اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدSubgraphs Satisfying MSO Properties on z-Topologically Orderable Digraphs
139
0
0.0
(
0
)
نشر من قبل
Mateus de Oliveira Oliveira
تاريخ النشر
2013
مجال البحث
الهندسة المعلوماتية
والبحث باللغة
English
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We introduce the notion of z-topological orderings for digraphs. We prove that given a digraph G on n vertices admitting a z-topological order- ing, together with such an ordering, one may count the number of subgraphs of G that at the same time satisfy a monadic second order formula {phi} and are the union of k directed paths, in time f ({phi}, k, z) * n^O(k*z) . Our result implies the polynomial time solvability of many natural counting problems on digraphs admitting z-topological orderings for constant values of z and k. Concerning the relationship between z-topological orderability and other digraph width measures, we observe that any digraph of directed path-width d has a z- topological ordering for z <= 2d + 1. On the other hand, there are digraphs on n vertices admitting a z-topological order for z = 2, but whose directed path-width is {Theta}(log n). Since graphs of bounded directed path-width can have both arbitrarily large undirected tree-width and arbitrarily large clique width, our result provides for the first time a suitable way of partially trans- posing metatheorems developed in the context of the monadic second order logic of graphs of constant undirected tree-width and constant clique width to the realm of digraph width measures that are closed under taking subgraphs and whose constant levels incorporate families of graphs of arbitrarily large undirected tree-width and arbitrarily large clique width.
قيم البحث
اقرأ أيضاً
We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry restriction. In the context of circuits computing polynomials defined on a matrix of variables, such as the determinant or the permanent, the restriction amo
unts to requiring that the shape of the circuit is invariant under simultaneous row and column permutations of the matrix. We establish unconditional exponential lower bounds on the size of any symmetric circuit for computing the permanent. In contrast, we show that there are polynomial-size symmetric circuits for computing the determinant over fields of characteristic zero.
We show that there are CNF formulas which can be refuted in resolution in both small space and small width, but for which any small-width proof must have space exceeding by far the linear worst-case upper bound. This significantly strengthens the spa
ce-width trade-offs in [Ben-Sasson 09]}, and provides one more example of trade-offs in the supercritical regime above worst case recently identified by [Razborov 16]. We obtain our results by using Razborovs new hardness condensation technique and combining it with the space lower bounds in [Ben-Sasson and Nordstrom 08].
Dawar and Wilsenach (ICALP 2020) introduce the model of symmetric arithmetic circuits and show an exponential separation between the sizes of symmetric circuits for computing the determinant and the permanent. The symmetry restriction is that the cir
cuits which take a matrix input are unchanged by a permutation applied simultaneously to the rows and columns of the matrix. Under such restrictions we have polynomial-size circuits for computing the determinant but no subexponential size circuits for the permanent. Here, we consider a more stringent symmetry requirement, namely that the circuits are unchanged by arbitrary even permutations applied separately to rows and columns, and prove an exponential lower bound even for circuits computing the determinant. The result requires substantial new machinery. We develop a general framework for proving lower bounds for symmetric circuits with restricted symmetries, based on a new support theorem and new two-player restricted bijection games. These are applied to the determinant problem with a novel construction of matrices that are bi-adjacency matrices of graphs based on the CFI construction. Our general framework opens the way to exploring a variety of symmetry restrictions and studying trade-offs between symmetry and other resources used by arithmetic circuits.
In the Survivable Network Design Problem (SNDP), the input is an edge-weighted (di)graph $G$ and an integer $r_{uv}$ for every pair of vertices $u,vin V(G)$. The objective is to construct a subgraph $H$ of minimum weight which contains $r_{uv}$ edge-
disjoint (or node-disjoint) $u$-$v$ paths. This is a fundamental problem in combinatorial optimization that captures numerous well-studied problems in graph theory and graph algorithms. In this paper, we consider the version of the problem where we are given a $lambda$-edge connected (di)graph $G$ with a non-negative weight function $w$ on the edges and an integer $k$, and the objective is to find a minimum weight spanning subgraph $H$ that is also $lambda$-edge connected, and has at least $k$ fewer edges than $G$. In other words, we are asked to compute a maximum weight subset of edges, of cardinality up to $k$, which may be safely deleted from $G$. Motivated by this question, we investigate the connectivity properties of $lambda$-edge connected (di)graphs and obtain algorithmically significant structural results. We demonstrate the importance of our structural results by presenting an algorithm running in time $2^{O(k log k)} |V(G)|^{O(1)}$ for $lambda$-ECS, thus proving its fixed-parameter tractability. We follow up on this result and obtain the {em first polynomial compression} for $lambda$-ECS on unweighted graphs. As a consequence, we also obtain the first fixed parameter tractable algorithm, and a polynomial kernel for a parameterized version of the classic Mininum Equivalent Graph problem. We believe that our structural results are of independent interest and will play a crucial role in the design of algorithms for connectivity-constrained problems in general and the SNDP problem in particular.
Let CMSO denote the counting monadic second order logic of graphs. We give a constructive proof that for some computable function $f$, there is an algorithm $mathfrak{A}$ that takes as input a CMSO sentence $varphi$, a positive integer $t$, and a con
nected graph $G$ of maximum degree at most $Delta$, and determines, in time $f(|varphi|,t)cdot 2^{O(Delta cdot t)}cdot |G|^{O(t)}$, whether $G$ has a supergraph $G$ of treewidth at most $t$ such that $Gmodels varphi$. The algorithmic metatheorem described above sheds new light on certain unresolved questions within the framework of graph completion algorithms. In particular, using this metatheorem, we provide an explicit algorithm that determines, in time $f(d)cdot 2^{O(Delta cdot d)}cdot |G|^{O(d)}$, whether a connected graph of maximum degree $Delta$ has a planar supergraph of diameter at most $d$. Additionally, we show that for each fixed $k$, the problem of determining whether $G$ has an $k$-outerplanar supergraph of diameter at most $d$ is strongly uniformly fixed parameter tractable with respect to the parameter $d$. This result can be generalized in two directions. First, the diameter parameter can be replaced by any contraction-closed effectively CMSO-definable parameter $mathbf{p}$. Examples of such parameters are vertex-cover number, dominating number, and many other contraction-bidimensional parameters. In the second direction, the planarity requirement can be relaxed to bounded genus, and more generally, to bounded local treewidth.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
