16,415 research outputs found
Finding Almost Tight Witness Trees
This paper addresses a graph optimization problem, called the Witness Tree problem, which seeks a spanning tree of a graph minimizing a certain non-linear objective function. This problem is of interest because it plays a crucial role in the analysis of the best approximation algorithms for two fundamental network design problems: Steiner Tree and Node-Tree Augmentation. We will show how a wiser choice of witness trees leads to an improved approximation for Node-Tree Augmentation, and for Steiner Tree in special classes of graphs
Choose your witnesses wisely
This paper addresses a graph optimization problem, called the Witness Tree
problem, which seeks a spanning tree of a graph minimizing a certain non-linear
objective function. This problem is of interest because it plays a crucial role
in the analysis of the best approximation algorithms for two fundamental
network design problems: Steiner Tree and Node-Tree Augmentation. We will show
how a wiser choice of witness trees leads to an improved approximation for
Node-Tree Augmentation, and for Steiner Tree in special classes of graphs.Comment: 33 pages, 7 figures, submitted to IPCO 202
How to Secure Matchings Against Edge Failures
Suppose we are given a bipartite graph that admits a perfect matching and an adversary may delete any edge from the graph with the intention of destroying all perfect matchings. We consider the task of adding a minimum cost edge-set to the graph, such that the adversary never wins. We show that this problem is equivalent to covering a digraph with non-trivial strongly connected components at minimal cost. We provide efficient exact and approximation algorithms for this task. In particular, for the unit-cost problem, we give a log_2 n-factor approximation algorithm and a polynomial-time algorithm for chordal-bipartite graphs. Furthermore, we give a fixed parameter algorithm for the problem parameterized by the treewidth of the input graph. For general non-negative weights we give tight upper and lower approximation bounds relative to the Directed Steiner Forest problem. Additionally we prove a dichotomy theorem characterizing minor-closed graph classes which allow for a polynomial-time algorithm. To obtain our results, we exploit a close relation to the classical Strong Connectivity Augmentation problem as well as directed Steiner problems
Approximating k-Forest with Resource Augmentation: A Primal-Dual Approach
In this paper, we study the -forest problem in the model of resource
augmentation. In the -forest problem, given an edge-weighted graph ,
a parameter , and a set of demand pairs , the
objective is to construct a minimum-cost subgraph that connects at least
demands. The problem is hard to approximate---the best-known approximation
ratio is . Furthermore, -forest is as hard to
approximate as the notoriously-hard densest -subgraph problem.
While the -forest problem is hard to approximate in the worst-case, we
show that with the use of resource augmentation, we can efficiently approximate
it up to a constant factor.
First, we restate the problem in terms of the number of demands that are {\em
not} connected. In particular, the objective of the -forest problem can be
viewed as to remove at most demands and find a minimum-cost subgraph that
connects the remaining demands. We use this perspective of the problem to
explain the performance of our algorithm (in terms of the augmentation) in a
more intuitive way.
Specifically, we present a polynomial-time algorithm for the -forest
problem that, for every , removes at most demands and has
cost no more than times the cost of an optimal algorithm
that removes at most demands
A -Approximation Algorithm for Weighted Connectivity Augmentation
Connectivity augmentation problems are among the most elementary questions in
Network Design. Many of these problems admit natural -approximation
algorithms, often through various classic techniques, whereas it remains open
whether approximation factors below can be achieved. One of the most basic
examples thereof is the Weighted Connectivity Augmentation Problem (WCAP). In
WCAP, one is given an undirected graph together with a set of additional
weighted candidate edges, and the task is to find a cheapest set of candidate
edges whose addition to the graph increases its edge-connectivity. We present a
-approximation algorithm for WCAP, showing for the first
time that factors below are achievable.
On a high level, we design a well-chosen local search algorithm, inspired by
recent advances for Weighted Tree Augmentation. To measure progress, we
consider a directed weakening of WCAP and show that it has highly structured
planar solutions. Interpreting a solution of the original problem as one of
this directed weakening allows us to describe local exchange steps in a clean
and algorithmically amenable way. Leveraging these insights, we show that we
can efficiently search for good exchange steps within a component class for
link sets that is closely related to bounded treewidth subgraphs of circle
graphs. Moreover, we prove that an optimum solution can be decomposed into
smaller components, at least one of which leads to a good local search step as
long as we did not yet achieve the claimed approximation guarantee
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