Finding 2-Edge and 2-Vertex Strongly Connected Components in Quadratic Time

We present faster algorithms for computing the 2-edge and 2-vertex strongly connected components of a directed graph, which are straightforward generalizations of strongly connected components. While in undirected graphs the 2-edge and 2-vertex connected components can be found in linear time, in directed graphs only rather simple $O(m n)$-time algorithms were known. We use a hierarchical sparsification technique to obtain algorithms that run in time $O(n^2)$. For 2-edge strongly connected components our algorithm gives the first running time improvement in 20 years. Additionally we present an $O(m^2 / \log{n})$-time algorithm for 2-edge strongly connected components, and thus improve over the $O(m n)$ running time also when $m = O(n)$. Our approach extends to k-edge and k-vertex strongly connected components for any constant k with a running time of $O(n^2 \log^2 n)$ for edges and $O(n^3)$ for vertices

Orienting Graphs to Optimize Reachability

The paper focuses on two problems: (i) how to orient the edges of an undirected graph in order to maximize the number of ordered vertex pairs (x,y) such that there is a directed path from x to y, and (ii) how to orient the edges so as to minimize the number of such pairs. The paper describes a quadratic-time algorithm for the first problem, and a proof that the second problem is NP-hard to approximate within some constant 1+epsilon > 1. The latter proof also shows that the second problem is equivalent to ``comparability graph completion''; neither problem was previously known to be NP-hard

Time Complexity of Decentralized Fixed-Mode Verification

Given an interconnected system, this note is concerned with the time complexity of verifying whether an unrepeated mode of the system is a decentralized fixed mode (DFM). It is shown that checking the decentralized fixedness of any distinct mode is tantamount to testing the strong connectivity of a digraph formed based on the system. It is subsequently proved that the time complexity of this decision problem using the proposed approach is the same as the complexity of matrix multiplication. This work concludes that the identification of distinct DFMs (by means of a deterministic algorithm, rather than a randomized one) is computationally very easy, although the existing algorithms for solving this problem would wrongly imply that it is cumbersome. This note provides not only a complexity analysis, but also an efficient algorithm for tackling the underlying problem

Symbolic Algorithms for Graphs and Markov Decision Processes with Fairness Objectives

Given a model and a specification, the fundamental model-checking problem asks for algorithmic verification of whether the model satisfies the specification. We consider graphs and Markov decision processes (MDPs), which are fundamental models for reactive systems. One of the very basic specifications that arise in verification of reactive systems is the strong fairness (aka Streett) objective. Given different types of requests and corresponding grants, the objective requires that for each type, if the request event happens infinitely often, then the corresponding grant event must also happen infinitely often. All $\omega$-regular objectives can be expressed as Streett objectives and hence they are canonical in verification. To handle the state-space explosion, symbolic algorithms are required that operate on a succinct implicit representation of the system rather than explicitly accessing the system. While explicit algorithms for graphs and MDPs with Streett objectives have been widely studied, there has been no improvement of the basic symbolic algorithms. The worst-case numbers of symbolic steps required for the basic symbolic algorithms are as follows: quadratic for graphs and cubic for MDPs. In this work we present the first sub-quadratic symbolic algorithm for graphs with Streett objectives, and our algorithm is sub-quadratic even for MDPs. Based on our algorithmic insights we present an implementation of the new symbolic approach and show that it improves the existing approach on several academic benchmark examples.Comment: Full version of the paper. To appear in CAV 201

Orientation-Constrained Rectangular Layouts

We construct partitions of rectangles into smaller rectangles from an input consisting of a planar dual graph of the layout together with restrictions on the orientations of edges and junctions of the layout. Such an orientation-constrained layout, if it exists, may be constructed in polynomial time, and all orientation-constrained layouts may be listed in polynomial time per layout.Comment: To appear at Algorithms and Data Structures Symposium, Banff, Canada, August 2009. 12 pages, 5 figure

Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts

A skew-symmetric graph $(D=(V,A),\sigma)$ is a directed graph $D$ with an involution $\sigma$ on the set of vertices and arcs. In this paper, we introduce a separation problem, $d$-Skew-Symmetric Multicut, where we are given a skew-symmetric graph $D$, a family of $\cal T$ of $d$-sized subsets of vertices and an integer $k$. The objective is to decide if there is a set $X\subseteq A$ of $k$ arcs such that every set $J$ in the family has a vertex $v$ such that $v$ and $\sigma(v)$ are in different connected components of $D'=(V,A\setminus (X\cup \sigma(X))$. In this paper, we give an algorithm for this problem which runs in time $O((4d)^{k}(m+n+\ell))$, where $m$ is the number of arcs in the graph, $n$ the number of vertices and $\ell$ the length of the family given in the input. Using our algorithm, we show that Almost 2-SAT has an algorithm with running time $O(4^kk^4\ell)$ and we obtain algorithms for {\sc Odd Cycle Transversal} and {\sc Edge Bipartization} which run in time $O(4^kk^4(m+n))$ and $O(4^kk^5(m+n))$ respectively. This resolves an open problem posed by Reed, Smith and Vetta [Operations Research Letters, 2003] and improves upon the earlier almost linear time algorithm of Kawarabayashi and Reed [SODA, 2010]. We also show that Deletion q-Horn Backdoor Set Detection is a special case of 3-Skew-Symmetric Multicut, giving us an algorithm for Deletion q-Horn Backdoor Set Detection which runs in time $O(12^kk^5\ell)$. This gives the first fixed-parameter tractable algorithm for this problem answering a question posed in a paper by a superset of the authors [STACS, 2013]. Using this result, we get an algorithm for Satisfiability which runs in time $O(12^kk^5\ell)$ where $k$ is the size of the smallest q-Horn deletion backdoor set, with $\ell$ being the length of the input formula