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 Wahlström (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 Bollobás's Two-Families Theorem in extremal combinatorics and introduce the use of total orderings into Kratsch and Wahlström's methods. This new perspective allows us to construct a sparsifier $H$ of $Θ(k^2)$ 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 $O(n^{ω+1})$. 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 $Ω(k^2)$ is required, both for DAGs and for undirected edge cuts.
In this paper, we consider the problem of designing cut sparsifiers and sketches for directed graphs. To bypass known lower bounds, we allow the sparsifier/sketch to depend on the balance of the input graph, which smoothly interpolates between undirected and directed graphs. We give nearly matching upper and lower bounds for both for-all (cf. Benczúr and Karger, STOC 1996) and for-each (Andoni et al., ITCS 2016) cut sparsifiers/sketches as a function of cut balance, defined the maximum ratio of the cut value in the two directions of a directed graph (Ene et al., STOC 2016). We also show an int…
Boltzmann sampling and optimal exact-size sampling for directed acyclic graphs
Wrocław University of Science and Technology（弗罗茨瓦夫科技大学）；Université Caen Normandie, ENSICAEN, CNRS, Normandie Univ, GREYC UMR 6072（卡昂诺曼底大学，ENSICAEN，CNRS，诺曼底大学，GREYC联合研究实验室）
In some applications of matching, the structural or hierarchical properties of the two graphs being aligned must be maintained. The hierarchical properties are induced by the direction of the edges in the two directed graphs. These structural relationships defined by the hierarchy in the graphs act as a constraint on the alignment. In this paper, we formalize the above problem as the weighted alignment between two directed acyclic graphs. We prove that this problem is NP-complete, show several upper bounds for approximating the solution, and finally introduce polynomial time algorithms for sub…
Directed acyclic graphs (DAGs) can be characterised as directed graphs whose strongly connected components are isolated vertices. Using this restriction on the strong components, we discover that when $m = cn$, where $m$ is the number of directed edges, $n$ is the number of vertices, and $c < 1$, the asymptotic probability that a random digraph is acyclic is an explicit function $p(c)$, such that $p(0) = 1$ and $p(1) = 0$. When $m = n(1 + μn^{-1/3})$, the asymptotic behaviour changes, and the probability that a digraph is acyclic becomes $n^{-1/3} C(μ)$, where $C(μ)$ is an explicit function of…
Given a pair of distinct vertices u, v in a graph G, we say that s is a junction of u, v if there are in G internally vertex disjoint directed paths from s to u and from s to v. We show how to characterize junctions in directed acyclic graphs. We also consider the two problems in the following and derive efficient algorithms to solve them. Given a directed acyclic graph G and a vertex s in G, how can we find all pairs of vertices of G such that s is a junction of them? And given a directed acyclic graph G and k pairs of vertices of G, how can we preprocess G such that all junctions of k given …
We present a linear-time algorithm for simplifying flow networks on directed planar graphs: Given a directed planar graph on $n$ vertices, a source vertex $s$ and a sink vertex $t$, our algorithm removes all the arcs that do not participate in any simple $s,t$-path in linear-time. The output graph produced by our algorithm satisfies the prerequisite needed by the $O(n\log n)$-time algorithm of Weihe [FOCS'94 \& JCSS'97] for computing maximum $s,t$-flow in directed planar graphs. Previously, Weihe's algorithm could not run in $O(n\log n)$-time due to the absence of the preprocessing step; all t…
An acyclic r-coloring of a directed graph G=(V,E) is a partition of the vertex set V into r acyclic sets. The dichromatic number of a directed graph G is the smallest r such that G allows an acyclic r-coloring. For symmetric digraphs the dichromatic number equals the well-known chromatic number of the underlying undirected graph. This allows us to carry over the W[1]-hardness and lower bounds for running times of the chromatic number problem parameterized by clique-width to the dichromatic number problem parameterized by directed clique-width. We introduce the first polynomial-time algorithm f…
We consider a well known model of random directed acyclic graphs of order $n$, obtained by recursively adding vertices, where each new vertex has a fixed outdegree $d\ge2$ and the endpoints of the $d$ edges from it are chosen uniformly at random among previously existing vertices. Our main results concern the number $X$ of vertices that are descendants of $n$. We show that $X/\sqrt n$ converges in distribution; the limit distribution is, up to a constant factor, given by the $d$th root of a Gamma distributed variable. $Γ(d/(d-1))$. When $d=2$, the limit distribution can also be described as a …
We study distributed algorithms built around minor-based vertex sparsifiers, and give the first algorithm in the CONGEST model for solving linear systems in graph Laplacian matrices to high accuracy. Our Laplacian solver has a round complexity of $O(n^{o(1)}(\sqrt{n}+D))$, and thus almost matches the lower bound of $\widetildeΩ(\sqrt{n}+D)$, where $n$ is the number of nodes in the network and $D$ is its diameter. We show that our distributed solver yields new sublinear round algorithms for several cornerstone problems in combinatorial optimization. This is achieved by leveraging the powerful a…
Graph Sparsification aims at compressing large graphs into smaller ones while preserving important characteristics of the input graph. In this work we study Vertex Sparsifiers, i.e., sparsifiers whose goal is to reduce the number of vertices. We focus on the following notions: (1) Given a digraph $G=(V,E)$ and terminal vertices $K \subset V$ with $|K| = k$, a (vertex) reachability sparsifier of $G$ is a digraph $H=(V_H,E_H)$, $K \subset V_H$ that preserves all reachability information among terminal pairs. In this work we introduce the notion of reachability-preserving minors (RPMs) , i.e., we…
Given a directed acyclic graph with labeled vertices, we consider the problem of finding the most common label sequences ("traces") among all paths in the graph (of some maximum length m). Since the number of paths can be huge, we propose novel algorithms whose time complexity depends only on the size of the graph, and on the relative frequency epsilon of the most frequent traces. In addition, we apply techniques from streaming algorithms to achieve space usage that depends only on epsilon, and not on the number of distinct traces. The abstract problem considered models a variety of tasks conc…
A directed acyclic hypergraph is a generalized concept of a directed acyclic graph, where each hyperedge can contain an arbitrary number of tails and heads. Directed hypergraphs can be used to model data flow and execution dependencies in streaming applications. Thus, hypergraph partitioning algorithms can be used to obtain efficient parallelizations for multiprocessor architectures. However, an acyclicity constraint on the partition is necessary when mapping streaming applications to embedded multiprocessors due to resource restrictions on this type of hardware. The acyclic hypergraph partiti…
Given a directed acyclic graph with positive edge-weights, two vertices s and t, and a threshold-weight L, we present a fully-polynomial time approximation-scheme for the problem of counting the s-t paths of length at most L. We extend the algorithm for the case of two (or more) instances of the same problem. That is, given two graphs that have the same vertices and edges and differ only in edge-weights, and given two threshold-weights L_1 and L_2, we show how to approximately count the s-t paths that have length at most L_1 in the first graph and length at most L_2 in the second graph. We bel…
We investigate the odd multiway node (edge) cut problem where the input is a graph with a specified collection of terminal nodes and the goal is to find a smallest subset of nonterminal nodes (edges) to delete so that the terminal nodes do not have an odd length path between them. In an earlier work, Lokshtanov and Ramanujan showed that both odd multiway node cut and odd multiway edge cut are fixed-parameter tractable (FPT) when parameterized by the size of the solution in undirected graphs. In this work, we focus on directed acyclic graphs (DAGs) and design a fixed-parameter algorithm. Our ma…
An oriented graph is a directed graph without directed 2-cycles. Poljak and Turzík (1986) proved that every connected oriented graph $G$ on $n$ vertices and $m$ arcs contains an acyclic subgraph with at least $\frac{m}{2}+\frac{n-1}{4}$ arcs. Raman and Saurabh (2006) gave another proof of this result and left it as an open question to establish the parameterized complexity of the following problem: does $G$ have an acyclic subgraph with least $\frac{m}{2}+\frac{n-1}{4}+k$ arcs, where $k$ is the parameter? We answer this question by showing that the problem can be solved by an algorithm of runt…
Motion planning is a fundamental problem of robotics with applications in many areas of computer science and beyond. Its restriction to graphs has been investigated in the literature for it allows to concentrate on the combinatorial problem abstracting from geometric considerations. In this paper, we consider motion planning over directed graphs, which are of interest for asymmetric communication networks. Directed graphs generalize undirected graphs, while introducing a new source of complexity to the motion planning problem: moves are not reversible. We first consider the class of acyclic di…
A motif is a frequently occurring subgraph of a given directed or undirected graph $G$. Motifs capture higher order organizational structure of $G$ beyond edge relationships, and, therefore, have found wide applications such as in graph clustering, community detection, and analysis of biological and physical networks to name a few. In these applications, the cut structure of motifs plays a crucial role as vertices are partitioned into clusters by cuts whose conductance is based on the number of instances of a particular motif, as opposed to just the number of edges, crossing the cuts. In this …
We analyze a new property of directed acyclic graphs (DAGs), called layerwidth, arising from a class of DAGs proposed by Eiter and Lukasiewicz. This class of DAGs permits certain problems of structural model-based causality and explanation to be tractably solved. In this paper, we first address an open question raised by Eiter and Lukasiewicz - the computational complexity of deciding whether a given graph has a bounded layerwidth. After proving that this problem is NP-complete, we proceed by proving numerous important properties of layerwidth that are helpful in efficiently computing the opti…
A straightforward dynamic programming method for the single-source shortest paths problem (SSSP) in an edge-weighted directed acyclic graph (DAG) processes the vertices in a topologically sorted order. First, we similarly iterate this method alternatively in a breadth-first search sorted order and the reverse order on an input directed graph with both positive and negative real edge weights, $n$ vertices and $m$ edges. For a positive integer $t,$ after $O(t)$ iterations in $O(tm)$ time, we obtain for each vertex $v$ a path distance from the source to $v$ not exceeding that yielded by the short…
The MULTICUT problem, given a graph G, a set of terminal pairs T={(s_i,t_i) | 1 <= i <= r} and an integer p, asks whether one can find a cutset consisting of at most p non-terminal vertices that separates all the terminal pairs, i.e., after removing the cutset, t_i is not reachable from s_i for each 1 <= i <= r. The fixed-parameter tractability of MULTICUT in undirected graphs, parameterized by the size of the cutset only, has been recently proven by Marx and Razgon (STOC'11) and, independently, by Bousquet et al. (STOC'11), after resisting attacks as a long-standing open problem. In this pape…
In recent years, hypergraph generalizations of many graph cut problems have been introduced and analyzed as a way to better explore and understand complex systems and datasets characterized by multiway relationships. Recent work has made use of a generalized hypergraph cut function which for a hypergraph $\mathcal{H} = (V,E)$ can be defined by associating each hyperedge $e \in E$ with a splitting function ${\bf w}_e$, which assigns a penalty to each way of separating the nodes of $e$. When each ${\bf w}_e$ is a submodular cardinality-based splitting function, meaning that ${\bf w}_e(S) = g(|S|…
This work addresses the NP-Hard problem of acyclic directed acyclic graph (DAG) partitioning problem. The acyclic partitioning problem is defined as partitioning the vertex set of a given directed acyclic graph into disjoint and collectively exhaustive subsets (parts). Parts are to be assigned such that the total sum of the vertex weights within each part satisfies a common upper bound and the total sum of the edge costs that connect nodes across different parts is minimized. Additionally, the quotient graph, i.e., the induced graph where all nodes that are assigned to the same part are contra…
Modern problems of concept annotation associate an object of interest (gene, individual, text document) with a set of interrelated textual descriptors (functions, diseases, topics), often organized in concept hierarchies or ontologies. Most ontologies can be seen as directed acyclic graphs, where nodes represent concepts and edges represent relational ties between these concepts. Given an ontology graph, each object can only be annotated by a consistent subgraph; that is, a subgraph such that if an object is annotated by a particular concept, it must also be annotated by all other concepts tha…
