145,572 research outputs found
Pre-Reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs
The study of graph products is a major research topic and typically concerns
the term , e.g., to show that . In this paper, we
study graph products in a non-standard form where is a
"reduction", a transformation of any graph into an instance of an intended
optimization problem. We resolve some open problems as applications.
(1) A tight -approximation hardness for the minimum
consistent deterministic finite automaton (DFA) problem, where is the
sample size. Due to Board and Pitt [Theoretical Computer Science 1992], this
implies the hardness of properly learning DFAs assuming (the
weakest possible assumption).
(2) A tight hardness for the edge-disjoint paths (EDP)
problem on directed acyclic graphs (DAGs), where denotes the number of
vertices.
(3) A tight hardness of packing vertex-disjoint -cycles for large .
(4) An alternative (and perhaps simpler) proof for the hardness of properly
learning DNF, CNF and intersection of halfspaces [Alekhnovich et al., FOCS 2004
and J. Comput.Syst.Sci. 2008]
Nice labeling problem for event structures: a counterexample
In this note, we present a counterexample to a conjecture of Rozoy and
Thiagarajan from 1991 (called also the nice labeling problem) asserting that
any (coherent) event structure with finite degree admits a labeling with a
finite number of labels, or equivalently, that there exists a function such that an event structure with degree
admits a labeling with at most labels. Our counterexample is based on
the Burling's construction from 1965 of 3-dimensional box hypergraphs with
clique number 2 and arbitrarily large chromatic numbers and the bijection
between domains of event structures and median graphs established by
Barth\'elemy and Constantin in 1993
On the phase transitions of graph coloring and independent sets
We study combinatorial indicators related to the characteristic phase
transitions associated with coloring a graph optimally and finding a maximum
independent set. In particular, we investigate the role of the acyclic
orientations of the graph in the hardness of finding the graph's chromatic
number and independence number. We provide empirical evidence that, along a
sequence of increasingly denser random graphs, the fraction of acyclic
orientations that are `shortest' peaks when the chromatic number increases, and
that such maxima tend to coincide with locally easiest instances of the
problem. Similar evidence is provided concerning the `widest' acyclic
orientations and the independence number
Cubicity of interval graphs and the claw number
Let be a simple, undirected graph where is the set of vertices
and is the set of edges. A -dimensional cube is a Cartesian product
, where each is a closed interval of
unit length on the real line. The \emph{cubicity} of , denoted by \cub(G)
is the minimum positive integer such that the vertices in can be mapped
to axis parallel -dimensional cubes in such a way that two vertices are
adjacent in if and only if their assigned cubes intersect. Suppose
denotes a star graph on nodes. We define \emph{claw number} of
the graph to be the largest positive integer such that is an induced
subgraph of . It can be easily shown that the cubicity of any graph is at
least \ceil{\log_2\psi(G)}.
In this paper, we show that, for an interval graph
\ceil{\log_2\psi(G)}\le\cub(G)\le\ceil{\log_2\psi(G)}+2. Till now we are
unable to find any interval graph with \cub(G)>\ceil{\log_2\psi(G)}. We also
show that, for an interval graph , \cub(G)\le\ceil{\log_2\alpha}, where
is the independence number of . Therefore, in the special case of
, \cub(G) is exactly \ceil{\log_2\alpha}.
The concept of cubicity can be generalized by considering boxes instead of
cubes. A -dimensional box is a Cartesian product , where each is a closed interval on the real
line. The \emph{boxicity} of a graph, denoted , is the minimum
such that is the intersection graph of -dimensional boxes. It is clear
that box(G)\le\cub(G). From the above result, it follows that for any graph
, \cub(G)\le box(G)\ceil{\log_2\alpha}
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