16 research outputs found
Efficient Möbius Transformations and their applications to D-S Theory
International audienceDempster-Shafer Theory (DST) generalizes Bayesian probability theory, offering useful additional information, but suffers from a high computational burden. A lot of work has been done to reduce the complexity of computations used in information fusion with Demp-ster's rule. The main approaches exploit either the structure of Boolean lattices or the information contained in belief sources. Each has its merits depending on the situation. In this paper, we propose sequences of graphs for the computation of the zeta and Möbius transformations that optimally exploit both the structure of distributive lattices and the information contained in belief sources. We call them the Efficient Möbius Transformations (EMT). We show that the complexity of the EMT is always inferior to the complexity of algorithms that consider the whole lattice, such as the Fast Möbius Transform (FMT) for all DST transformations. We then explain how to use them to fuse two belief sources. More generally, our EMTs apply to any function in any finite distributive lattice, focusing on a meet-closed or join-closed subset
The Fine-Grained Complexity of Computing the Tutte Polynomial of a Linear Matroid
We show that computing the Tutte polynomial of a linear matroid of dimension
on points over a field of elements requires
time unless the \#ETH---a counting extension of the Exponential
Time Hypothesis of Impagliazzo and Paturi [CCC 1999] due to Dell {\em et al.}
[ACM TALG 2014]---is false. This holds also for linear matroids that admit a
representation where every point is associated to a vector with at most two
nonzero coordinates. We also show that the same is true for computing the Tutte
polynomial of a binary matroid of dimension on points with at
most three nonzero coordinates in each point's vector. This is in sharp
contrast to computing the Tutte polynomial of a -vertex graph (that is, the
Tutte polynomial of a {\em graphic} matroid of dimension ---which is
representable in dimension over the binary field so that every vector has
two nonzero coordinates), which is known to be computable in
time [Bj\"orklund {\em et al.}, FOCS 2008]. Our lower-bound proofs proceed via
(i) a connection due to Crapo and Rota [1970] between the number of tuples of
codewords of full support and the Tutte polynomial of the matroid associated
with the code; (ii) an earlier-established \#ETH-hardness of counting the
solutions to a bipartite -CSP on vertices in time; and
(iii) new embeddings of such CSP instances as questions about codewords of full
support in a linear code. We complement these lower bounds with two algorithm
designs. The first design computes the Tutte polynomial of a linear matroid of
dimension~ on points in operations. The second design
generalizes the Bj\"orklund~{\em et al.} algorithm and runs in
time for linear matroids of dimension defined over the
-element field by points with at most two nonzero coordinates
each.Comment: This version adds Theorem
Fast Algorithms for Join Operations on Tree Decompositions
Treewidth is a measure of how tree-like a graph is. It has many important
algorithmic applications because many NP-hard problems on general graphs become
tractable when restricted to graphs of bounded treewidth. Algorithms for
problems on graphs of bounded treewidth mostly are dynamic programming
algorithms using the structure of a tree decomposition of the graph. The
bottleneck in the worst-case run time of these algorithms often is the
computations for the so called join nodes in the associated nice tree
decomposition.
In this paper, we review two different approaches that have appeared in the
literature about computations for the join nodes: one using fast zeta and
M\"obius transforms and one using fast Fourier transforms. We combine these
approaches to obtain new, faster algorithms for a broad class of vertex subset
problems known as the [\sigma,\rho]-domination problems. Our main result is
that we show how to solve [\sigma,\rho]-domination problems in arithmetic operations. Here, t is the treewidth, s is the
(fixed) number of states required to represent partial solutions of the
specific [\sigma,\rho]-domination problem, and n is the number of vertices in
the graph. This reduces the polynomial factors involved compared to the
previously best time bound (van Rooij, Bodlaender, Rossmanith, ESA 2009) of arithmetic operations. In particular, this removes
the dependence of the degree of the polynomial on the fixed number of
states~.Comment: An earlier version appeared in "Treewidth, Kernels, and Algorithms.
Essays Dedicated to Hans L. Bodlaender on the Occasion of His 60th Birthday"
LNCS 1216
Computing Generalized Convolutions Faster Than Brute Force
In this paper, we consider a general notion of convolution.
Let be a finite domain and let be the set of -length vectors
(tuples) of . Let be a function and
let be a coordinate-wise application of . The -Convolution of two
functions is
\begin{displaymath}
(g \circledast_f h)(v) := \sum_{\substack{v_g,v_h \in
D^n\\ \text{s.t. } v = v_g \oplus_f v_h}} g(v_g) \cdot h(v_h)
\end{displaymath}
for every .
This problem generalizes many fundamental convolutions
such as Subset Convolution, XOR Product, Covering Product or Packing Product,
etc.
For arbitrary function and domain we can compute -Convolution via brute-force enumeration
in time.
Our main result is an improvement over this naive algorithm. We show that -Convolution
can be computed exactly in for constant when has even cardinality. Our main observation is that a
\emph{cyclic partition} of a function can
be used to speed up the computation of -Convolution, and we show that an appropriate
cyclic partition exists for every .
Furthermore, we demonstrate that a single entry of the -Convolution can be computed
more efficiently. In this variant, we are given two functions alongside with a vector and the task of
the -Query problem is to compute integer . This is a
generalization of the well-known Orthogonal Vectors problem. We show that
-Query can be computed in time, where is the exponent of currently fastest matrix multiplication
algorithm
Tight Algorithm for Connected Odd Cycle Transversal Parameterized by Clique-width
Recently, Bojikian and Kratsch [2023] have presented a novel approach to
tackle connectivity problems parameterized by clique-width
(), based on counting small representations of partial
solutions (modulo two). Using this technique, they were able to get a tight
bound for the Steiner Tree problem, answering an open question posed by
Hegerfeld and Kratsch [ESA, 2023]. We use the same technique to solve the
Connected Odd Cycle Transversal problem in time
. We define a new representation of
partial solutions by separating the connectivity requirement from the
2-colorability requirement of this problem. Moreover, we prove that our result
is tight by providing SETH-based lower bound excluding algorithms with running
time even when
parameterized by linear clique-width. This answers the second question posed by
Hegerfeld and Kratsch in the same paper