### Approximating Fault-Tolerant Group-Steiner Problems

Rohit Khandekar, Guy Kortsarz & Zeev Nutov
In this paper, we initiate the study of designing approximation algorithms for {\sf Fault-Tolerant Group-Steiner} ({\sf FTGS}) problems. The motivation is to protect the well-studied group-Steiner networks from edge or vertex failures. In {\sf Fault-Tolerant Group-Steiner} problems, we are given a graph with edge- (or vertex-) costs, a root vertex, and a collection of subsets of vertices called groups. The objective is to find a minimum-cost subgraph that has two edge- (or vertex-) disjoint paths...

### Sound Lemma Generation for Proving Inductive Validity of Equations

Takahito Aoto
In many automated methods for proving inductive theorems, finding a suitable generalization of a conjecture is a key for the success of proof attempts. On the other hand, an obtained generalized conjecture may not be a theorem, and in this case hopeless proof attempts for the incorrect conjecture are made, which is against the success and efficiency of theorem proving. Urso and Kounalis (2004) proposed a generalization method for proving inductive validity of equations, called...

### 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...

### Non-Local Box Complexity and Secure Function Evaluation

Marc Kaplan, Iordanis Kerenidis, Sophie Laplante & Jérémie Roland
A non-local box is an abstract device into which Alice and Bob input bits $x$ and $y$ respectively and receive outputs $a$ and $b$ respectively, where $a,b$ are uniformly distributed and $a \oplus b = x \wedge y$. Such boxes have been central to the study of quantum or generalized non-locality as well as the simulation of non-signaling distributions. In this paper, we start by studying how many non-local boxes Alice and Bob need in...

### Randomness on Computable Probability Spaces - A Dynamical Point of View

Peter Gacs, Mathieu Hoyrup & Cristobal Rojas
We extend the notion of randomness (in the version introduced by Schnorr) to computable Probability Spaces and compare it to a \emph{dynamical} notion of randomness: typicality. Roughly, a point is \emph{typical} for some dynamic, if it follows the statistical behavior of the system (Birkhoff's pointwise ergodic theorem). We prove that a point is Schnorr random if and only if it is typical for every \emph{mixing} computable dynamics. To prove the result we develop some tools...

### The Complexity of Tree Transducer Output Languages

Kazuhiro Inaba & Sebastian Maneth
Two complexity results are shown for the output languages generated by compositions of macro tree transducers. They are in $\NSPACE(n)$ and hence are context-sensitive, and the class is NP-complete.

### Generating Shorter Bases for Hard Random Lattices

Joel Alwen & Chris Peikert
We revisit the problem of generating a hard'' random lattice together with a basis of relatively short vectors. This problem has gained in importance lately due to new cryptographic schemes that use such a procedure for generating public/secret key pairs. In these applications, a shorter basis directly corresponds to milder underlying complexity assumptions and smaller key sizes. The contributions of this work are twofold. First, using the \emph{Hermite normal form} as an organizing principle, we...

### Improved Approximations for Guarding 1.5-Dimensional Terrains

Khaled Elbassioni, Erik Krohn, Domagoj Matijevic, Julian Mestre & Domagoj Severdija
We present a 4-approximation algorithm for the problem of placing the fewest guards on a 1.5D terrain so that every point of the terrain is seen by at least one guard. This improves on the currently best approximation factor of 5 (J. King, 2006). Unlike most of the previous techniques, our method is based on rounding the linear programming relaxation of the corresponding covering problem. Besides the simplicity of the analysis, which mainly relies on...

### Recurrence and Transience for Probabilistic Automata

Mathieu Tracol, Christel Baier & Marcus Grösser
In a context of $\omega$-regular specifications for infinite execution sequences, the classical B\"uchi condition, or repeated liveness condition, asks that an accepting state is visited infinitely often. In this paper, we show that in a probabilistic context it is relevant to strengthen this infinitely often condition. An execution path is now accepting if the \emph{proportion} of time spent on an accepting state does not go to zero as the length of the path goes to...

### Economical Caching

Matthias Englert, Heiko Röglin, Jacob Spönemann & Berthold Vöcking
We study the management of buffers and storages in environments with unpredictably varying prices in a competitive analysis. In the economical caching problem, there is a storage with a certain capacity. For each time step, an online algorithm is given a price from the interval $[1,\alpha]$, a consumption, and possibly a buying limit. The online algorithm has to decide the amount to purchase from some commodity, knowing the parameter $\alpha$ but without knowing how the...

### Synthesis of Finite-state and Definable Winning Strategies

Alexander Rabinovich
Church's Problem asks for the construction of a procedure which, given a logical specification $\varphi$ on sequence pairs, realizes for any input sequence $I$ an output sequence $O$ such that $(I,O)$ satisfies $\varphi$. McNaughton reduced Church's Problem to a problem about two-player$\omega$-games. B\"uchi and Landweber gave a solution for Monadic Second-Order Logic of Order ($\MLO$) specifications in terms of finite-state strategies. We consider two natural generalizations of the Church problem to countable ordinals: the first...

### Kolmogorov Complexity in Randomness Extraction

John M. Hitchcock, Aduri Pavan & N. V. Vinodchandran
We clarify the role of Kolmogorov complexity in the area of randomness extraction. We show that a computable function is an almost randomness extractor if and only if it is a Kolmogorov complexity extractor, thus establishing a fundamental equivalence between two forms of extraction studied in the literature: Kolmogorov extraction and randomness extraction. We present a distribution ${\cal M}_k$ based on Kolmogorov complexity that is complete for randomness extraction in the sense that a computable...

### An Order on Sets of Tilings Corresponding to an Order on Languages

Nathalie Aubrun & Mathieu Sablik
Traditionally a tiling is defined with a finite number of finite forbidden patterns. We can generalize this notion considering any set of patterns. Generalized tilings defined in this way can be studied with a dynamical point of view, leading to the notion of subshift. In this article we establish a correspondence between an order on subshifts based on dynamical transformations on them and an order on languages of forbidden patterns based on computability properties.

### 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...

### A Comparison of Techniques for Sampling Web Pages

Eda Baykan, Monika Henzinger, Stefan F. Keller, Sebastian De Castelberg & Markus Kinzler
As the World Wide Web is growing rapidly, it is getting increasingly challenging to gather representative information about it. Instead of crawling the web exhaustively one has to resort to other techniques like sampling to determine the properties of the web. A uniform random sample of the web would be useful to determine the percentage of web pages in a specific language, on a topic or in a top level domain. Unfortunately, no approach has...

### 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...

### 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...

### Harnessing the Multicores: Nested Data Parallelism in Haskell

Simon Peyton Jones, Roman Leshchinskiy, Gabriele Keller & Manuel M T Chakravarty
If you want to program a parallel computer, a purely functional language like Haskell is a promising starting point. Since the language is pure, it is by-default safe for parallel evaluation, whereas imperative languages are by-default unsafe. But that doesn\'t make it easy! Indeed it has proved quite difficult to get robust, scalable performance increases through parallel functional programming, especially as the number of processors increases. A particularly promising and well-studied approach to employing large...

### Kernel(s) for Problems with No Kernel: On Out-Trees with Many Leaves

Henning Fernau, Fedor V. Fomin, Daniel Lokshtanov, Daniel Raible, Saket Saurabh & Yngve Villanger
The {\sc $k$-Leaf Out-Branching} problem is to find an out-branching, that is a rooted oriented spanning tree, with at least $k$ leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the {\sc $k$-Leaf-Out-Branching} problem. We give the first polynomial kernel for {\sc Rooted $k$-Leaf-Out-Branching}, a variant of {\sc $k$-Leaf-Out-Branching} where the root of the tree searched for is...

### Deterministic Automata and Extensions of Weak MSO

Mikolaj Bojanczyk & Szymon Torunczyk
We introduce a new class of automata on infinite words, called min-automata. We prove that min-automata have the same expressive power as weak monadic second-order logic (weak MSO) extended with a new quantifier, the recurrence quantifier. These results are dual to a framework presented in \cite{max-automata}, where max-automata were proved equivalent to weak MSO extended with an unbounding quantifier. We also present a general framework, which tries to explain which types of automata on infinite...

### 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.

### Qualitative Reachability in Stochastic BPA Games

Tomas Brazdil, Vaclav Brozek, Antonin Kucera & Jan Obdrzalek
We consider a class of infinite-state stochastic games generated by stateless pushdown automata (or, equivalently, 1-exit recursive state machines), where the winning objective is specified by a regular set of target configurations and a qualitative probability constraint ${>}0$' or ${=}1$'. The goal of one player is to maximize the probability of reaching the target set so that the constraint is satisfied, while the other player aims at the opposite. We show that the winner in...

### Deductive Verification of Continuous Dynamical Systems

Ankur Taly & Ashish Tiwari
We define the notion of inductive invariants for continuous dynamical systems and use it to present inference rules for safety verification of polynomial continuous dynamical systems. We present two different sound and complete inference rules, but neither of these rules can be effectively applied. We then present several simpler and practical inference rules that are sound and relatively complete for different classes of inductive invariants. The simpler inference rules can be effectively checked when all...

### Randomness extractors -- applications and constructions

Avi Wigderson
Randomness extractors are efficient algorithms which convert weak random sources into nearly perfect ones. While such purification of randomness was the original motivation for constructing extractors, these constructions turn out to have strong pseudorandom properties which found applications in diverse areas of computer science and combinatorics. We will highlight some of the applications, as well as recent constructions achieving near-optimal extraction.

### The Power of Depth 2 Circuits over Algebras

Chandan Saha, Ramprasad Saptharishi & Nitin Saxena
We study the problem of polynomial identity testing (PIT) for depth $2$ arithmetic circuits over matrix algebra. We show that identity testing of depth $3$ ($\Sigma \Pi \Sigma$) arithmetic circuits over a field $\F$ is polynomial time equivalent to identity testing of depth $2$ ($\Pi \Sigma$) arithmetic circuits over $\mathsf{U}_2(\mathbb{F})$, the algebra of upper-triangular $2\times 2$ matrices with entries from $\F$. Such a connection is a bit surprising since we also show that, as computational...

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