526 research outputs found
A Greedy Partition Lemma for Directed Domination
A directed dominating set in a directed graph is a set of vertices of
such that every vertex has an adjacent vertex
in with directed to . The directed domination number of , denoted
by , is the minimum cardinality of a directed dominating set in .
The directed domination number of a graph , denoted , which is
the maximum directed domination number over all orientations of
. The directed domination number of a complete graph was first studied by
Erd\"{o}s [Math. Gaz. 47 (1963), 220--222], albeit in disguised form. In this
paper we prove a Greedy Partition Lemma for directed domination in oriented
graphs. Applying this lemma, we obtain bounds on the directed domination
number. In particular, if denotes the independence number of a graph
, we show that .Comment: 12 page
On the heapability of finite partial orders
We investigate the partitioning of partial orders into a minimal number of
heapable subsets. We prove a characterization result reminiscent of the proof
of Dilworth's theorem, which yields as a byproduct a flow-based algorithm for
computing such a minimal decomposition. On the other hand, in the particular
case of sets and sequences of intervals we prove that this minimal
decomposition can be computed by a simple greedy-type algorithm. The paper ends
with a couple of open problems related to the analog of the Ulam-Hammersley
problem for decompositions of sets and sequences of random intervals into
heapable sets
A note on the greedy approximation algorithm for the unweighted set covering problem
Bibliography: leaf 9.Abhay K. Parekh
From Bandits to Experts: A Tale of Domination and Independence
We consider the partial observability model for multi-armed bandits,
introduced by Mannor and Shamir. Our main result is a characterization of
regret in the directed observability model in terms of the dominating and
independence numbers of the observability graph. We also show that in the
undirected case, the learner can achieve optimal regret without even accessing
the observability graph before selecting an action. Both results are shown
using variants of the Exp3 algorithm operating on the observability graph in a
time-efficient manner
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