14 research outputs found
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
An O(1)-Approximation for Minimum Spanning Tree Interdiction
Network interdiction problems are a natural way to study the sensitivity of a
network optimization problem with respect to the removal of a limited set of
edges or vertices. One of the oldest and best-studied interdiction problems is
minimum spanning tree (MST) interdiction. Here, an undirected multigraph with
nonnegative edge weights and positive interdiction costs on its edges is given,
together with a positive budget B. The goal is to find a subset of edges R,
whose total interdiction cost does not exceed B, such that removing R leads to
a graph where the weight of an MST is as large as possible. Frederickson and
Solis-Oba (SODA 1996) presented an O(log m)-approximation for MST interdiction,
where m is the number of edges. Since then, no further progress has been made
regarding approximations, and the question whether MST interdiction admits an
O(1)-approximation remained open.
We answer this question in the affirmative, by presenting a 14-approximation
that overcomes two main hurdles that hindered further progress so far.
Moreover, based on a well-known 2-approximation for the metric traveling
salesman problem (TSP), we show that our O(1)-approximation for MST
interdiction implies an O(1)-approximation for a natural interdiction version
of metric TSP
Efficient algorithms for robustness in resource allocation and scheduling problems
AbstractThe robustness function of an optimization (minimization) problem measures the maximum increase in the value of its optimal solution that can be produced by spending a given amount of resources increasing the values of the elements in its input. We present efficient algorithms for computing the robustness function of resource allocation and scheduling problems that can be modeled with partition and scheduling matroids. For the case of scheduling matroids, we give an O(m2n2) time algorithm for computing a complete description of the robustness function, where m is the number of elements in the matroid and n is its rank. For partition matroids, we give two algorithms: one that computes the complete robustness function in O(mlogm) time, and other that optimally evaluates the robustness function at only a specified point
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Improving spanning trees by upgrading nodes
We study budget constrained optimal network upgrading problems. Such problems aim at finding optimal strategies for improving a network under some cost measure subject to certain budget constraints. A general problem in this setting is the following. We are given an edge weighted graph G = (V, E) where nodes represent processors and edges represent bidirectional communication links. The processor at a node v {element_of} V can be upgraded at a cost of c(v). Such an upgrade reduces the delay of each link emanating from v. The goal is to find a minimum cost set of nodes to be upgraded so that the resulting network has the best performance with respect to some measure. We consider the problem under two measures, namely, the weight of a minimum spanning tree and the bottleneck weight of a minimum bottleneck spanning tree. We present approximation and hardness results for the problem. Our results are tight to within constant factors. We also show that these approximation algorithms can be used to construct good approximation algorithms for the dual versions of the problems where there is a budget constraint on the upgrading cost and the objectives are minimum weight spanning tree and minimum bottleneck weight spanning tree respectively
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Node weighted network upgrade problems
Consider a network where nodes represent processors and edges represent bidirectional communication links. The processor at a node v can be upgraded at an expense of cost(v). Such an upgrade reduces the delay of each link emanating from v by a fixed factor x, where 0 < x < 1. The goal is to find a minimum cost set of nodes to be upgraded so that the resulting network has a spanning tree in which edge is of delay at most a given value {delta}. The authors provide both hardness and approximation results for the problem. They show that the problem is NP-hard and cannot be approximated within any factor {beta} < ln n, unless NP {improper_subset} DTIME(n{sup log log n}), where n is the number of nodes in the network. They then present the first polynomial time approximation algorithms for the problem. For the general case, the approximation algorithm comes within a factor of 2 ln n of the minimum upgrading cost. When the cost of upgrading each node is 1, they present an approximation algorithm with a performance guarantee of 4(2 + ln {Delta}), where {Delta} is the maximum node degree. Finally, they present a polynomial time algorithm for the class of treewidth-bounded graphs