Parameterized Complexity of Directed Spanner Problems

An approximation algorithm for the Generalized [formula omitted]-Multicut problem

On knot-free vertex deletion: Fine-grained parameterized complexity analysis of a deadlock resolution graph problem

Approximating minimum cost source location problems with local vertex-connectivity demands

We consider the Maximum Node Disjoint Paths (MNDP) problem in undirected graphs. The input consists of an undirected graph G = (V, E) and a collection {(s1, t1), …, (sk, tk)} of k source-sink pairs. The goal is to select a maximum cardinality subset of pairs that can be routed/connected via node-disjoint paths. A relaxed version of MNDP allows up to c paths to use a node, where c is the congestion parameter. We give a polynomial time algorithm that routes Ω(OPT/poly log k) pairs with O(1) congestion, where OPT is the value of an optimum fractional solution to a natural multicommodity flow relaxation. Our result builds on the recent breakthrough of Chuzhoy [17] who gave the first poly-logarithmic approximation with constant congestion for the Maximum Edge Disjoint Paths (MEDP) problem.

Connectivity related concepts are of fundamental interest in graph theory. The area has received extensive attention over four decades, but many problems remain unsolved, especially for directed graphs. A directed graph is 2-edge-connected (resp., 2-vertex-connected) if the removal of any edge (resp., vertex) leaves the graph strongly connected. In this paper we present improved algorithms for computing the maximal 2-edge- and 2-vertex-connected subgraphs of a given directed graph. These problems were first studied more than 35 years ago, with O(mn) time algorithms for graphs with m edges and n vertices being known since the late 1980s. In contrast, the same problems for undirected graphs are known to be solvable in linear time. Henzinger et al. [ICALP 2015] recently introduced O(n2) time algorithms for the directed case, thus improving the running times for dense graphs. Our new algorithms run in time O(m3/2), which further improves the running times for sparse graphs.The notion of 2-connectivity naturally generalizes to k-connectivity for k > 2. For constant values of k, we extend one of our algorithms to compute the maximal k-edge-connected in time O(m3/2 log n), improving again for sparse graphs the best known algorithm by Henzinger et al. [ICALP 2015] that runs in O(n2 log n) time.

CHAPTER 4 - Directed Graphs

We consider a shortest path problem for a directed graph with edges labeled with a cost and a capacity. The problem is to push an unsplittable flow $f$ from a specified source to all other vertices with the minimum cost for all f values. Let G = (V, E) with |V| = n and |E| = m. If there are t different capacity values, we can solve the single source shortest path problem t times for all f in O(tm + tn log n) time, which is O(m^2) when t = m. We improve this time to O(min{t, cn}m + cn^2), which is less than O(cmn) if edge costs are non-negative integers bounded by c. Our algorithm performs better for denser graphs.

On the Steenrod length of real projective spaces: finding longest chains in certain directed graphs

Extrema property of the [formula omitted]-ranking of directed paths and cycles

Let [Formula: see text] be a finite field with [Formula: see text] elements, [Formula: see text] a positive integer, [Formula: see text] the semigroup of all [Formula: see text] upper triangular matrices over [Formula: see text] under matrix multiplication, [Formula: see text] the group of all invertible matrices in [Formula: see text], [Formula: see text] the quotient group of [Formula: see text] by its center. The one-divisor graph of [Formula: see text], written as [Formula: see text], is defined to be a directed graph with [Formula: see text] as vertex set, and there is a directed edge from [Formula: see text] to [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text] and [Formula: see text] are, respectively, a left divisor and a right divisor of a rank one matrix in [Formula: see text]. The definition of [Formula: see text] is motivated by the definition of zero-divisor graph [Formula: see text] of [Formula: see text], which has vertex set of all nonzero zero-divisors in [Formula: see text] and there is a directed edge from a vertex [Formula: see text] to a vertex [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text]. The automorphism group of zero-divisor graph [Formula: see text] of [Formula: see text] was recently determined by Wang [A note on automorphisms of the zero-divisor graph of upper triangular matrices, Lin. Alg. Appl. 465 (2015) 214–220.]. In this paper, we characterize the automorphism group of one-divisor graph [Formula: see text] of [Formula: see text], proving that [Formula: see text], where [Formula: see text] is the automorphism group of field [Formula: see text], [Formula: see text] is a direct product of some symmetric groups. Besides, an application of automorphisms of [Formula: see text] is given in this paper.

A local partitioning algorithm finds a set with small conductance near a specified seed vertex. In this paper, we present a generalization of a local partitioning algorithm for undirected graphs to strongly connected directed graphs. In particular, we prove that by computing a personalized PageRank vector in a directed graph, starting from a single seed vertex within a set _S_ that has conductance at most α, and by performing a sweep over that vector, we can obtain a set of vertices _S_′ with conductance <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uinm_a_10129297_o_ilf0001.gif"></inline-graphic>. Here, the conductance function Φ_<em>M</em> is defined in terms of the stationary distribution of a random walk in the directed graph. In addition, we describe how this algorithm may be applied to the PageRank Markov chain of an arbitrary directed graph, which provides a way to partition directed graphs that are not strongly connected.

We present a study of multithreaded implementations of Thorup's algorithm for solving the Single Source Shortest Path (SSSP) problemfor undirected graphs. Our implementations leverage thefledgling MultiThreaded Graph Library (MTGL) to perform operations such as finding connected components and extracting induced subgraphs. To achieve good parallel performance from this algorithm, we deviate from several theoretically optimal algorithmic steps. In this paper; we present simplifications that perform better in practice, and we describe details of the multithreaded implementation that were necessary for scalability. We study synthetic graphs that model unstructured networks, such as social networks and economic transaction networks. Most of the recent progress in shortest path algorithms relies on structure that these networks do not have. In this work, we take a step back and explore the synergy between an elegant theoretical algorithm and an elegant computer architecture. Finally, we conclude with a prediction that this work will become relevant to shortest path computation on structured networks.

A rotator graph was proposed as a topology for interconnection networks of parallel computers, and it is promising because of its small diameter and small degree. However, a rotator graph is a directed graph that sometimes behaves harmful when it is applied to actual problems. A bi-rotator graph is obtained by making each edge of a rotator graph bi-directional. A bi-rotator has a Hamilton cycle and it is also pan-cyclic. In this paper, we give an algorithm for the node-to-set disjoint paths problem in bi-rotator graphs with its evaluation results. The solution achieves some fault tolerance such as file distribution based information dispersal technique. The algorithm is of polynomial order of n for an n-bi-rotator graph. It is based on recursion and divided into three cases according to the distribution of destination nodes in classes into which all the nodes in a bi-rotator graph are categorized. The sum of lengths of paths obtained and the time complexity of the algorithm are estimated. Average performance of the algorithm is also evaluated by computer experiments.

Given a directed graph $G = (V,E)$ and an integer $k \geq 1$, a $k$-transitive-closure-spanner ($k$-TC-spanner) of $G$ is a directed graph $H = (V, E_H)$ that has (1) the same transitive-closure as $G$ and (2) diameter at most $k$. These spanners were implicitly studied in the context of circuit complexity, data structures, property testing, and access control, and properties of these spanners have been rediscovered over the span of 20 years. We abstract the common task implicitly tackled in these diverse applications as the problem of constructing sparse TC-spanners. We initiate the study of approximability of the size of the sparsest $k$-TC-spanner of a given directed graph. We completely resolve the approximability of $2$-TC-spanners, showing that it is $\Theta(\log n)$ unless $\textsf{P} = \textsf{NP}$. For $k>2$, we present a polynomial time algorithm that finds a $k$-TC-spanner with size within $O((n \log n)^{1-1/k})$ of the optimum. Our techniques also yield algorithms with the first nontrivial approximation ratio for well-studied problems on directed spanners when $k>3$: Directed $k$-Spanner, Client/Server Directed $k$-Spanner, and $k$-Diameter Spanning Subgraph. For constant $k \geq 3$, we show that the size of the sparsest $k$-TC-spanner is hard to approximate within a factor of $2^{\log^{1-\eps} n}$ for any $\eps \in (0,1)$ unless $\NP \subseteq \text{DTIME}(n^{\polylog n})$. Finally, we study the size of the sparsest $k$-TC-spanners for $H$-minor-free graph families. Combining our constructions with our insight that 2-TC-spanners can be used for designing property testers, we obtain a monotonicity tester with $O(\log^2 n /\eps)$ queries for any poset whose transitive reduction, when viewed as an undirected graph, is free of a fixed minor. Previously, the best upper bound on the query complexity for such graphs was $O(\sqrt{n/\eps})$.

Vertex-frequency analysis (VFA) is a useful technique in graph signal processing to extract the correspondence between frequencies and vertices. VFA can be calculated by the windowed graph Fourier transform (WGFT) and the localized graph Fourier transform (LGFT). However, since the inability of the classical graph Fourier transform (GFT) to generate orthogonal bases for directed graphs, the two approaches are confined to undirected graphs. In order to address the absence of VFA on directed graphs, we extend the fundamental concepts of VFA, shifted vertex window function and band-pass transfer filters in spectral domain to directed graphs. We first propose WDGFT based on the shifted window function in vertex domain. The relevant properties, reconstruction formulas, and simulation results of vertex-frequency representation are given. Then, we propose the localized directed graph Fourier transform (LDGFT), where the band-pass transfer filters can capture the localized spectral domain. The transfer functions satisfying the reconstruction condition are discussed. The vertex-frequency representation obtained by LDGFT is provided through numerical examples. Furthermore, we present a polynomial approximation technique to decrease computational cost. The LDGFT can be calculated without any matrix decomposition. Finally, we evaluate the proposed directed VFA framework with two kinds of applications, clustering and malfunction detection. We demonstrate that the proposed VFA framework is a powerful tool for directed graph signal processing.