96 research outputs found
Tight Bounds for Monotone Minimal Perfect Hashing
The monotone minimal perfect hash function (MMPHF) problem is the following
indexing problem. Given a set of distinct keys from
a universe of size , create a data structure that answers the
following query:
Solutions to the MMPHF problem are in widespread use in both theory and
practice.
The best upper bound known for the problem encodes in bits and performs queries in time. It has been an open problem
to either improve the space upper bound or to show that this somewhat odd
looking bound is tight.
In this paper, we show the latter: specifically that any data structure
(deterministic or randomized) for monotone minimal perfect hashing of any
collection of elements from a universe of size requires expected bits to answer every query correctly.
We achieve our lower bound by defining a graph where the nodes
are the possible inputs and where two nodes are adjacent if
they cannot share the same . The size of is then lower bounded by the
log of the chromatic number of . Finally, we show that the
fractional chromatic number (and hence the chromatic number) of is
lower bounded by
Defective Coloring on Classes of Perfect Graphs
In Defective Coloring we are given a graph and two integers ,
and are asked if we can -color so that the maximum
degree induced by any color class is at most . We show that this
natural generalization of Coloring is much harder on several basic graph
classes. In particular, we show that it is NP-hard on split graphs, even when
one of the two parameters , is set to the smallest possible
fixed value that does not trivialize the problem ( or ). Together with a simple treewidth-based DP algorithm this completely
determines the complexity of the problem also on chordal graphs. We then
consider the case of cographs and show that, somewhat surprisingly, Defective
Coloring turns out to be one of the few natural problems which are NP-hard on
this class. We complement this negative result by showing that Defective
Coloring is in P for cographs if either or is fixed; that
it is in P for trivially perfect graphs; and that it admits a sub-exponential
time algorithm for cographs when both and are unbounded
Partitioning random graphs into monochromatic components
Erd\H{o}s, Gy\'arf\'as, and Pyber (1991) conjectured that every -colored
complete graph can be partitioned into at most monochromatic components;
this is a strengthening of a conjecture of Lov\'asz (1975) in which the
components are only required to form a cover. An important partial result of
Haxell and Kohayakawa (1995) shows that a partition into monochromatic
components is possible for sufficiently large -colored complete graphs.
We start by extending Haxell and Kohayakawa's result to graphs with large
minimum degree, then we provide some partial analogs of their result for random
graphs. In particular, we show that if , then a.a.s. in every -coloring of there exists
a partition into two monochromatic components, and for if , then a.a.s. there exists an -coloring
of such that there does not exist a cover with a bounded number of
components. Finally, we consider a random graph version of a classic result of
Gy\'arf\'as (1977) about large monochromatic components in -colored complete
graphs. We show that if , then a.a.s. in every
-coloring of there exists a monochromatic component of order at
least .Comment: 27 pages, 2 figures. Appears in Electronic Journal of Combinatorics
Volume 24, Issue 1 (2017) Paper #P1.1
On Vizing's problem for triangle-free graphs
We prove that for any triangle-free
graph of maximum degree provided . This gives
tangible progress towards an old problem of Vizing, in a form cast by Reed. We
use a method of Hurley and Pirot, which in turn relies on a new counting
argument of the second author.Comment: 10 page
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