49 research outputs found

    The complexity of finding arc-disjoint branching flows

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    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 ss to all other vertices, generalizes the concept of arc-disjoint out-branchings (spanning out-trees) in a digraph. A pair of out-branchings Bs,1+,Bs,2+B_{s,1}^+,B_{s,2}^+ from a root ss in a digraph D=(V,A)D=(V,A) on nn vertices corresponds to arc-disjoint branching flows x1,x2x_1,x_2 (the arcs carrying flow in xix_i are those used in Bs,i+B_{s,i}^+, i=1,2i=1,2) in the network that we obtain from DD by giving all arcs capacity n−1n-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 ss.We prove that for every fixed integer k≄2k \geq 2 it is\begin{itemize}\item an NP-complete problem to decide whether a network N=(V,A,u){\cal N}=(V,A,u) where uij=ku_{ij}=k for every arc ijij has two arc-disjoint branching flows rooted at ss.\item a polynomial problem to decide whether a network N=(V,A,u){\cal N}=(V,A,u) on nn vertices and uij=n−ku_{ij}=n-k for every arc ijij has two arc-disjoint branching flows rooted at ss.\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 Ï”>0\epsilon{}>0 and for every k(n)k(n) with (log⁥(n))1+ϔ≀k(n)≀n2(\log{}(n))^{1+\epsilon}\leq k(n)\leq \frac{n}{2} (and for every large ii we have k(n)=ik(n)=i for some nn) 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 n−k(n)n-k(n)

    Determinantal Sieving

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    We introduce determinantal sieving, a new, remarkably powerful tool in the toolbox of algebraic FPT algorithms. Given a polynomial P(X)P(X) on a set of variables X={x1,
,xn}X=\{x_1,\ldots,x_n\} and a linear matroid M=(X,I)M=(X,\mathcal{I}) of rank kk, both over a field F\mathbb{F} of characteristic 2, in 2k2^k evaluations we can sieve for those terms in the monomial expansion of PP which are multilinear and whose support is a basis for MM. Alternatively, using 2k2^k evaluations of PP we can sieve for those monomials whose odd support spans MM. Applying this framework, we improve on a range of algebraic FPT algorithms, such as: 1. Solving qq-Matroid Intersection in time O∗(2(q−2)k)O^*(2^{(q-2)k}) and qq-Matroid Parity in time O∗(2qk)O^*(2^{qk}), improving on O∗(4qk)O^*(4^{qk}) (Brand and Pratt, ICALP 2021) 2. TT-Cycle, Colourful (s,t)(s,t)-Path, Colourful (S,T)(S,T)-Linkage in undirected graphs, and the more general Rank kk (S,T)(S,T)-Linkage problem, all in O∗(2k)O^*(2^k) time, improving on O∗(2k+∣S∣)O^*(2^{k+|S|}) respectively O∗(2∣S∣+O(k2log⁡(k+∣F∣)))O^*(2^{|S|+O(k^2 \log(k+|\mathbb{F}|))}) (Fomin et al., SODA 2023) 3. Many instances of the Diverse X paradigm, finding a collection of rr solutions to a problem with a minimum mutual distance of dd in time O∗(2r(r−1)d/2)O^*(2^{r(r-1)d/2}), improving solutions for kk-Distinct Branchings from time 2O(klog⁡k)2^{O(k \log k)} to O∗(2k)O^*(2^k) (Bang-Jensen et al., ESA 2021), and for Diverse Perfect Matchings from O∗(22O(rd))O^*(2^{2^{O(rd)}}) to O∗(2r2d/2)O^*(2^{r^2d/2}) (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

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    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

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    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 − 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 ≄ 2 it is ‱ 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. ‱ a polynomial problem to decide whether a network N = (V , A, u) on n vertices and u ij = n − 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 − k(n)

    Reconstructing perfect phylogenies via binary matrices, branchings in DAGs, and a generalization of Dilworth\u27s theorem

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    A time- and space-optimal algorithm for the many-visits TSP

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    The many-visits traveling salesperson problem (MV-TSP) asks for an optimal tour of nn cities that visits each city cc a prescribed number kck_c 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 nO(n)+O(n3log⁡∑ckc)n^{O(n)} + O(n^3 \log \sum_c k_c ) and requires nΘ(n)n^{\Theta(n)} space. An interesting feature of the Cosmadakis-Papadimitriou algorithm is its \emph{logarithmic} dependence on the total length ∑ckc\sum_c k_c 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 2O(n)2^{O(n)}, 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

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    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
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