52,395 research outputs found
First-Fit coloring of Cartesian product graphs and its defining sets
Let the vertices of a Cartesian product graph be ordered by an
ordering . By the First-Fit coloring of we mean the
vertex coloring procedure which scans the vertices according to the ordering
and for each vertex assigns the smallest available color. Let
be the number of colors used in this coloring. By
introducing the concept of descent we obtain a sufficient condition to
determine whether , where and
are arbitrary orders. We study and obtain some bounds for , where is any quasi-lexicographic ordering. The First-Fit
coloring of does not always yield an optimum coloring. A
greedy defining set of is a subset of vertices in the
graph together with a suitable pre-coloring of such that by fixing the
colors of the First-Fit coloring of yields an optimum
coloring. We show that the First-Fit coloring and greedy defining sets of
with respect to any quasi-lexicographic ordering (including the known
lexicographic order) are all the same. We obtain upper and lower bounds for the
smallest cardinality of a greedy defining set in , including some
extremal results for Latin squares.Comment: Accepted for publication in Contributions to Discrete Mathematic
Online Contention Resolution Schemes
We introduce a new rounding technique designed for online optimization
problems, which is related to contention resolution schemes, a technique
initially introduced in the context of submodular function maximization. Our
rounding technique, which we call online contention resolution schemes (OCRSs),
is applicable to many online selection problems, including Bayesian online
selection, oblivious posted pricing mechanisms, and stochastic probing models.
It allows for handling a wide set of constraints, and shares many strong
properties of offline contention resolution schemes. In particular, OCRSs for
different constraint families can be combined to obtain an OCRS for their
intersection. Moreover, we can approximately maximize submodular functions in
the online settings we consider.
We, thus, get a broadly applicable framework for several online selection
problems, which improves on previous approaches in terms of the types of
constraints that can be handled, the objective functions that can be dealt
with, and the assumptions on the strength of the adversary. Furthermore, we
resolve two open problems from the literature; namely, we present the first
constant-factor constrained oblivious posted price mechanism for matroid
constraints, and the first constant-factor algorithm for weighted stochastic
probing with deadlines.Comment: 33 pages. To appear in SODA 201
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