990 research outputs found
Hypercontractive Inequality for Pseudo-Boolean Functions of Bounded Fourier Width
A function is called pseudo-Boolean.
It is well-known that each pseudo-Boolean function can be written as
where ${\cal F}\subseteq \{I:\
I\subseteq [n]\}[n]=\{1,2,...,n\}\chi_I(x)=\prod_{i\in I}x_i\hat{f}(I)f\max \{|I|:\ I\in {\cal
F}\}f\rhoi\in
[n]\rho\cal Fi\in [n]\mathbf{x}_i\mathbf{x}_jj\neq i.\mathbf{x}=(\mathbf{x}_1,...,\mathbf{x}_n)pf||f||_p=(\mathbb E[|f(\mathbf{x})|^p])^{1/p}p\ge 1||f||_q\ge ||f||_pq> p\ge 1ffdq> p>1 ||f||_q\le
(\frac{q-1}{p-1})^{d/2}||f||_p.d\rhoq> p\ge 2 ||f||_q\le
((2r)!\rho^{r-1})^{1/(2r)}||f||_p,r=\lceil q/2\rceilq=4p=2 ||f||_4\le (2\rho+1)^{1/4}||f||_2.
Transversals in -Uniform Hypergraphs
Let be a -regular -uniform hypergraph on vertices. The
transversal number of is the minimum number of vertices that
intersect every edge. Lai and Chang [J. Combin. Theory Ser. B 50 (1990),
129--133] proved that . Thomass\'{e} and Yeo [Combinatorica
27 (2007), 473--487] improved this bound and showed that .
We provide a further improvement and prove that , which is
best possible due to a hypergraph of order eight. More generally, we show that
if is a -uniform hypergraph on vertices and edges with maximum
degree , then , which proves a known
conjecture. We show that an easy corollary of our main result is that the total
domination number of a graph on vertices with minimum degree at least~4 is
at most , which was the main result of the Thomass\'{e}-Yeo paper
[Combinatorica 27 (2007), 473--487].Comment: 41 page
Kernels for Below-Upper-Bound Parameterizations of the Hitting Set and Directed Dominating Set Problems
In the {\sc Hitting Set} problem, we are given a collection of
subsets of a ground set and an integer , and asked whether has a
-element subset that intersects each set in . We consider two
parameterizations of {\sc Hitting Set} below tight upper bounds: and
. In both cases is the parameter. We prove that the first
parameterization is fixed-parameter tractable, but has no polynomial kernel
unless coNPNP/poly. The second parameterization is W[1]-complete,
but the introduction of an additional parameter, the degeneracy of the
hypergraph , makes the problem not only fixed-parameter
tractable, but also one with a linear kernel. Here the degeneracy of
is the minimum integer such that for each the
hypergraph with vertex set and edge set containing all edges of
without vertices in , has a vertex of degree at most
In {\sc Nonblocker} ({\sc Directed Nonblocker}), we are given an undirected
graph (a directed graph) on vertices and an integer , and asked
whether has a set of vertices such that for each vertex there is an edge (arc) from a vertex in to . {\sc Nonblocker} can be
viewed as a special case of {\sc Directed Nonblocker} (replace an undirected
graph by a symmetric digraph). Dehne et al. (Proc. SOFSEM 2006) proved that
{\sc Nonblocker} has a linear-order kernel. We obtain a linear-order kernel for
{\sc Directed Nonblocker}
Parameterized Rural Postman Problem
The Directed Rural Postman Problem (DRPP) can be formulated as follows: given
a strongly connected directed multigraph with nonnegative integral
weights on the arcs, a subset of and a nonnegative integer ,
decide whether has a closed directed walk containing every arc of and
of total weight at most . Let be the number of weakly connected
components in the the subgraph of induced by . Sorge et al. (2012) ask
whether the DRPP is fixed-parameter tractable (FPT) when parameterized by ,
i.e., whether there is an algorithm of running time where is a
function of only and the notation suppresses polynomial factors.
Sorge et al. (2012) note that this question is of significant practical
relevance and has been open for more than thirty years. Using an algebraic
approach, we prove that DRPP has a randomized algorithm of running time
when is bounded by a polynomial in the number of vertices in
. We also show that the same result holds for the undirected version of
DRPP, where is a connected undirected multigraph
Out-degree reducing partitions of digraphs
Let be a fixed integer. We determine the complexity of finding a
-partition of the vertex set of a given digraph such
that the maximum out-degree of each of the digraphs induced by , () is at least smaller than the maximum out-degree of . We show
that this problem is polynomial-time solvable when and -complete otherwise. The result for and answers a question
posed in \cite{bangTCS636}. We also determine, for all fixed non-negative
integers , the complexity of deciding whether a given digraph of
maximum out-degree has a -partition such that the digraph
induced by has maximum out-degree at most for . It
follows from this characterization that the problem of deciding whether a
digraph has a 2-partition such that each vertex has at
least as many neighbours in the set as in , for is
-complete. This solves a problem from \cite{kreutzerEJC24} on
majority colourings.Comment: 11 pages, 1 figur
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