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Register a new userNear-linear-time, Optimal Vertex Cut Sparsifiers in Directed Acyclic Graphs
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Zhiyang He
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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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.
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Cuts in graphs are a fundamental object of study, and play a central role in the study of graph algorithms. The problem of sparsifying a graph while approximately preserving its cut structure has been extensively studied and has many applications. In a seminal work, Benczur and Karger (1996) showed that given any $n$-vertex undirected weighted graph $G$ and a parameter $varepsilon in (0,1)$, there is a near-linear time algorithm that outputs a weighted subgraph $G$ of $G$ of size $tilde{O}(n/varepsilon^2)$ such that the weight of every cut in $G$ is preserved to within a $(1 pm varepsilon)$-factor in $G$. The graph $G$ is referred to as a {em $(1 pm varepsilon)$-approximate cut sparsifier} of $G$. A natural question is if such cut-preserving sparsifiers also exist for hypergraphs. Kogan and Krauthgamer (2015) initiated a study of this question and showed that given any weighted hypergraph $H$ where the cardinality of each hyperedge is bounded by $r$, there is a polynomial-time algorithm to find a $(1 pm varepsilon)$-approximate cut sparsifier of $H$ of size $tilde{O}(frac{nr}{varepsilon^2})$. Since $r$ can be as large as $n$, in general, this gives a hypergraph cut sparsifier of size $tilde{O}(n^2/varepsilon^2)$, which is a factor $n$ larger than the Benczur-Karger bound for graphs. It has been an open question whether or not Benczur-Karger bound is achievable on hypergraphs. In this work, we resolve this question in the affirmative by giving a new polynomial-time algorithm for creating hypergraph sparsifiers of size $tilde{O}(n/varepsilon^2)$.
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.
Let $G = (V,E,w)$ be a weighted undirected graph on $|V| = n$ vertices and $|E| = m$ edges, let $k ge 1$ be any integer, and let $epsilon < 1$ be any parameter. We present the following results on fast constructions of spanners with near-optimal sparsity and lightness, which culminate a long line of work in this area. (By near-optimal we mean optimal under ErdH{o}s girth conjecture and disregarding the $epsilon$-dependencies.) - There are (deterministic) algorithms for constructing $(2k-1)(1+epsilon)$-spanners for $G$ with a near-optimal sparsity of $O(n^{1/k} log(1/epsilon)/epsilon))$. The first algorithm can be implemented in the pointer-machine model within time $O(malpha(m,n) log(1/epsilon)/epsilon) + SORT(m))$, where $alpha( , )$ is the two-parameter inverse-Ackermann function and $SORT(m)$ is the time needed to sort $m$ integers. The second algorithm can be implemented in the WORD RAM model within time $O(m log(1/epsilon)/epsilon))$. - There is a (deterministic) algorithm for constructing a $(2k-1)(1+epsilon)$-spanner for $G$ that achieves a near-optimal bound of $O(n^{1/k}mathrm{poly}(1/epsilon))$ on both sparsity and lightness. This algorithm can be implemented in the pointer-machine model within time $O(malpha(m,n) mathrm{poly}(1/epsilon) + SORT(m))$ and in the WORD RAM model within time $O(m alpha(m,n) mathrm{poly}(1/epsilon))$. The previous fastest constructions of $(2k-1)(1+epsilon)$-spanners with near-optimal sparsity incur a runtime of is $O(min{m(n^{1+1/k}) + nlog n,k n^{2+1/k}})$, even regardless of the lightness. Importantly, the greedy spanner for stretch $2k-1$ has sparsity $O(n^{1/k})$ -- with no $epsilon$-dependence whatsoever, but its runtime is $O(m(n^{1+1/k} + nlog n))$. Moreover, the state-of-the-art lightness bound of any $(2k-1)$-spanner is poor, even regardless of the sparsity and runtime.
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. Benczur 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 interesting application of digraph sparsification via cut balance by using it to give a very short proof of a celebrated maximum flow result of Karger and Levine (STOC 2002).
In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph $D$ on $n$ vertices and $m$ edges, and an integer $k$. The objective is to determine whether there exists a set of at most $k$ vertices intersecting every directed cycle of $D$. Whether or not DFVS admits a fixed parameter tractable (FPT) algorithm was considered the most important open problem in parameterized complexity until Chen, Liu, Lu, OSullivan and Razgon [JACM 2008] answered the question in the affirmative. They gave an algorithm for the problem with running time $O(k!4^kk^4nm)$. Since then, no faster algorithm for the problem has been found. In this paper, we give an algorithm for DFVS with running time $O(k!4^kk^5(n+m))$. Our algorithm is the first algorithm for DFVS with linear dependence on input size. Furthermore, the asymptotic dependence of the running time of our algorithm on the parameter $k$ matches up to a factor $k$ the algorithm of Chen, Liu, Lu, OSullivan and Razgon. On the way to designing our algorithm for DFVS, we give a general methodology to shave off a factor of $n$ from iterative-compression based algorithms for a few other well-studied covering problems in parameterized complexity. We demonstrate the applicability of this technique by speeding up by a factor of $n$, the current best FPT algorithms for Multicut [STOC 2011, SICOMP 2014] and Directed Subset Feedback Vertex Set [ICALP 2012, TALG 2014].
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