4,562 research outputs found
Approximating the Minimum Equivalent Digraph
The MEG (minimum equivalent graph) problem is, given a directed graph, to
find a small subset of the edges that maintains all reachability relations
between nodes. The problem is NP-hard. This paper gives an approximation
algorithm with performance guarantee of pi^2/6 ~ 1.64. The algorithm and its
analysis are based on the simple idea of contracting long cycles. (This result
is strengthened slightly in ``On strongly connected digraphs with bounded cycle
length'' (1996).) The analysis applies directly to 2-Exchange, a simple ``local
improvement'' algorithm, showing that its performance guarantee is 1.75.Comment: conference version in ACM-SIAM Symposium on Discrete Algorithms
(1994
Parameterized Algorithms for Directed Maximum Leaf Problems
We prove that finding a rooted subtree with at least leaves in a digraph
is a fixed parameter tractable problem. A similar result holds for finding
rooted spanning trees with many leaves in digraphs from a wide family
that includes all strong and acyclic digraphs. This settles completely an open
question of Fellows and solves another one for digraphs in . Our
algorithms are based on the following combinatorial result which can be viewed
as a generalization of many results for a `spanning tree with many leaves' in
the undirected case, and which is interesting on its own: If a digraph of order with minimum in-degree at least 3 contains a rooted
spanning tree, then contains one with at least leaves
Graph-Theoretic Approaches to Two-Sender Index Coding
Consider a communication scenario over a noiseless channel where a sender is
required to broadcast messages to multiple receivers, each having side
information about some messages. In this scenario, the sender can leverage the
receivers' side information during the encoding of messages in order to reduce
the required transmissions. This type of encoding is called index coding. In
this paper, we study index coding with two cooperative senders, each with some
subset of messages, and multiple receivers, each requesting one unique message.
The index coding in this setup is called two-sender unicast index coding
(TSUIC). The main aim of TSUIC is to minimize the total number of transmissions
required by the two senders. Based on graph-theoretic approaches, we prove that
TSUIC is equivalent to single-sender unicast index coding (SSUIC) for some
special cases. Moreover, we extend the existing schemes for SSUIC, viz., the
cycle-cover scheme, the clique-cover scheme, and the local-chromatic scheme to
the corresponding schemes for TSUIC.Comment: To be presented at 2016 IEEE Global Communications Conference
(GLOBECOM 2016) Workshop on Network Coding and Applications (NetCod),
Washington, USA, 201
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