Dagstuhl Seminar “Dynamic Traffic Models in Transportation Science”
Open Problem: Complexity of Computing a Maximum Robust Flow
October 7, 2015
Jannik Matuschke, S. Thomas McCormick, Gianpaolo Oriolo, Britta Peis, Martin Skutella
Overview
Consider a game played on a max flow network G = (N, A) with capacities u ∈ RA
+ and source s
and sink t. Define P as the set of s–t directed paths. We are also given an integer number k (with
1 ≤ k ≤ |A|) of arcs that can be removed. The Flow Player moves first and establishes some feasible flow
P
P
0
x ∈ X := {x0 ∈ RP
+ :
P :a∈P xP ≤ ua ∀a ∈ A} on s–t paths with value val(x) =
P ∈P xP (x need not
maximize val(x)). Then the Cut Player chooses a subset C ⊆ A of k arcs to remove from the network.
When the Cut Player removes arc subset C, she removes the flow on all paths including an arc in C,
P
i.e., amount P :P ∩C6=∅ xP of flow. The Cut Player’s objective is to minimize the remaining amount of
flow, i.e., to
X
xP .
min val(x) −
C⊆A:|C|=k
P :P ∩C6=∅
Notice that when k is fixed and x is given, there are only O(|A|k ) possible arc subsets to check, and so
the Cut Player’s problem is easy. However, when k is not fixed, there is a straightforward reduction from
Set Cover showing that the Cut Player’s problem is NP Hard.
The Flow Player seeks to distribute the flow over the network as evenly as possible so that it is robust
against any possible attack by the Cut Player. Formally, the Flow Player wants to
X
max min val(x) −
xP .
x∈X C⊆A:|C|=k
P :P ∩C6=∅
The open problem concerns the complexity of the Flow Player’s problem (note that the above observations
on the Cut Player’s problem do not have any direct implications on the complexity of the Flow Player’s
problem).
When k = 1, Aneja et al. [1] show that the Flow Player’s problem can be solved in polynomial time
using parametric flow techniques, and Bertsimas et al. [3] note that in this case arc flows is equivalent to
path flows.
When k = 2, Du and Chandrasekaran [4] reason that the Flow Player’s problem is NP Hard along
these lines: Formulate the Flow Player’s problem as an (exponentially large) LP. If the Flow Player’s
problem was easy, then this LP would be easy, and in particular we could use the Ellipsoid Algorithm to
solve both Separation and Optimization problems on both the primal and dual LP. They show that the
Separation problem on the dual LP leads to an NP Hard variant of Shortest Path, and so conclude that
the problem must be hard.
However, there is a flaw in their reasoning: The equivalence of Separation and Optimization holds
only if we can optimize the LP for any reasonable objective function. In Du and Chandrasekaran’s dual
LP the objective coefficients of most of the variables are zero, and so we are concerned about the easiness
or hardness of this LP only for this specific class of objective functions. Hence the equivalence between
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Separation and Optimization breaks down, and so showing that Separation of the dual is NP Hard does
not imply that solving the primal is NP Hard. (We communicated with Du and Chandrasekaran, and in
an email of 27 September 2015 Du agreed that there is a problem with their proof.) This leads to our:
Open Problem: Is there a polynomial algorithm for solving the Flow Player’s problem when
k = 2, or is the problem NP Hard?
Note that Section 4 of [3] gives some good background on this. In particular, its Figure 6 (c) shows
an interesting example of an optimal solution of the Flow Player’s problem for k = 2. It would also be
interesting to figure out the hardness of this problem for fixed values of k larger than two. See also [2],
which gives an O(1/k)-approximation algorithm for general k.
While the proof in [4] might not imply that the Flow Player’s problem is NP Hard, it does have some
implications on the game described in the beginning: Any feasible solution to the dual LP corresponds
mixed strategy of the Cut Player, i.e., a probability distributions over arc sets of size k. Du and Chandrasekaran’s hardness proof for the separation problem thus implies that it is NP Hard to compute a
best response of the Flow Player to such a mixed strategy.
References
[1] Y.P. Aneja, R. Chandrasekaran, K.P.K. Nair (2001). Maximizing residual flow under an arc destruction. Networks, 38, 194–198.
[2] D. Bertsimas, E. Nasrabadi, J.B. Orlin (2015). On the power of randomization in network interdiction.
Working paper, arXiv:1312.3478v1.
[3] D. Bertsimas, E. Nasrabadi, S. Stiller (2013). Robust and Adaptive Network Flows. Operations
Research, 61, 1218–1242.
[4] D. Du and R. Chandrasekaran (2007). The maximum residual flow problem: NP-hardness with twoarc destruction. Networks, 50, 181–182.
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