b-stable set interdiction on bipartite graphs
Stephen Chestnut & Rico Zenklusen
IFOR, ETH Zurich
Swiss OR Days
May 7, 2015
Forests and stable sets
Barahona, Weintraub, and Epstein. “Habitat dispersion in forest planning and the
stable set problem”. Operations Research. 1992.
Forests and stable sets
Barahona, Weintraub, and Epstein. “Habitat dispersion in forest planning and the
stable set problem”. Operations Research. 1992.
Forests and stable sets
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2
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Barahona, Weintraub, and Epstein. “Habitat dispersion in forest planning and the
stable set problem”. Operations Research. 1992.
Forests and stable sets
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Barahona, Weintraub, and Epstein. “Habitat dispersion in forest planning and the
stable set problem”. Operations Research. 1992.
Interdiction overview
Goal Inhibit solutions to an optimization problem by limiting the
feasible set
Example Forbid vertices to minimize the maximum stable set
Interdiction overview
Goal Inhibit solutions to an optimization problem by limiting the
feasible set
Example Forbid vertices to minimize the maximum stable set
Interdiction overview
Goal Inhibit solutions to an optimization problem by limiting the
feasible set
Example Forbid vertices to minimize the maximum stable set
Our results
This talk
Polynomial-time (1 + )-approximation algorithm for
b-stable set interdiction on bipartite graphs
More generally
2-approximation
or
super-optimal with cost ≤ 2B
and
improvements for some problems
Extending techniques of
Burch, Carr, Krumke, Marathe, Phillips, & Sundberg. “A decomposition
based pseudoapproximation algorithm for network flow inhibition.” 2003.
b-stable set interdiction in bipartite graphs
G
b
= (V = I ∪ J, E ) bipartite graph
∈ NE
bounds
4
1
3
x ∈ NV is b-stable if
2
x(u) + x(v ) ≤ buv , for all uv ∈ E .
2
b-stable set interdiction in bipartite graphs
G
b
4
0
2
1
3
0
x ∈ NV is b-stable if
1
2
x(u) + x(v ) ≤ buv , for all uv ∈ E .
2
1
= (V = I ∪ J, E ) bipartite graph
∈ NE
bounds
2
b-stable set interdiction in bipartite graphs
G
b
= (V = I ∪ J, E ) bipartite graph
∈ NE
bounds
4
1
3
x ∈ NV is b-stable if
2
x(u) + x(v ) ≤ buv , for all uv ∈ E .
2
b-stable set interdiction in bipartite graphs
4
5
=
∈
∈
∈
(V = I ∪ J, E )
NE
NV
N
bipartite graph
bounds
interdiction costs
interdiction budget
1
3
1
4
G
b
c
B
x ∈ NV is b-stable if
2
2
x(u) + x(v ) ≤ buv , for all uv ∈ E .
2
2
3
Interdict R ⊆ V by enforcing
x(v ) = 0, for all v ∈ R.
R is feasible if c(R) ≤ B.
Formulation
3: Dualize the max LP
1: Integer formulation
OPT =
min max 1T x
r ∈{0,1}V
T
c T r ≤B A x
x
rT x
≤ b
≥ 0
= 0
2: Interdiction to the objective
OPT =
min max (1 − r )T x
r ∈{0,1}V
T
c T r ≤B A x
x
≤ b
≥ 0
OPT = min b T y
Ay + r ≥
y ≥
r ∈
cT r ≤
1
0
{0, 1}V
B
4: Relax
OPT ≥ min b T y
Ay + r
y
r
cT r
≥
≥
≥
≤
1
0
0
B
Sufficient conditions recap
Feasible set is down-closed
Objective is max 1T x
Integral LP description is box-Totally Dual Integral
Pseudoapproximation
2
1
OPT ≥ M = min b T y
Ay + r
cT r
y, r
OPT
≥ 1
≤ B
≥ 0
Pseudoapproximation guarantee
For one of i = 1 or i = 2
b T y i ≤ 2M
and
cT ri ≤ B
bT y i ≤ M
and
c T r i ≤ 2B
OR
Exploiting dual adjacency
PTAS for bipartite b-stable set interdiction
Edge Cover adjacency
Augmented graph
min b T y
+

A I
 0 χI
0 χJ
0T r
0T y
cT r
+
+ 0rLR
 
0
y


1
r  ≥ 1
rLR
1
+ 0rLR
≤ B
y , r , rLR
≥ 0
G 0 = (V ∪ {L, R}, E 0 ) with
⇐ incidence matrix
4
5
2
2
2
Theorem (Hurkens)
If edge covers F1 , F2 ⊆ E 0 are adj., then
F1 4F2 is an alternating path or cycle.
1
3
1
4
2
3
L
0
R
Exploiting Edge Cover adjacency for a PTAS
Observation 1
F1 4F2
c(F2 ) ≤ B + 2 maxv c(v )
F1 : under budget, sub-optimal
F2 : over budget, super-optimal
Observation 2
b(F1 ) ≤ OPT + 2 maxe b(e)
If 2 maxe b(e) ≤ OPT, then F1
is a (1 + )-approximation.
If not, guess the d 2 e edges in the
solution with largest b-value.
Result
(1 + )-approximation!
L
R
Interdiction summary
2-approximation or super-optimal with cost ≤ 2B
Objective is maximization
Model interdiction in the objective (down-closed)
LP formulation with a dual integrality property & {0,1}-objective
Works for: matroid intersection, network flow, indep. systems with TU LP
improvements for some problems:
b-stable set (1 + )-approximation
weighted rank matroids, intersection of two matroids
weighted rank matroids with submodular costs
1-stable set interdiction in polynomial time
max stable set G [V \ R ∗ ] = |V | − |R ∗ | − max matching G [V \ R ∗ ]