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Register a new userCovering Pairs in Directed Acyclic Graphs
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Riccardo Dondi
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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The Minimum Path Cover problem on directed acyclic graphs (DAGs) is a classical problem that provides a clear and simple mathematical formulation for several applications in different areas and that has an efficient algorithmic solution. In this paper, we study the computational complexity of two constrained variants of Minimum Path Cover motivated by the recent introduction of next-generation sequencing technologies in bioinformatics. The first problem (MinPCRP), given a DAG and a set of pairs of vertices, asks for a minimum cardinality set of paths covering all the vertices such that both vertices of each pair belong to the same path. For this problem, we show that, while it is NP-hard to compute if there exists a solution consisting of at most three paths, it is possible to decide in polynomial time whether a solution consisting of at most two paths exists. The second problem (MaxRPSP), given a DAG and a set of pairs of vertices, asks for a path containing the maximum number of the given pairs of vertices. We show its NP-hardness and also its W[1]-hardness when parametrized by the number of covered pairs. On the positive side, we give a fixed-parameter algorithm when the parameter is the maximum overlapping degree, a natural parameter in the bioinformatics applications of the problem.
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A directed graph $D$ is semicomplete if for every pair $x,y$ of vertices of $D,$ there is at least one arc between $x$ and $y.$ viol{Thus, a tournament is a semicomplete digraph.} In the Directed Component Order Connectivity (DCOC) problem, given a digraph $D=(V,A)$ and a pair of natural numbers $k$ and $ell$, we are to decide whether there is a subset $X$ of $V$ of size $k$ such that the largest strong connectivity component in $D-X$ has at most $ell$ vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for $ell=1.$ We study parametered complexity of DCOC for general and semicomplete digraphs with the following parameters: $k, ell,ell+k$ and $n-ell$. In particular, we prove that DCOC with parameter $k$ on semicomplete digraphs can be solved in time $O^*(2^{16k})$ but not in time $O^*(2^{o(k)})$ unless the Exponential Time Hypothesis (ETH) fails. gutin{The upper bound $O^*(2^{16k})$ implies the upper bound $O^*(2^{16(n-ell)})$ for the parameter $n-ell.$ We complement the latter by showing that there is no algorithm of time complexity $O^*(2^{o({n-ell})})$ unless ETH fails.} Finally, we improve viol{(in dependency on $ell$)} the upper bound of G{{o}}ke, Marx and Mnich (2019) for the time complexity of DCOC with parameter $ell+k$ on general digraphs from $O^*(2^{O(kelllog (kell))})$ to $O^*(2^{O(klog (kell))}).$ Note that Drange, Dregi and van t Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time $O^*(2^{o(klog ell)})$ unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity $O^*(2^{o(klog k)}).$
Let $G$ be a graph and $S, T subseteq V(G)$ be (possibly overlapping) sets of terminals, $|S|=|T|=k$. We are interested in computing a vertex sparsifier for terminal cuts in $G$, i.e., a graph $H$ on a smallest possible number of vertices, where $S cup T subseteq V(H)$ and such that for every $A subseteq S$ and $B subseteq T$ the size of a minimum $(A,B)$-vertex cut is the same in $G$ as in $H$. We assume that our graphs are unweighted and that terminals may be part of the min-cut. In previous work, Kratsch and Wahlstrom (FOCS 2012/JACM 2020) used connections to matroid theory to show that a vertex sparsifier $H$ with $O(k^3)$ vertices can be computed in randomized polynomial time, even for arbitrary digraphs $G$. However, since then, no improvements on the size $O(k^3)$ have been shown. In this paper, we draw inspiration from the renowned Bollobass Two-Families Theorem in extremal combinatorics and introduce the use of total orderings into Kratsch and Wahlstroms methods. This new perspective allows us to construct a sparsifier $H$ of $Theta(k^2)$ vertices for the case that $G$ is a DAG. We also show how to compute $H$ in time near-linear in the size of $G$, improving on the previous $O(n^{omega+1})$. Furthermore, $H$ recovers the closest min-cut in $G$ for every partition $(A,B)$, which was not previously known. Finally, we show that a sparsifier of size $Omega(k^2)$ is required, both for DAGs and for undirected edge cuts.
We examine the effect of bounding the diameter for well-studied variants of the Colouring problem. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring. The last problem is also known as $L(1,1)$-Labelling and we also consider the framework of $L(a,b)$-Labelling. We prove a number of (almost-)complete complexity classifications. In particular, we show that for graphs of diameter at most $d$, Acyclic $3$-Colouring is polynomial-time solvable if $dleq 2$ but NP-complete if $dgeq 4$, and Star $3$-Colouring is polynomial-time solvable if $dleq 3$ but NP-complete for $dgeq 8$. As far as we are aware, Star $3$-Colouring is the first problem that exhibits a complexity jump for some $dgeq 3$. Our third main result is that $L(1,2)$-Labelling is NP-complete for graphs of diameter $2$; we relate the latter problem to a special case of Hamiltonian Path.
Missing data is a pervasive problem in data analyses, resulting in datasets that contain censored realizations of a target distribution. Many approaches to inference on the target distribution using censored observed data, rely on missing data models represented as a factorization with respect to a directed acyclic graph. In this paper we consider the identifiability of the target distribution within this class of models, and show that the most general identification strategies proposed so far retain a significant gap in that they fail to identify a wide class of identifiable distributions. To address this gap, we propose a new algorithm that significantly generalizes the types of manipulations used in the ID algorithm, developed in the context of causal inference, in order to obtain identification.
We introduce a structure for the directed acyclic graph (DAG) and a mechanism design based on that structure so that peers can reach consensus at large scale based on proof of work (PoW). We also design a mempool transaction assignment method based on the DAG structure to render negligible the probability that a transaction being processed by more than one miners. The result is a significant scale-up of the capacity without sacrificing security and decentralization.
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