134 Works

The Wadge Hierarchy of Max-Regular Languages

Jérémie Cabessa, Jacques Duparc, Alessandro Facchini & Filip Murlak
Recently, Miko{\l}aj Boja{\'n}czyk introduced a class of max-regular languages, an extension of regular languages of infinite words preserving manyof its usual properties. This new class can be seen as a different way of generalising the notion of regularity from finite to infinite words. This paper compares regular and max-regular languages in terms of topological complexity.It is proved that up to Wadge equivalence the classes coincide. Moreover, when restricted to $\mathbf{\Delta}^0_2$-languages, the classes contain virtually the...

Domination Problems in Nowhere-Dense Classes

Anuj Dawar & Stephan Kreutzer
We investigate the parameterized complexity of generalisations and variations of the dominating set problem on classes of graphs that are nowhere dense. In particular, we show that the distance-$d$ dominating-set problem, also known as the $(k,d)$-centres problem, is fixed-parameter tractable on any class that is nowhere dense and closed under induced subgraphs. This generalises known results about the dominating set problem on $H$-minor free classes, classes with locally excluded minors and classes of graphs of...

Verification and Refutation of Probabilistic Specifications via Games

Mark Kattenbelt & Michael Huth
We develop an abstraction-based framework to check probabilistic specifications of Markov Decision Processes (MDPs) using the stochastic two-player game abstractions (\ie ``games'') developed by Kwiatkowska et al.\ as a foundation. We define an abstraction preorder for these game abstractions which enables us to identify many new game abstractions for each MDP --- ranging from compact and imprecise to complex and precise. This added ability to trade precision for efficiency is crucial for scalable software model...

A Fine-grained Analysis of a Simple Independent Set Algorithm

Joachim Kneis, Alexander Langer & Peter Rossmanith
We present a simple exact algorithm for the \is\ problem with a runtime bounded by $O(\rt^n \poly(n))$. This bound is obtained by, firstly, applying a new branching rule and, secondly, by a distinct, computer-aided case analysis. The new branching rule uses the concept of satellites and has previously only been used in an algorithm for sparse graphs. The computer-aided case analysis allows us to capture the behavior of our algorithm in more detail than in...

Using Elimination Theory to construct Rigid Matrices

Abhinav Kumar, Satyanarayana V. Lokam, Vijay M. Patankar & Jayalal Sarma M. N.
The rigidity of a matrix $A$ for target rank $r$ is the minimum number of entries of $A$ that must be changed to ensure that the rank of the altered matrix is at most $r$. Since its introduction by Valiant \cite{Val77}, rigidity and similar rank-robustness functions of matrices have found numerous applications in circuit complexity, communication complexity, and learning complexity. Almost all $\nbyn$ matrices over an infinite field have a rigidity of $(n-r)^2$. It is...

On Nondeterministic Unranked Tree Automata with Sibling Constraints

Christof Löding & Karianto Wong
We continue the study of bottom-up unranked tree automata with equality and disequality constraints between direct subtrees. In particular, we show that the emptiness problem for the nondeterministic automata is decidable. In addition, we show that the universality problem, in contrast, is undecidable.

Pruning 2-Connected Graphs

Chandra Chekuri & Nitish Korula
Given an edge-weighted undirected graph $G$ with a specified set of terminals, let the \emph{density} of any subgraph be the ratio of its weight/cost to the number of terminals it contains. If $G$ is 2-connected, does it contain smaller 2-connected subgraphs of density comparable to that of $G$? We answer this question in the affirmative by giving an algorithm to \emph{prune} $G$ and find such subgraphs of any desired size, at the cost of only...

Single-Sink Network Design with Vertex Connectivity Requirements

Chandra Chekuri & Nitish Korula
We study single-sink network design problems in undirected graphs with vertex connectivity requirements. The input to these problems is an edge-weighted undirected graph $G=(V,E)$, a sink/root vertex $r$, a set of terminals $T \subseteq V$, and integer $k$. The goal is to connect each terminal $t \in T$ to $r$ via $k$ \emph{vertex-disjoint} paths. In the {\em connectivity} problem, the objective is to find a min-cost subgraph of $G$ that contains the desired paths. There...

3-connected Planar Graph Isomorphism is in Log-space

Samir Datta, Nutan Limaye & Prajakta Nimbhorkar
We consider the isomorphism and canonization problem for $3$-connected planar graphs. The problem was known to be \Log-hard and in \ULcoUL\ \cite{TW07}. In this paper, we give a deterministic log-space algorithm for $3$-connected planar graph isomorphism and canonization. This gives an \Log-completeness result, thereby settling its complexity. \par The algorithm uses the notion of universal exploration sequences from \cite{koucky01} and \cite{Rei05}. To our knowledge, this is a completely new approach to graph canonization.

An Optimal Construction of Finite Automata from Regular Expressions

Stefan Gulan & Henning Fernau
We consider the construction of finite automata from their corresponding regular expressions by a series of digraph-transformations along the expression\'s structure. Each intermediate graph represents an extended finite automaton accepting the same language. The character of our construction allows a fine-grained analysis of the emerging automaton\'s size, eventually leading to an optimality result.

Explicit Muller Games are PTIME

Florian Horn
Regular games provide a very useful model for the synthesis of controllers in reactive systems. The complexity of these games depends on the representation of the winning condition: if it is represented through a win-set, a coloured condition, a Zielonka-DAG or Emerson-Lei formulae, the winner problem is \pspace-complete; if the winning condition is represented as a Zielonka tree, the winner problem belongs to \np and \conp. In this paper, we show that explicit Muller games...

Complexity Analysis of Term Rewriting Based on Matrix and Context Dependent Interpretations

Georg Moser, Andreas Schnabl & Johannes Waldmann
For a given (terminating) term rewriting system one can often estimate its \emph{derivational complexity} indirectly by looking at the proof method that established termination. In this spirit we investigate two instances of the interpretation method: \emph{matrix interpretations} and \emph{context dependent interpretations}. We introduce a subclass of matrix interpretations, denoted as \emph{triangular matrix interpretations}, which induce polynomial derivational complexity and establish tight correspondence results between a subclass of context dependent interpretations and restricted triangular matrix interpretations....

A Hierarchy of Semantics for Non-deterministic Term Rewriting Systems

Juan Rodriguez-Hortala
Formalisms involving some degree of nondeterminism are frequent in computer science. In particular, various programming or specification languages are based on term rewriting systems where confluence is not required. In this paper we examine three concrete possible semantics for non-determinism that can be assigned to those programs. Two of them --call-time choice and run-time choice-- are quite well-known, while the third one --plural semantics-- is investigated for the first time in the context of term...

Quantum Query Complexity of Multilinear Identity Testing

Vikraman Arvind & Partha Mukhopadhyay
Motivated by the quantum algorithm for testing commutativity of black-box groups (Magniez and Nayak, 2007), we study the following problem: Given a black-box finite ring by an additive generating set and a multilinear polynomial over that ring, also accessed as a black-box function (we allow the indeterminates of the polynomial to be commuting or noncommuting), we study the problem of testing if the polynomial is an \emph{identity} for the given ring. We give a quantum...

Kolmogorov Complexity and Solovay Functions

Laurent Bienvenu & Rod Downey
Solovay (1975) proved that there exists a computable upper bound~$f$ of the prefix-free Kolmogorov complexity function~$K$ such that $f(x)=K(x)$ for infinitely many~$x$. In this paper, we consider the class of computable functions~$f$ such that $K(x) \leq f(x)+O(1)$ for all~$x$ and $f(x) \leq K(x)+O(1)$ for infinitely many~$x$, which we call Solovay functions. We show that Solovay functions present interesting connections with randomness notions such as Martin-L\"of randomness and K-triviality.

Nonclairvoyant Speed Scaling for Flow and Energy

Ho-Leung Chan, Jeff Edmonds, Tak-Wah Lam, Lap-Kei Lee, Alberto Marchetti-Spaccamela & Kirk Pruhs
We study online nonclairvoyant speed scaling to minimize total flow time plus energy. We first consider the traditional model where the power function is $P(s)=s^\alpha$. We give a nonclairvoyant algorithm that is shown to be $O(\alpha^3)$-competitive. We then show an $\Omega( \alpha^{1/3-\epsilon} )$ lower bound on the competitive ratio of any nonclairvoyant algorithm. We also show that there are power functions for which no nonclairvoyant algorithm can be $O(1)$-competitive.

Fragments of First-Order Logic over Infinite Words

Volker Diekert & Manfred Kufleitner
We give topological and algebraic characterizations as well as language theoretic descriptions of the following subclasses of first-order logic $\mathrm{FO}[

On the Average Complexity of Moore's State Minimization Algorithm

Frederique Bassino, Julien David & Cyril Nicaud
We prove that, for any arbitrary finite alphabet and for the uniform distribution over deterministic and accessible automata with $n$ states, the average complexity of Moore's state minimization algorithm is in $\mathcal{O}(n \log n)$. Moreover this bound is tight in the case of unary automata.

Testing Linear-Invariant Non-Linear Properties

Arnab Bhattacharyya, Victor Chen, Madhu Sudan & Ning Xie
We consider the task of testing properties of Boolean functions that are invariant under linear transformations of the Boolean cube. Previous work in property testing, including the linearity test and the test for Reed-Muller codes, has mostly focused on such tasks for linear properties. The one exception is a test due to Green for {}``triangle freeness'': A function $f:\mathbb{F}_{2}^{n}\to\mathbb{F}_{2}$ satisfies this property if $f(x),f(y),f(x+y)$ do not all equal $1$, for any pair $x,y\in\mathbb{F}_{2}^{n}$. Here we...

Computing Graph Roots Without Short Cycles

Babak Farzad, Lap Chi Lau, Van Bang Le & Nguyen Ngoc Tuy
Graph $G$ is the square of graph $H$ if two vertices $x,y$ have an edge in $G$ if and only if $x,y$ are of distance at most two in $H$. Given $H$ it is easy to compute its square $H^2$, however Motwani and Sudan proved that it is NP-complete to determine if a given graph $G$ is the square of some graph $H$ (of girth $3$). In this paper we consider the characterization and recognition...

Semi-Online Preemptive Scheduling: One Algorithm for All Variants

Tomas Ebenlendr & Jiri Sgall
We present a unified optimal semi-online algorithm for preemptive scheduling on uniformly related machines with the objective to minimize the makespan. This algorithm works for all types of semi-online restrictions, including the ones studied before, like sorted (decreasing) jobs, known sum of processing times, known maximal processing time, their combinations, and so on. Based on the analysis of this algorithm, we derive some global relations between various semi-online restrictions and tight bounds on the approximation...

Forward Analysis for WSTS, Part I: Completions

Alain Finkel & Jean Goubault-Larrecq
Well-structured transition systems provide the right foundation to compute a finite basis of the set of predecessors of the upward closure of a state. The dual problem, to compute a finite representation of the set of successors of the downward closure of a state, is harder: Until now, the theoretical framework for manipulating downward-closed sets was missing. We answer this problem, using insights from domain theory (dcpos and ideal completions), from topology (sobrifications), and shed...

Optimal Cache-Aware Suffix Selection

Gianni Franceschini, Roberto Grossi & S. Muthukrishnan
Given string $S[1..N]$ and integer $k$, the {\em suffix selection} problem is to determine the $k$th lexicographically smallest amongst the suffixes $S[i\ldots N]$, $1 \leq i \leq N$. We study the suffix selection problem in the cache-aware model that captures two-level memory inherent in computing systems, for a \emph{cache} of limited size $M$ and block size $B$. The complexity of interest is the number of block transfers. We present an optimal suffix selection algorithm in...

Random Fruits on the Zielonka Tree

Florian Horn
Stochastic games are a natural model for the synthesis of controllers confronted to adversarial and/or random actions. In particular, $\omega$-regular games of infinite length can represent reactive systems which are not expected to reach a correct state, but rather to handle a continuous stream of events. One critical resource in such applications is the memory used by the controller. In this paper, we study the amount of memory that can be saved through the use...

On the Borel Inseparability of Game Tree Languages

Szczepan Hummel, Henryk Michalewski & Damian Niwinski
The game tree languages can be viewed as an automata-theoretic counterpart of parity games on graphs. They witness the strictness of the index hierarchy of alternating tree automata, as well as the fixed-point hierarchy over binary trees. We consider a game tree language of the first non-trivial level, where Eve can force that 0 repeats from some moment on, and its dual, where Adam can force that 1 repeats from some moment on. Both these...

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