49 research outputs found
The complexity of finding arc-disjoint branching flows
The concept of arc-disjoint flows in networks was recently introduced in \cite{bangTCSflow}. This is a very general framework within which many well-known and important problems can be formulated. In particular, the existence of arc-disjoint branching flows, that is, flows which send one unit of flow from a given source to all other vertices, generalizes the concept of arc-disjoint out-branchings (spanning out-trees) in a digraph. A pair of out-branchings from a root in a digraph on vertices corresponds to arc-disjoint branching flows (the arcs carrying flow in are those used in , ) in the network that we obtain from by giving all arcs capacity .It is then a natural question to ask how much we can lower the capacities on the arcs and still have, say, two arc-disjoint branching flows from the given root .We prove that for every fixed integer it is\begin{itemize}\item an NP-complete problem to decide whether a network where for every arc has two arc-disjoint branching flows rooted at .\item a polynomial problem to decide whether a network on vertices and for every arc has two arc-disjoint branching flows rooted at .\end{itemize}The algorithm for the later result generalizes the polynomial algorithm, due to Lov\'asz, for deciding whether a given input digraph has two arc-disjoint out-branchings rooted at a given vertex.Finally we prove that under the so-called Exponential Time Hypothesis (ETH), for every and for every with (and for every large we have for some ) there is no polynomial algorithm for deciding whether a given digraph contains twoarc-disjoint branching flows from the same root so that no arc carries flow larger than
Determinantal Sieving
We introduce determinantal sieving, a new, remarkably powerful tool in the
toolbox of algebraic FPT algorithms. Given a polynomial on a set of
variables and a linear matroid of
rank , both over a field of characteristic 2, in
evaluations we can sieve for those terms in the monomial expansion of which
are multilinear and whose support is a basis for . Alternatively, using
evaluations of we can sieve for those monomials whose odd support
spans . Applying this framework, we improve on a range of algebraic FPT
algorithms, such as:
1. Solving -Matroid Intersection in time and -Matroid
Parity in time , improving on (Brand and Pratt,
ICALP 2021)
2. -Cycle, Colourful -Path, Colourful -Linkage in undirected
graphs, and the more general Rank -Linkage problem, all in
time, improving on respectively (Fomin et al., SODA 2023)
3. Many instances of the Diverse X paradigm, finding a collection of
solutions to a problem with a minimum mutual distance of in time
, improving solutions for -Distinct Branchings from time
to (Bang-Jensen et al., ESA 2021), and for Diverse
Perfect Matchings from to (Fomin et al.,
STACS 2021)
All matroids are assumed to be represented over a field of characteristic 2.
Over general fields, we achieve similar results at the cost of using
exponential space by working over the exterior algebra. For a class of
arithmetic circuits we call strongly monotone, this is even achieved without
any loss of running time. However, the odd support sieving result appears to be
specific to working over characteristic 2
Dominators in Directed Graphs: A Survey of Recent Results, Applications, and Open Problems
The computation of dominators is a central tool in program optimization and code generation, and it has applications in other diverse areas includingconstraint programming, circuit testing, and biology. In this paper we survey recent results, applications, and open problems related to the notion of dominators in directed graphs,including dominator verification and certification, computing independent spanning trees, and connectivity and path-determination problems in directed graphs
The complexity of finding arc-disjoint branching flows
International audienceThe concept of arc-disjoint flows in networks was recently introduced in Bang-Jensen and Bessy (2014). This is a very general framework within which many well-known and important problems can be formulated. In particular, the existence of arc-disjoint branching flows, that is, flows which send one unit of flow from a given source s to all other vertices, generalizes the concept of arc-disjoint out-branchings (spanning out-trees) in a digraph. A pair of out-branchings B + s,1 , B + s,2 from a root s in a digraph D = (V , A) on n vertices corresponds to arc-disjoint branching flows x 1 , x 2 (the arcs carrying flow in x i are those used in B + s,i , i = 1, 2) in the network that we obtain from D by giving all arcs capacity n â 1. It is then a natural question to ask how much we can lower the capacities on the arcs and still have, say, two arc-disjoint branching flows from the given root s. We prove that for every fixed integer k â„ 2 it is âą an NP-complete problem to decide whether a network N = (V , A, u) where u ij = k for every arc ij has two arc-disjoint branching flows rooted at s. âą a polynomial problem to decide whether a network N = (V , A, u) on n vertices and u ij = n â k for every arc ij has two arc-disjoint branching flows rooted at s. The algorithm for the later result generalizes the polynomial algorithm, due to LovĂĄsz, for deciding whether a given input digraph has two arc-disjoint out-branchings rooted at a given vertex. Finally we prove that under the so-called Exponential Time Hypothesis (ETH), for every Ï” > 0 and for every k(n) with (log(n)) 1+Ï” †k(n) †n 2 (and for every large i we have k(n) = i for some n) there is no polynomial algorithm for deciding whether a given digraph contains two arc-disjoint branching flows from the same root so that no arc carries flow larger than n â k(n)
A time- and space-optimal algorithm for the many-visits TSP
The many-visits traveling salesperson problem (MV-TSP) asks for an optimal
tour of cities that visits each city a prescribed number of
times. Travel costs may be asymmetric, and visiting a city twice in a row may
incur a non-zero cost. The MV-TSP problem finds applications in scheduling,
geometric approximation, and Hamiltonicity of certain graph families.
The fastest known algorithm for MV-TSP is due to Cosmadakis and Papadimitriou
(SICOMP, 1984). It runs in time and
requires space. An interesting feature of the
Cosmadakis-Papadimitriou algorithm is its \emph{logarithmic} dependence on the
total length of the tour, allowing the algorithm to handle
instances with very long tours. The \emph{superexponential} dependence on the
number of cities in both the time and space complexity, however, renders the
algorithm impractical for all but the narrowest range of this parameter.
In this paper we improve upon the Cosmadakis-Papadimitriou algorithm, giving
an MV-TSP algorithm that runs in time , i.e.\
\emph{single-exponential} in the number of cities, using \emph{polynomial}
space. Our algorithm is deterministic, and arguably both simpler and easier to
analyse than the original approach of Cosmadakis and Papadimitriou. It involves
an optimization over directed spanning trees and a recursive, centroid-based
decomposition of trees.Comment: Small fixes, journal versio
Matter-antimatter asymmetry restrains the dimensionality of neural representations: quantum decryption of large-scale neural coding
Projections from the study of the human universe onto the study of the
self-organizing brain are herein leveraged to address certain concerns raised
in latest neuroscience research, namely (i) the extent to which neural codes
are multidimensional; (ii) the functional role of neural dark matter; (iii) the
challenge to traditional model frameworks posed by the needs for accurate
interpretation of large-scale neural recordings linking brain and behavior. On
the grounds of (hyper-)self-duality under (hyper-)mirror supersymmetry,
inter-relativistic quantum principles are introduced, whose consolidation, as
spin-geometrical pillars of a network- and game-theoretical construction, is
conducive to (i) the high-precision reproduction and reinterpretation of core
experimental observations on neural coding in the self-organizing brain, with
the instantaneous geometric dimensionality of neural representations of a
spontaneous behavioral state being proven to be at most 16, unidirectionally;
(ii) a possible role for spinor (co-)representations, as the latent building
blocks of self-organizing cortical circuits subserving (co-)behavioral states;
(iii) an early crystallization of pertinent multidimensional synaptic
(co-)architectures, whereby Lorentz (co-)partitions are in principle
verifiable; and, ultimately, (iv) potentially inverse insights into
matter-antimatter asymmetry. New avenues for the decryption of large-scale
neural coding in health and disease are being discussed.Comment: 33 pages;3 figures;1 table;minor edit