18 research outputs found
The Chromatic Number of Random Regular Graphs
Given any integer d >= 3, let k be the smallest integer such that d < 2k log
k. We prove that with high probability the chromatic number of a random
d-regular graph is k, k+1, or k+2, and that if (2k-1) \log k < d < 2k \log k
then the chromatic number is either k+1 or k+2
Balanced Interval Coloring
We consider the discrepancy problem of coloring intervals with colors
such that at each point on the line, the maximal difference between the number
of intervals of any two colors is minimal. Somewhat surprisingly, a coloring
with maximal difference at most one always exists. Furthermore, we give an
algorithm with running time for its construction.
This is in particular interesting because many known results for discrepancy
problems are non-constructive. This problem naturally models a load balancing
scenario, where tasks with given start- and endtimes have to be distributed
among servers. Our results imply that this can be done ideally balanced.
When generalizing to -dimensional boxes (instead of intervals), a solution
with difference at most one is not always possible. We show that for any and any it is NP-complete to decide if such a solution exists,
which implies also NP-hardness of the respective minimization problem.
In an online scenario, where intervals arrive over time and the color has to
be decided upon arrival, the maximal difference in the size of color classes
can become arbitrarily high for any online algorithm.Comment: Accepted at STACS 201
Improved Bounds and Schemes for the Declustering Problem
The declustering problem is to allocate given data on parallel working
storage devices in such a manner that typical requests find their data evenly
distributed on the devices. Using deep results from discrepancy theory, we
improve previous work of several authors concerning range queries to
higher-dimensional data. We give a declustering scheme with an additive error
of independent of the data size, where is the
dimension, the number of storage devices and does not exceed the
smallest prime power in the canonical decomposition of into prime powers.
In particular, our schemes work for arbitrary in dimensions two and three.
For general , they work for all that are powers of two.
Concerning lower bounds, we show that a recent proof of a
bound contains an error. We close the gap in
the proof and thus establish the bound.Comment: 19 pages, 1 figur
Polychromatic Colorings of Geometric Hypergraphs via Shallow Hitting Sets
A range family is a family of subsets of , like
all halfplanes, or all unit disks. Given a range family , we
consider the -uniform range capturing hypergraphs
whose vertex-sets are finite sets of points
in with any vertices forming a hyperedge whenever for some . Given additionally an integer ,
we seek to find the minimum such that every
admits a polychromatic -coloring of its
vertices, that is, where every hyperedge contains at least one point of each
color. Clearly, and the gold standard is an upper
bound that is linear in .
A -shallow hitting set in is a subset such that for each hyperedge ; i.e.,
every hyperedge is hit at least once but at most times by . We show for
several range families the existence of -shallow hitting sets
in every with being a constant only
depending on . This in particular proves that in such cases, improving previous polynomial bounds in .
Particularly, we prove this for the range families of all axis-aligned strips
in , all bottomless and topless rectangles in , and
for all unit-height axis-aligned rectangles in
The condensation phase transition in random graph coloring
Based on a non-rigorous formalism called the "cavity method", physicists have
put forward intriguing predictions on phase transitions in discrete structures.
One of the most remarkable ones is that in problems such as random -SAT or
random graph -coloring, very shortly before the threshold for the existence
of solutions there occurs another phase transition called "condensation"
[Krzakala et al., PNAS 2007]. The existence of this phase transition appears to
be intimately related to the difficulty of proving precise results on, e.g.,
the -colorability threshold as well as to the performance of message passing
algorithms. In random graph -coloring, there is a precise conjecture as to
the location of the condensation phase transition in terms of a distributional
fixed point problem. In this paper we prove this conjecture for exceeding a
certain constant