem: � � � w v y i w v − min v v ∈ V i ∈ K v ∈ V � y i v ≤ 1 v ∈ V , i ∈ K y i u + y j v ≤ 1 i � = j ∈ K , uv ∈ E , � y i v ≥ 1 i ∈ K , v ∈ V y i v ∈ { 0 , 1 } i ∈ K , v ∈ V . Drawbacks: LP-optimal solution is zero (set all y i v = 1 / k ), symmetries, etc. On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 9
Bilevel approach On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 10
Bilevel approach Property A graph G has at least k (not empty) components if and only if any cycle-free subgraph of G contains at most | V | − k edges. Example with | V | = 9 and k = 3: On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 11
Bilevel approach Property A graph G has at least k (not empty) components if and only if any cycle-free subgraph of G contains at most | V | − k edges. Example with | V | = 9 and k = 3: On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 12
Bilevel approach The k -vertex cut problem can be seen as a Stackelberg game: the leader searches the smallest subset of vertices V 0 to delete; the follower maximizes the size of the cycle-free subgraph on the residual graph. Property The solution V 0 ⊂ V of the leader is feasible if and only if the value of the optimal follower’s response (i.e., the size of the maximum cycle-free subgraph in the remaining graph) is at most | V | − | V 0 | − k . On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 13
Bilevel approach The leader decisions: � 1 if vertex v is in the k -vertex cut x v = v ∈ V 0 otherwise For the decisions of the follower, we use additional binary variables associated with the edges of G : � 1 if edge uv is selected to be in the cycle-free subgraph uv ∈ E e uv = 0 otherwise On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 14
Bilevel approach The Bilevel ILP formulation of the k -vertex cut problem reads as follows: � min x v v ∈ V � Φ( x ) ≤ | V | − x v − k v ∈ V x v ∈ { 0 , 1 } v ∈ V . Φ( x ) is the optimal solution value of the follower subproblem for a given x . Value Function Reformulation . Value function Φ( x ) is neither convex, nor concave, nor connected... On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 15
How do we calculate Φ( x )? For a solution x ∗ of the leader, which denotes a set V 0 of interdicted vertices, the follower’s subproblem is: � Φ( x ∗ ) = max e uv uv ∈ E e ( S ) ≤ | S | − 1 S ⊆ V , S � = ∅ , e uv ≤ 1 − x ∗ uv ∈ E , u e uv ≤ 1 − x ∗ uv ∈ E , v e uv ∈ { 0 , 1 } uv ∈ E . On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 16
Bilevel approach We can prove that the follower’s subproblem is equivalently restated as: � Φ( x ∗ ) = z uv (1 − x ∗ u − x ∗ max v ) uv ∈ E z ( S ) ≤ | S | − 1 S ⊆ V , S � = ∅ z uv ∈ { 0 , 1 } uv ∈ E . Convexification of the value function Φ( x ) On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 17
Bilevel approach Since the space of feasible solutions of the redefined follower subproblem does not depend on the leader anymore, the non-linear constraint from the BILP formulation: � Φ( x ) ≤ | V | − x v − k v ∈ V can now be replaced by the following exponential family of inequalities: � � (1 − x u − x v ) ≤ | V | − x v − k T ∈ T uv ∈ E ( T ) v ∈ V where T denote the set of all cycle-free subgraphs of G . On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 18
Natural Formulation The following single-level formulation, denoted as Natural Formulation , is a valid model for the k -vertex cut problem: � min w v x v v ∈ V � [deg T ( v ) − 1] x v ≥ k − | V | + | E ( T ) | T ∈ T , v ∈ V x v ∈ { 0 , 1 } v ∈ V . On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 19
Natural formulation � [deg T ( v ) − 1] x v ≥ k − | V | + | E ( T ) | v ∈ V On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 20
Natural formulation � [deg T ( v ) − 1] x v ≥ k − | V | + | E ( T ) | v ∈ V 2 x 2 + x 4 + x 5 ≥ 2 On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 21
Natural formulation � [deg T ( v ) − 1] x v ≥ k − | V | + | E ( T ) | v ∈ V 2 x 2 + x 4 + x 5 ≥ 2 On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 22
Natural formulation � [deg T ( v ) − 1] x v ≥ k − | V | + | E ( T ) | v ∈ V 2 x 2 + x 4 + x 5 ≥ 2 − x 2 + 2 x 4 + 2 x 6 ≥ 1 On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 23
Natural formulation � [deg T ( v ) − 1] x v ≥ k − | V | + | E ( T ) | v ∈ V 2 x 2 + x 4 + x 5 ≥ 2 − x 2 + 2 x 4 + 2 x 6 ≥ 1 On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 24
Separation procedure Let x ∗ be the current solution. We define edge-weights as w ∗ uv = 1 − x ∗ u − x ∗ v , uv ∈ E and search for the maximum-weighted cycle-free subgraph in G . Let W ∗ denote the weight of the obtained subgraph; if W ∗ > | V | − k − � v ∈ V x ∗ v , we have detected a violated inequality. The separation procedure can be performed in polynomial time by running an adaptation of Kruskal’s algorithm for minimum-spanning trees. On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 25
A Hybrid Approach On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 26
Representative Variables Observation A graph G admits a k -vertex cut if and only if α ( G ) ≥ k . To each component we associate a vertex from the stable set - a representative . We introduce a set of binary variable to select which vertices are representative: � 1 if vertex v is the representative of a component z v = v ∈ V 0 otherwise On Integer and Bilevel Formulations for the k -Vertex Cut Problem (I. Ljubi´ c, F. Furini, E. Malaguti, P. Paronuzzi) 27
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