اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدAn Algorithmic Metatheorem for Directed Treewidth
125
0
0.0
(
0
)
نشر من قبل
Mateus de Oliveira Oliveira
تاريخ النشر
2014
مجال البحث
الهندسة المعلوماتية
والبحث باللغة
English
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
The notion of directed treewidth was introduced by Johnson, Robertson, Seymour and Thomas [Journal of Combinatorial Theory, Series B, Vol 82, 2001] as a first step towards an algorithmic metatheory for digraphs. They showed that some NP-complete properties such as Hamiltonicity can be decided in polynomial time on digraphs of constant directed treewidth. Nevertheless, despite more than one decade of intensive research, the list of hard combinatorial problems that are known to be solvable in polynomial time when restricted to digraphs of constant directed treewidth has remained scarce. In this work we enrich this list by providing for the first time an algorithmic metatheorem connecting the monadic second order logic of graphs to directed treewidth. We show that most of the known positive algorithmic results for digraphs of constant directed treewidth can be reformulated in terms of our metatheorem. Additionally, we show how to use our metatheorem to provide polynomial time algorithms for two classes of combinatorial problems that have not yet been studied in the context of directed width measures. More precisely, for each fixed $k,w in mathbb{N}$, we show how to count in polynomial time on digraphs of directed treewidth $w$, the number of minimum spanning strong subgraphs that are the union of $k$ directed paths, and the number of maximal subgraphs that are the union of $k$ directed paths and satisfy a given minor closed property. To prove our metatheorem we devise two technical tools which we believe to be of independent interest. First, we introduce the notion of tree-zig-zag number of a digraph, a new directed width measure that is at most a constant times directed treewidth. Second, we introduce the notion of $z$-saturated tree slice language, a new formalism for the specification and manipulation of infinite sets of digraphs.
قيم البحث
اقرأ أيضاً
We consider the convex hull $P_{varphi}(G)$ of all satisfying assignments of a given MSO formula $varphi$ on a given graph $G$. We show that there exists an extended formulation of the polytope $P_{varphi}(G)$ that can be described by $f(|varphi|,tau
)cdot n$ inequalities, where $n$ is the number of vertices in $G$, $tau$ is the treewidth of $G$ and $f$ is a computable function depending only on $varphi$ and $tau.$ In other words, we prove that the extension complexity of $P_{varphi}(G)$ is linear in the size of the graph $G$, with a constant depending on the treewidth of $G$ and the formula $varphi$. This provides a very general yet very simple meta-theorem about the extension complexity of polytopes related to a wide class of problems and graphs. As a corollary of our main result, we obtain an analogous result % for the weaker MSO$_1$ logic on the wider class of graphs of bounded cliquewidth. Furthermore, we study our main geometric tool which we term the glued product of polytopes. While the glued product of polytopes has been known since the 90s, we are the first to show that it preserves decomposability and boundedness of treewidth of the constraint matrix. This implies that our extension of $P_varphi(G)$ is decomposable and has a constraint matrix of bounded treewidth; so far only few classes of polytopes are known to be decomposable. These properties make our extension useful in the construction of algorithms.
We study the approximability of the Maximum Independent Set (MIS) problem in $H$-free graphs (that is, graphs which do not admit $H$ as an induced subgraph). As one motivation we investigate the following conjecture: for every fixed graph $H$, there
exists a constant $delta > 0$ such that MIS can be $n^{1 - delta}$-approximated in $H$-free graphs, where $n$ denotes the number of vertices of the input graph. We first prove that a constructive version of the celebrated ErdH{o}s-Hajnal conjecture implies ours. We then prove that the set of graphs $H$ satisfying our conjecture is closed under the so-called graph substitution. This, together with the known polynomial-time algorithms for MIS in $H$-free graphs (e.g. $P_6$-free and fork-free graphs), implies that our conjecture holds for many graphs $H$ for which the ErdH{o}s-Hajnal conjecture is still open. We then focus on improving the constant $delta$ for some graph classes: we prove that the classical Local Search algorithm provides an $OPT^{1-frac{1}{t}}$-approximation in $K_{t,t}$-free graphs (hence a $sqrt{OPT}$-approximation in $C_4$-free graphs), and, while there is a simple $sqrt{n}$-approximation in triangle-free graphs, it cannot be improved to $n^{frac{1}{4}-varepsilon}$ for any $varepsilon > 0$ unless $NP subseteq BPP$. More generally, we show that there is a constant $c$ such that MIS in graphs of girth $gamma$ cannot be $n^{frac{c}{gamma}}$-approximated. Up to a constant factor in the exponent, this matches the ratio of a known approximation algorithm by Monien and Speckenmeyer, and by Murphy. To the best of our knowledge, this is the first strong (i.e., $Omega(n^delta)$ for some $delta > 0$) inapproximability result for Maximum Independent Set in a proper hereditary class.
The Subgraph Isomorphism problem is of considerable importance in computer science. We examine the problem when the pattern graph H is of bounded treewidth, as occurs in a variety of applications. This problem has a well-known algorithm via color-cod
ing that runs in time $O(n^{tw(H)+1})$ [Alon, Yuster, Zwick95], where $n$ is the number of vertices of the host graph $G$. While there are pattern graphs known for which Subgraph Isomorphism can be solved in an improved running time of $O(n^{tw(H)+1-varepsilon})$ or even faster (e.g. for $k$-cliques), it is not known whether such improvements are possible for all patterns. The only known lower bound rules out time $n^{o(tw(H) / log(tw(H)))}$ for any class of patterns of unbounded treewidth assuming the Exponential Time Hypothesis [Marx07]. In this paper, we demonstrate the existence of maximally hard pattern graphs $H$ that require time $n^{tw(H)+1-o(1)}$. Specifically, under the Strong Exponential Time Hypothesis (SETH), a standard assumption from fine-grained complexity theory, we prove the following asymptotic statement for large treewidth $t$: For any $varepsilon > 0$ there exists $t ge 3$ and a pattern graph $H$ of treewidth $t$ such that Subgraph Isomorphism on pattern $H$ has no algorithm running in time $O(n^{t+1-varepsilon})$. Under the more recent 3-uniform Hyperclique hypothesis, we even obtain tight lower bounds for each specific treewidth $t ge 3$: For any $t ge 3$ there exists a pattern graph $H$ of treewidth $t$ such that for any $varepsilon>0$ Subgraph Isomorphism on pattern $H$ has no algorithm running in time $O(n^{t+1-varepsilon})$. In addition to these main results, we explore (1) colored and uncolored problem variants (and why they are equivalent for most cases), (2) Subgraph Isomorphism for $tw < 3$, (3) Subgraph Isomorphism parameterized by pathwidth, and (4) a weighted problem variant.
The Cut & Count technique and the rank-based approach have lead to single-exponential FPT algorithms parameterized by treewidth, that is, running in time $2^{O(tw)}n^{O(1)}$, for Feedback Vertex Set and connect
Learning in the presence of outliers is a fundamental problem in statistics. Until recently, all known efficient unsupervised learning algorithms were very sensitive to outliers in high dimensions. In particular, even for the task of robust mean esti
mation under natural distributional assumptions, no efficient algorithm was known. Recent work in theoretical computer science gave the first efficient robust estimators for a number of fundamental statistical tasks, including mean and covariance estimation. Since then, there has been a flurry of research activity on algorithmic high-dimensional robust estimation in a range of settings. In this survey article, we introduce the core ideas and algorithmic techniques in the emerging area of algorithmic high-dimensional robust statistics with a focus on robust mean estimation. We also provide an overview of the approaches that have led to computationally efficient robust estimators for a range of broader statistical tasks and discuss new directions and opportunities for future work.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
