110 research outputs found
Correlation Clustering and Two-edge-connected Augmentation for Planar Graphs
In correlation clustering, the input is a graph with edge-weights, where every edge is labelled either + or - according to similarity of its endpoints. The goal is to produce a partition of the vertices that disagrees with the edge labels as little as possible.
In two-edge-connected augmentation, the input is a graph with edge-weights and a subset R of edges of the graph. The goal is to produce a minimum weight subset S of edges of the graph, such that for every edge in R, its endpoints are two-edge-connected in Rcup S.
For planar graphs, we prove that correlation clustering reduces to two-edge-connected augmentation, and that both problems have a polynomial-time approximation scheme
Approximation Algorithms for Union and Intersection Covering Problems
In a classical covering problem, we are given a set of requests that we need
to satisfy (fully or partially), by buying a subset of items at minimum cost.
For example, in the k-MST problem we want to find the cheapest tree spanning at
least k nodes of an edge-weighted graph. Here nodes and edges represent
requests and items, respectively.
In this paper, we initiate the study of a new family of multi-layer covering
problems. Each such problem consists of a collection of h distinct instances of
a standard covering problem (layers), with the constraint that all layers share
the same set of requests. We identify two main subfamilies of these problems: -
in a union multi-layer problem, a request is satisfied if it is satisfied in at
least one layer; - in an intersection multi-layer problem, a request is
satisfied if it is satisfied in all layers. To see some natural applications,
consider both generalizations of k-MST. Union k-MST can model a problem where
we are asked to connect a set of users to at least one of two communication
networks, e.g., a wireless and a wired network. On the other hand, intersection
k-MST can formalize the problem of connecting a subset of users to both
electricity and water.
We present a number of hardness and approximation results for union and
intersection versions of several standard optimization problems: MST, Steiner
tree, set cover, facility location, TSP, and their partial covering variants
A Constant-Factor Approximation for Quasi-bipartite Directed Steiner Tree on Minor-Free Graphs
We give the first constant-factor approximation algorithm for quasi-bipartite
instances of Directed Steiner Tree on graphs that exclude fixed minors. In
particular, for -minor-free graphs our approximation guarantee is
and, further, for planar graphs our approximation
guarantee is 20.
Our algorithm uses the primal-dual scheme. We employ a more involved method
of determining when to buy an edge while raising dual variables since, as we
show, the natural primal-dual scheme fails to raise enough dual value to pay
for the purchased solution. As a consequence, we also demonstrate integrality
gap upper bounds on the standard cut-based linear programming relaxation for
the Directed Steiner Tree instances we consider.Comment: 24 page
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