1,023 research outputs found
Wiretapping a hidden network
We consider the problem of maximizing the probability of hitting a
strategically chosen hidden virtual network by placing a wiretap on a single
link of a communication network. This can be seen as a two-player win-lose
(zero-sum) game that we call the wiretap game. The value of this game is the
greatest probability that the wiretapper can secure for hitting the virtual
network. The value is shown to equal the reciprocal of the strength of the
underlying graph.
We efficiently compute a unique partition of the edges of the graph, called
the prime-partition, and find the set of pure strategies of the hider that are
best responses against every maxmin strategy of the wiretapper. Using these
special pure strategies of the hider, which we call
omni-connected-spanning-subgraphs, we define a partial order on the elements of
the prime-partition. From the partial order, we obtain a linear number of
simple two-variable inequalities that define the maxmin-polytope, and a
characterization of its extreme points.
Our definition of the partial order allows us to find all equilibrium
strategies of the wiretapper that minimize the number of pure best responses of
the hider. Among these strategies, we efficiently compute the unique strategy
that maximizes the least punishment that the hider incurs for playing a pure
strategy that is not a best response. Finally, we show that this unique
strategy is the nucleolus of the recently studied simple cooperative spanning
connectivity game
Approximating ATSP by Relaxing Connectivity
The standard LP relaxation of the asymmetric traveling salesman problem has
been conjectured to have a constant integrality gap in the metric case. We
prove this conjecture when restricted to shortest path metrics of node-weighted
digraphs. Our arguments are constructive and give a constant factor
approximation algorithm for these metrics. We remark that the considered case
is more general than the directed analog of the special case of the symmetric
traveling salesman problem for which there were recent improvements on
Christofides' algorithm.
The main idea of our approach is to first consider an easier problem obtained
by significantly relaxing the general connectivity requirements into local
connectivity conditions. For this relaxed problem, it is quite easy to give an
algorithm with a guarantee of 3 on node-weighted shortest path metrics. More
surprisingly, we then show that any algorithm (irrespective of the metric) for
the relaxed problem can be turned into an algorithm for the asymmetric
traveling salesman problem by only losing a small constant factor in the
performance guarantee. This leaves open the intriguing task of designing a
"good" algorithm for the relaxed problem on general metrics.Comment: 25 pages, 2 figures, fixed some typos in previous versio
Approximability of Connected Factors
Finding a d-regular spanning subgraph (or d-factor) of a graph is easy by
Tutte's reduction to the matching problem. By the same reduction, it is easy to
find a minimal or maximal d-factor of a graph. However, if we require that the
d-factor is connected, these problems become NP-hard - finding a minimal
connected 2-factor is just the traveling salesman problem (TSP).
Given a complete graph with edge weights that satisfy the triangle
inequality, we consider the problem of finding a minimal connected -factor.
We give a 3-approximation for all and improve this to an
(r+1)-approximation for even d, where r is the approximation ratio of the TSP.
This yields a 2.5-approximation for even d. The same algorithm yields an
(r+1)-approximation for the directed version of the problem, where r is the
approximation ratio of the asymmetric TSP. We also show that none of these
minimization problems can be approximated better than the corresponding TSP.
Finally, for the decision problem of deciding whether a given graph contains
a connected d-factor, we extend known hardness results.Comment: To appear in the proceedings of WAOA 201
Setting Parameters by Example
We introduce a class of "inverse parametric optimization" problems, in which
one is given both a parametric optimization problem and a desired optimal
solution; the task is to determine parameter values that lead to the given
solution. We describe algorithms for solving such problems for minimum spanning
trees, shortest paths, and other "optimal subgraph" problems, and discuss
applications in multicast routing, vehicle path planning, resource allocation,
and board game programming.Comment: 13 pages, 3 figures. To be presented at 40th IEEE Symp. Foundations
of Computer Science (FOCS '99
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
An Improved Algorithm for Finding Maximum Outerplanar Subgraphs
We study the NP-complete Maximum Outerplanar Subgraph problem. The previous
best known approximation ratio for this problem is 2/3. We propose a new
approximation algorithm which improves the ratio to 7/10
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