New integer and Bilevel Formulations for the k-Vertex Cut Problem … · 2020. 11. 13. · New...
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New integer and Bilevel Formulations for the k-VertexCut Problem
Ivana Ljubic•,joint work with F. Furini◦, E. Malaguti∗ and P. Paronuzzi∗
∗DEI “Guglielmo Marconi”, University of Bologna
◦IASI-CNR, Rome
•ESSEC Business School, Paris
SPOC22 Meeting, Oct 30 2020, online edition
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Outline
1 Problem setting and motivation
2 Compact Model
3 Representative Formulation
4 Bilevel approach
5 A Hybrid Approach?
6 Computational experiments
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Based on the paper
Fabio Furini, Ivana Ljubic, Enrico Malaguti, Paolo Paronuzzi:On integer and bilevel formulations for the k-vertex cut problem.
Math. Program. Comput. 12(2): 133-164 (2020)
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Problem setting and motivation
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Problem setting
A k-vertex cut is a subset of vertices whose removal disconnects thegraph in at least k (not-empty) components.
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Problem setting
Example of a 3-vertex cut:
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Problem setting
Another example of a 3-vertex cut:
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The k-Vertex Cut Problem
Definition
Given an undirected graph G = (V ,E ) with vertex weights wv , v ∈ V ,and a integer k ≥ 2, find a subset of vertices of minimum weight whoseremoval disconnects G in at least k (not-empty) components.
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Motivation
Family of Critical Node Detection Problems (M. Lalou, M. A.Tahraoui, and H. Kheddouci. The critical node detection problem innetworks: A survey. Computer Science Review, 2018);
Defender-Attacker model: k-vertex cut are vertices to be defended(protected, vaccinated)Attacker-Defender model: k-vertex cut are vertices to be attacked
Decomposition method for linear equation systems: vertices arecolumns of a matrix, and an edge connects two vertices if there aretwo non-zero entries in the same raw.
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Problem Complexity
Vertex k-cut
k = 2: polynomial (Ben Ameur & Biha, 2012).
k ≥ 3: NP-hard (Berger et al, 2014), even for a fixed value of k(Cornaz et al, 2019)
Edge k-cut
For any fixed value of k : polynomial (Goldschmidt & Hochbaum,1994).
2-approximation algorithms exist.
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Compact Model
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Compact formulation
We associate a binary variable y iv to all vertices v ∈ V and for all integersi ∈ K , such that:
y iv =
{1 if vertex v belongs to component i
0 otherwisei ∈ K , v ∈ V .
If∑
i∈K y iv = 0 ⇒ v belongs to the k-vertex cut.
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Compact Model
Compact ILP formulation for k-Vertex-Cut Problem:
min∑v∈V
wv −∑i∈K
∑v∈V
wvyiv∑
i∈Ky iv ≤ 1 v ∈ V ,
y iu +∑
j∈K\{i}
y jv ≤ 1 i ∈ K , uv ∈ E ,
∑v∈V
y iv ≥ 1 i ∈ K ,
y iv ∈ {0, 1} i ∈ K , v ∈ V .
Drawbacks: LP-optimal solution is zero (set all y iv = 1/k), symmetries,etc.
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Representative Formulation
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Representative formulation
Observation
A graph G = (V ,E ) admits ak-vertex cut if and only if its stabilitynumber α(G ) is at least k .
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Representative formulation
In the Multi-Terminal Vertex Separator problem a set of representativevertices for each component are given.Similar model is proposed in Y. Magnouche’s Ph.D. thesis (2017).
We introduce a set of binary variable to select which vertices arerepresentative:
zv =
{1 if vertex v is the representative of a component
0 otherwisev ∈ V
and we use the same set of binary variables denoting whether a vertex is inthe k-vertex cut:
xv =
{1 if vertex v is in the k-vertex cut
0 otherwisev ∈ V
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Representative formulation
The Representative Formulation reads as follows:
min∑v∈V
xv∑v∈V
zv = k v ∈ V ,
zu + zv ≤ 1 uv ∈ E ,∑w∈V (P)\{u,v}
xw ≥ zu + zv − 1 u, v ∈ V ,P ∈ Πuv , uv 6∈ E
xv , zv ∈ {0, 1} v ∈ V .
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Representative formulation: Path Inequalities
∑w∈V (P)\{u,v}
xw ≥ zu + zv − 1
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Representative formulation: Path Inequalities
∑w∈V (P)\{u,v}
xw ≥ zu + zv − 1
x2 + x5 ≥ z1 + z7 − 1
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Representative formulation: Path Inequalities
∑w∈V (P)\{u,v}
xw ≥ zu + zv − 1
x2 + x5 ≥ z1 + z7 − 1
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Representative formulation: Path Inequalities
∑w∈V (P)\{u,v}
xw ≥ zu + zv − 1
x2 + x5 ≥ z1 + z7 − 1
x4 + x6 ≥ z1 + z7 − 1
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Representative formulation: Path Inequalities
∑w∈V (P)\{u,v}
xw ≥ zu + zv − 1
x2 + x5 ≥ z1 + z7 − 1
x4 + x6 ≥ z1 + z7 − 1
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Separation of Path Inequalities
Given a solution x∗, z∗ ∈ [0, 1]V , the separation problem asks for finding apair of vertices u, v such that there is a path P∗ ∈ Πuv with
zu + zv >∑
w∈V (P∗)\{u,v}
xw − 1.
We can search for such a path in polynomial time by solving a shortestpath problem (for each pair of not adjacent vertices) on graph G , wherewe define the length of each edge (u, v) ∈ E as:
luv =x∗u + x∗v
2
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Representative formulation
Valid constraints in polynomial number:
xu + zu ≤ 1 u ∈ V ,
zu +∑
v∈N(u)
zv ≤ 1 + (deg(u)− 1)xu u ∈ V .
Strengthened Path Inequalities:∑w∈V (P)\{u,v}
xw ≥ zu + zv +∑
w∈V (P)\{u,v}
zw − 1
Clique-Path Inequalities... Each z on the RHS is replaced by aclique...
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Bilevel approach
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Bilevel approach
Property
A graph G has at least k (not empty) components if and only if anycycle-free subgraph of G contains at most |V | − k edges.
Example with |V | = 9 and k = 3:
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Bilevel approach
Property
A graph G has at least k (not empty) components if and only if anycycle-free subgraph of G contains at most |V | − k edges.
Example with |V | = 9 and k = 3:
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Bilevel approach
The k-vertex cut problem can be seen as a Stackelberg game:
the leader searches the smallest subset of vertices V0 to delete;
the follower maximizes the size of the cycle-free subgraph on theresidual graph.
Property
The solution V0 ⊂ V of the leader is feasible if and only if the value of theoptimal follower’s response (i.e., the size of the maximum cycle-freesubgraph in the remaining graph) is at most |V | − |V0| − k.
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Bilevel approach
The leader decisions:
xv =
{1 if vertex v is in the k-vertex cut
0 otherwisev ∈ V
For the decisions of the follower, we use additional binary variablesassociated with the edges of G :
euv =
{1 if edge uv is selected to be in the cycle-free subgraph
0 otherwiseuv ∈ E
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Bilevel approach
The Bilevel ILP formulation of the k-vertex cut problem reads as follows:
min∑v∈V
xv
Φ(x) ≤ |V | −∑v∈V
xv − k
xv ∈ {0, 1} v ∈ V .
Φ(x) is the optimal solution value of the follower subproblem for agiven x .
Value Function Reformulation.
Value function Φ(x) is neither convex, nor concave, nor connected...
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How do we calculate Φ(x)?
For a solution x∗ of the leader, which denotes a set V0 of interdictedvertices, the follower’s subproblem is:
Φ(x∗) = max∑uv∈E
euv
e(S) ≤ |S | − 1 S ⊆ V ,S 6= ∅,euv ≤ 1− x∗u uv ∈ E ,
euv ≤ 1− x∗v uv ∈ E ,
euv ∈ {0, 1} uv ∈ E .
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Bilevel approach
We can prove that the follower’s subproblem is equivalently restated as:
Φ(x∗) = max∑uv∈E
zuv (1− x∗u − x∗v )
z(S) ≤ |S | − 1 S ⊆ V ,S 6= ∅zuv ∈ {0, 1} uv ∈ E .
Convexification of the value function Φ(x)
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Bilevel approach
Since the space of feasible solutions of the redefined follower subproblemdoes not depend on the leader anymore, the non-linear constraint from theBILP formulation:
Φ(x) ≤ |V | −∑v∈V
xv − k
can now be replaced by the following exponential family of inequalities:
∑uv∈E(T )
(1− xu − xv ) ≤ |V | −∑v∈V
xv − k T ∈ T
where T denote the set of all cycle-free subgraphs of G .
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Natural Formulation
The following single-level formulation, denoted as Natural Formulation, isa valid model for the k-vertex cut problem:
min∑v∈V
wvxv∑v∈V
[degT (v)− 1]xv ≥ k − |V |+ |E (T )| T ∈ T ,
xv ∈ {0, 1} v ∈ V .
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Natural formulation
∑v∈V
[degT (v)− 1]xv ≥ k − |V |+ |E (T )|
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Natural formulation
∑v∈V
[degT (v)− 1]xv ≥ k − |V |+ |E (T )|
2x2 + x4 + x5 ≥ 2
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Natural formulation
∑v∈V
[degT (v)− 1]xv ≥ k − |V |+ |E (T )|
2x2 + x4 + x5 ≥ 2
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Natural formulation
∑v∈V
[degT (v)− 1]xv ≥ k − |V |+ |E (T )|
2x2 + x4 + x5 ≥ 2
−x2 + 2x4 + 2x6 ≥ 1
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Natural formulation
∑v∈V
[degT (v)− 1]xv ≥ k − |V |+ |E (T )|
2x2 + x4 + x5 ≥ 2
−x2 + 2x4 + 2x6 ≥ 1
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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:
adaptation of Kruskal’s algorithm for minimum-spanning trees(fractional points), or
BFS (integer points) on the graph from which xv = 1 vertices areremoved. Extended to spanning subgraphs (dominating cuts).
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A Hybrid Approach?
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Model Strength
Theorem
For k ≤ n/2 we have:
vLP(NAT ) ≤ vLP(REP)
and there exist instances where the strict inequality holds.
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Natural Formulation + Representative Constraints
∑v∈V
zv = k
zu + zv ≤ 1 uv ∈ E ,
xu + zu ≤ 1 u ∈ V ,
zu +∑
v∈N(u)
zv ≤ 1 + (deg(u)− 1)xu u ∈ V .
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Computational experiments
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Computational Experiments
We considered two sets of graph instances from the 2nd DIMACS and10th DIMACS challenges.For all the instances we tested four different values of k (5, 10, 15, 20).
Compared Methods (time limit of 1 hour):
COMP: Compact model (solved by CPLEX 12.7.1);
BP: State-of-the-art Branch-and-Price solving an Extendedformulation (Cornaz, D., Furini, F., Lacroix, M., Malaguti, E.,Mahjoub, A. R., & Martin, S. The Vertex k-cut Problem, DiscreteOptimization, 2018.);
HYB: Hybrid approach
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Comparing REP, NAT and HYB Formulation
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Computational Experiments
Case with k = 5
0
20
40
60
80
100
1 10 102 103 104
�(�)
� (k=5)
COMPBP
HYB
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Computational Experiments
Case with k = 15
0
20
40
60
80
100
1 10 102 103 104
�(�)
� (k=15)
COMPBP
HYB
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Computational Experiments
Case with k = 20
0
20
40
60
80
100
1 10 102 103 104
�(�)
� (k=20)
COMPBP
HYB
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Conclusions and future work
Our hybrid formulation outperforms both CPLEX and B&P;
It is a thin formulation, with O(n) variables
We partially exploit a hereditary property on G (if a subset of edgesis cycle-free, any subset of it is cycle-free too) to convexify Φ(x)
This allows us to derive an ILP formulation in the natural space (wasopen for some time)
Where else can we exploit similar ideas?
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Conclusions and future work
Thank you for your attention.
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