280 research outputs found
Interdiction Problems on Planar Graphs
Interdiction problems are leader-follower games in which the leader is
allowed to delete a certain number of edges from the graph in order to
maximally impede the follower, who is trying to solve an optimization problem
on the impeded graph. We introduce approximation algorithms and strong
NP-completeness results for interdiction problems on planar graphs. We give a
multiplicative -approximation for the maximum matching
interdiction problem on weighted planar graphs. The algorithm runs in
pseudo-polynomial time for each fixed . We also show that
weighted maximum matching interdiction, budget-constrained flow improvement,
directed shortest path interdiction, and minimum perfect matching interdiction
are strongly NP-complete on planar graphs. To our knowledge, our
budget-constrained flow improvement result is the first planar NP-completeness
proof that uses a one-vertex crossing gadget.Comment: 25 pages, 9 figures. Extended abstract in APPROX-RANDOM 201
Parameterized Complexity of Edge Interdiction Problems
We study the parameterized complexity of interdiction problems in graphs. For
an optimization problem on graphs, one can formulate an interdiction problem as
a game consisting of two players, namely, an interdictor and an evader, who
compete on an objective with opposing interests. In edge interdiction problems,
every edge of the input graph has an interdiction cost associated with it and
the interdictor interdicts the graph by modifying the edges in the graph, and
the number of such modifications is constrained by the interdictor's budget.
The evader then solves the given optimization problem on the modified graph.
The action of the interdictor must impede the evader as much as possible. We
focus on edge interdiction problems related to minimum spanning tree, maximum
matching and shortest paths. These problems arise in different real world
scenarios. We derive several fixed-parameter tractability and W[1]-hardness
results for these interdiction problems with respect to various parameters.
Next, we show close relation between interdiction problems and partial cover
problems on bipartite graphs where the goal is not to cover all elements but to
minimize/maximize the number of covered elements with specific number of sets.
Hereby, we investigate the parameterized complexity of several partial cover
problems on bipartite graphs
Non-Uniform Robust Network Design in Planar Graphs
Robust optimization is concerned with constructing solutions that remain
feasible also when a limited number of resources is removed from the solution.
Most studies of robust combinatorial optimization to date made the assumption
that every resource is equally vulnerable, and that the set of scenarios is
implicitly given by a single budget constraint. This paper studies a robustness
model of a different kind. We focus on \textbf{bulk-robustness}, a model
recently introduced~\cite{bulk} for addressing the need to model non-uniform
failure patterns in systems.
We significantly extend the techniques used in~\cite{bulk} to design
approximation algorithm for bulk-robust network design problems in planar
graphs. Our techniques use an augmentation framework, combined with linear
programming (LP) rounding that depends on a planar embedding of the input
graph. A connection to cut covering problems and the dominating set problem in
circle graphs is established. Our methods use few of the specifics of
bulk-robust optimization, hence it is conceivable that they can be adapted to
solve other robust network design problems.Comment: 17 pages, 2 figure
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