126 research outputs found
Edge covering with budget constrains
We study two related problems: finding a set of k vertices and minimum number
of edges (kmin) and finding a graph with at least m' edges and minimum number
of vertices (mvms).
Goldschmidt and Hochbaum \cite{GH97} show that the mvms problem is NP-hard
and they give a 3-approximation algorithm for the problem. We improve
\cite{GH97} by giving a ratio of 2. A 2(1+\epsilon)-approximation for the
problem follows from the work of Carnes and Shmoys \cite{CS08}. We improve the
approximation ratio to 2. algorithm for the problem. We show that the natural
LP for \kmin has an integrality gap of 2-o(1). We improve the NP-completeness
of \cite{GH97} by proving the pronlem are APX-hard unless a well-known instance
of the dense k-subgraph admits a constant ratio. The best approximation
guarantee known for this instance of dense k-subgraph is O(n^{2/9})
\cite{BCCFV}. We show that for any constant \rho>1, an approximation guarantee
of \rho for the \kmin problem implies a \rho(1+o(1)) approximation for \mwms.
Finally, we define we give an exact algorithm for the density version of kmin.Comment: 17 page
How Bad is Forming Your Own Opinion?
The question of how people form their opinion has fascinated economists and
sociologists for quite some time. In many of the models, a group of people in a
social network, each holding a numerical opinion, arrive at a shared opinion
through repeated averaging with their neighbors in the network. Motivated by
the observation that consensus is rarely reached in real opinion dynamics, we
study a related sociological model in which individuals' intrinsic beliefs
counterbalance the averaging process and yield a diversity of opinions.
By interpreting the repeated averaging as best-response dynamics in an
underlying game with natural payoffs, and the limit of the process as an
equilibrium, we are able to study the cost of disagreement in these models
relative to a social optimum. We provide a tight bound on the cost at
equilibrium relative to the optimum; our analysis draws a connection between
these agreement models and extremal problems that lead to generalized
eigenvalues. We also consider a natural network design problem in this setting:
which links can we add to the underlying network to reduce the cost of
disagreement at equilibrium
Dial a Ride from k-forest
The k-forest problem is a common generalization of both the k-MST and the
dense--subgraph problems. Formally, given a metric space on vertices
, with demand pairs and a ``target'' ,
the goal is to find a minimum cost subgraph that connects at least demand
pairs. In this paper, we give an -approximation
algorithm for -forest, improving on the previous best ratio of
by Segev & Segev.
We then apply our algorithm for k-forest to obtain approximation algorithms
for several Dial-a-Ride problems. The basic Dial-a-Ride problem is the
following: given an point metric space with objects each with its own
source and destination, and a vehicle capable of carrying at most objects
at any time, find the minimum length tour that uses this vehicle to move each
object from its source to destination. We prove that an -approximation
algorithm for the -forest problem implies an
-approximation algorithm for Dial-a-Ride. Using our
results for -forest, we get an -
approximation algorithm for Dial-a-Ride. The only previous result known for
Dial-a-Ride was an -approximation by Charikar &
Raghavachari; our results give a different proof of a similar approximation
guarantee--in fact, when the vehicle capacity is large, we give a slight
improvement on their results.Comment: Preliminary version in Proc. European Symposium on Algorithms, 200
Duality between Feature Selection and Data Clustering
The feature-selection problem is formulated from an information-theoretic
perspective. We show that the problem can be efficiently solved by an extension
of the recently proposed info-clustering paradigm. This reveals the fundamental
duality between feature selection and data clustering,which is a consequence of
the more general duality between the principal partition and the principal
lattice of partitions in combinatorial optimization
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