59,824 research outputs found
Connect Four and Graph Decomposition
We introduce the standard decomposition, a way of decomposing a labeled graph
into a sum of certain labeled subgraphs. We motivate this graph-theoretic
concept by relating it to Connect Four decompositions of standard sets. We
prove that all standard decompositions can be generated in polynomial time,
which implies that all Connect Four decompositions can be generated in
polynomial time
Components of Gr\"obner strata in the Hilbert scheme of points
We fix the lexicographic order on the polynomial ring
over a ring . We define \Hi^{\prec\Delta}_{S/k},
the moduli space of reduced Gr\"obner bases with a given finite standard set
, and its open subscheme \Hi^{\prec\Delta,\et}_{S/k}, the moduli
space of families of #\Delta points whose attached ideal has the standard set
. We determine the number of irreducible and connected components of
the latter scheme; we show that it is equidimensional over ; and
we determine its relative dimension over . We show that analogous
statements do not hold for the scheme \Hi^{\prec\Delta}_{S/k}. Our results
prove a version of a conjecture by Bernd Sturmfels.Comment: 49 page
On the Equivalence among Problems of Bounded Width
In this paper, we introduce a methodology, called decomposition-based
reductions, for showing the equivalence among various problems of
bounded-width.
First, we show that the following are equivalent for any :
* SAT can be solved in time,
* 3-SAT can be solved in time,
* Max 2-SAT can be solved in time,
* Independent Set can be solved in time, and
* Independent Set can be solved in time, where
tw and cw are the tree-width and clique-width of the instance, respectively.
Then, we introduce a new parameterized complexity class EPNL, which includes
Set Cover and Directed Hamiltonicity, and show that SAT, 3-SAT, Max 2-SAT, and
Independent Set parameterized by path-width are EPNL-complete. This implies
that if one of these EPNL-complete problems can be solved in time,
then any problem in EPNL can be solved in time.Comment: accepted to ESA 201
Line-graphs of cubic graphs are normal
A graph is called normal if its vertex set can be covered by cliques and also
by stable sets, such that every such clique and stable set have non-empty
intersection. This notion is due to Korner, who introduced the class of normal
graphs as an extension of the class of perfect graphs. Normality has also
relevance in information theory. Here we prove, that the line graphs of cubic
graphs are normal.Comment: 16 pages, 10 figure
Treewidth, crushing, and hyperbolic volume
We prove that there exists a universal constant such that any closed
hyperbolic 3-manifold admits a triangulation of treewidth at most times its
volume. The converse is not true: we show there exists a sequence of hyperbolic
3-manifolds of bounded treewidth but volume approaching infinity. Along the
way, we prove that crushing a normal surface in a triangulation does not
increase the carving-width, and hence crushing any number of normal surfaces in
a triangulation affects treewidth by at most a constant multiple.Comment: 20 pages, 12 figures. V2: Section 4 has been rewritten, as the former
argument (in V1) used a construction that relied on a wrong theorem. Section
5.1 has also been adjusted to the new construction. Various other arguments
have been clarifie
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