Rastislav Královič

Mathematical Foundations of Computer Science 2006

Invited Talks.- A Core Calculus for Scala Type Checking.- Tree Exploration with an Oracle.- Distributed Data Structures: A Survey on Informative Labeling Schemes.- From Deduction Graphs to Proof Nets: Boxes and Sharing in the Graphical Presentation of Deductions.- The Structure of Tractable Constraint Satisfaction Problems.- On the Representation of Kleene Algebras with Tests.- From Three Ideas in TCS to Three Applications in Bioinformatics.- Contributed Papers.- Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-triangles.- Approximate Shortest Path Queries on Weighted Polyhedral Surfaces.- A Unified Construction of the Glushkov, Follow, and Antimirov Automata.- Algebraic Characterizations of Unitary Linear Quantum Cellular Automata.- A Polynomial Time Nilpotence Test for Galois Groups and Related Results.- The Multiparty Communication Complexity of Exact-T: Improved Bounds and New Problems.- Crochemore Factorization of Sturmian and Other Infinite Words.- Equations on Partial Words.- Concrete Multiplicative Complexity of Symmetric Functions.- On the Complexity of Limit Sets of Cellular Automata Associated with Probability Measures.- Coloring Random 3-Colorable Graphs with Non-uniform Edge Probabilities.- The Kleene Equality for Graphs.- On the Repetition Threshold for Large Alphabets.- Improved Parameterized Upper Bounds for Vertex Cover.- On Comparing Sums of Square Roots of Small Integers.- A Combinatorial Approach to Collapsing Words.- Optimal Linear Arrangement of Interval Graphs.- The Lempel-Ziv Complexity of Fixed Points of Morphisms.- Partially Commutative Inverse Monoids.- Learning Bayesian Networks Does Not Have to Be NP-Hard.- Lower Bounds for the Transition Complexity of NFAs.- Smart Robot Teams Exploring Sparse Trees.- k-Sets of Convex Inclusion Chains of Planar Point Sets.- Toward the Eigenvalue Power Law.- Multicast Transmissions in Non-cooperative Networks with a Limited Number of Selfish Moves.- Very Sparse Leaf Languages.- On the Correlation Between Parity and Modular Polynomials.- Optimally Fast Data Gathering in Sensor Networks.- Magic Numbers in the State Hierarchy of Finite Automata.- Online Single Machine Batch Scheduling.- Machines that Can Output Empty Words.- Completeness of Global Evaluation Logic.- NOF-Multiparty Information Complexity Bounds for Pointer Jumping.- Dimension Characterizations of Complexity Classes.- Approximation Algorithms and Hardness Results for Labeled Connectivity Problems.- An Expressive Temporal Logic for Real Time.- On Matroid Representability and Minor Problems.- Non-cooperative Tree Creation.- Guarantees for the Success Frequency of an Algorithm for Finding Dodgson-Election Winners.- Reductions for Monotone Boolean Circuits.- Generalised Integer Programming Based on Logically Defined Relations.- Probabilistic Length-Reducing Automata.- Sorting Long Sequences in a Single Hop Radio Network.- Systems of Equations over Finite Semigroups and the #CSP Dichotomy Conjecture.- Valiants Model: From Exponential Sums to Exponential Products.- A Reachability Algorithm for General Petri Nets Based on Transition Invariants.- Approximability of Bounded Occurrence Max Ones.- Fast Iterative Arrays with Restricted Inter-cell Communication: Constructions and Decidability.- Faster Algorithm for Bisimulation Equivalence of Normed Context-Free Processes.- Quantum Weakly Nondeterministic Communication Complexity.- Minimal Chordal Sense of Direction and Circulant Graphs.- Querying and Embedding Compressed Texts.- Lempel-Ziv Dimension for Lempel-Ziv Compression.- Characterizing Valiants Algebraic Complexity Classes.- The Price of Defense.- The Data Complexity of MDatalog in Basic Modal Logics.- The Complexity of Counting Functions with Easy Decision Version.- On Non-Interactive Zero-Knowledge Proofs of Knowledge in the Shared Random String Model.- Constrained Minimum Enclosing Circle with Center on a Query Line Segment.- Hierarchical Unambiguity.- An Efficient Algorithm Finds Noticeable Trends and Examples Concerning the ?erny Conjecture.- On Genome Evolution with Innovation.

On the Advice Complexity of Online Problems

In this paper, we investigate to what extent the solution quality of online algorithms can be improved by allowing the algorithm to extract a given amount of information about the input. We consider the recently introduced notion of advice complexity where the algorithm, in addition to being fed the requests one by one, has access to a tape of advice bits that were computed by some oracle function from the complete input. The advice complexity is the number of advice bits read. We introduce an improved model of advice complexity and investigate the connections of advice complexity to the competitive ratio of both deterministic and randomized online algorithms using the paging problem, job shop scheduling, and the routing problem on a line as sample problems. We provide both upper and lower bounds on the advice complexity of all three problems. Our results for all of these problems show that very small advice (only three bits in the case of paging) already suffices to significantly improve over the best deterministic algorithm. Moreover, to achieve the same competitive ratio as any randomized online algorithm, a logarithmic number of advice bits is sufficient. On the other hand, to obtain optimality, much larger advice is necessary.

On time versus size for monotone dynamic monopolies in regular topologies

We consider a well-known distributed colouring game played on a simple connected graph: initially, each vertex is coloured black or white; at each round, each vertex simultaneously recolours itself by the colour of the simple (strong) majority of its neighbours. A set of vertices M is said to be a dynamo, if starting the game with only the vertices of M coloured black, the computation eventually reaches an all-black configuration.The importance of this game follows from the fact that it models the spread of faults in point-to-point systems with majority-based voting; in particular, dynamos correspond to those sets of initial failures which will lead the entire system to fail. Investigations on dynamos have been extensive but restricted to establishing tight bounds on the size (i.e., how small a dynamic monopoly might be).In this paper we start to study dynamos systematically with respect to both the size and the time (i.e., how many rounds are needed to reach all-black configuration) in various models and topologies.We derive tight tradeoffs between the size and the time for a number of regular graphs, including rings, complete d-ary trees, tori, wrapped butterflies, cube connected cycles and hypercubes. In addition, we determine optimal size bounds of irreversible dynamos for butterflies and shuffle-exchange using simple majority and for DeBruijn using strong majority rules. Finally, we make some observations concerning irreversible versus reversible monotone models and slow complete computations from minimal dynamos.

Information complexity of online problems

What is information? Frequently spoken about in many contexts, yet nobody has ever been able to define it with mathematical rigor. The best we are left with so far is the concept of entropy by Shannon, and the concept of information content of binary strings by Chaitin and Kolmogorov. While these are doubtlessly great research instruments, they are hardly helpful in measuring the amount of information contained in particular objects. In a pursuit to overcome these limitations, we propose the notion of information content of algorithmic problems. We discuss our approaches and their possible usefulness in understanding the basic concepts of informatics, namely the concept of algorithms and the concept of computational complexity.

Measuring the problem-relevant information in input

We propose a new way of characterizing the complexity of online problems. Instead of measuring the degradation of the output quality caused by the ignorance of the future we choose to quantify the amount of additional global information needed for an online algorithm to solve the problem optimally. In our model, the algorithm cooperates with an oracle that can see the whole input. We define the advice complexity of the problem to be the minimal number of bits (normalized per input request, and minimized over all algorithm-oracle pairs) communicated by the algorithm to the oracle in order to solve the problem optimally. Hence, the advice complexity measures the amount of problem-relevant information contained in the input. We introduce two modes of communication between the algorithm and the oracle based on whether the oracle offers an advice spontaneously (helper) or on request (answerer). We analyze the Paging and DiffServ problems in terms of advice complexity and deliver upper and lower bounds in both communication modes; in the case of DiffServ problem in helper mode the bounds are tight.

On the advice complexity of the k-server problem

Competitive analysis is the established tool for measuring the output quality of algorithms that work in an online environment. Recently, the model of advice complexity has been introduced as an alternative measurement which allows for a more fine-grained analysis of the hardness of online problems. In this model, one tries to measure the amount of information an online algorithm is lacking about the future parts of the input. This concept was investigated for a number of well-known online problems including the k-server problem. In this paper, we first extend the analysis of the k-server problem by giving both a lower bound on the advice needed to obtain an optimal solution, and upper bounds on algorithms for the general k-server problem on metric graphs and the special case of dealing with the Euclidean plane. In the general case, we improve the previously known results by an exponential factor, in the Euclidean case we design an algorithm which achieves a constant competitive ratio for a very small (i. e., constant) number of advice bits per request. Furthermore, we investigate the relation between advice complexity and randomized online computations by showing how lower bounds on the advice complexity can be used for proving lower bounds for the competitive ratio of randomized online algorithms.

How much information about the future is needed

We propose a new way of characterizing the complexity of online problems. Instead of measuring the degradation of output quality caused by the ignorance of the future we choose to quantify the amount of additional global information needed for an online algorithm to solve the problem optimally. In our model, the algorithm cooperates with an oracle that can see the whole input. We define the advice complexity of the problem to be the minimal number of bits (normalized per input request, and minimized over all algorithm-oracle pairs) communicated between the algorithm and the oracle in order to solve the problem optimally. Hence, the advice complexity measures the amount of problem-relevant information contained in the input. We introduce two modes of communication between the algorithm and the oracle based on whether the oracle offers an advice spontaneously (helper) or on request (answerer). We analyze the Paging and DiffServ problems in terms of advice complexity and deliver tight bounds in both communication modes.

Online graph exploration with advice

We study the problem of exploring an unknown undirected graph with non-negative edge weights. Starting at a distinguished initial vertex s, an agent must visit every vertex of the graph and return to s. Upon visiting a node, the agent learns all incident edges, their weights and endpoints. The goal is to find a tour with minimal cost of traversed edges. This variant of the exploration problem has been introduced by Kalyanasundaram and Pruhs in [18] and is known as a fixed graph scenario. There have been recent advances by Megow, Mehlhorn, and Schweitzer ([19]), however the main question whether there exists a deterministic algorithm with constant competitive ratio (w.r.t. to offline algorithm knowing the graph) working on all graphs and with arbitrary edge weights remains open. In this paper we study this problem in the context of advice complexity, investigating the tradeoff between the amount of advice available to the deterministic agent, and the quality of the solution. We show that Ω(n logn) bits of advice are necessary to achieve a competitive ratio of 1 (w.r.t. an optimal algorithm knowing the graph topology). Furthermore, we give a deterministic algorithm which uses O(n) bits of advice and achieves a constant competitive ratio on any graph with arbitrary weights. Finally, going back to the original problem, we prove a lower bound of 5/2−e for deterministic algorithms working with no advice, improving the best previous lower bound of 2−e of Miyazaki, Morimoto, and Okabe from [20]. In this case, significantly more elaborate technique was needed to achieve the result.

Exploring an Unknown Graph to Locate a Black Hole Using Tokens

Consider a team of (one or more) mobile agents operating in a graph G. Unaware of the graph topology and starting from the same node, the team must explore the graph. This problem, known as graph exploration, was initially formulated by Shannon in 1951, and has been extensively studied since under a variety of conditions. The existing investigations have all assumed that the network is safe for the agents, and the solutions presented in the literature succeed in their task only under this assumption.

Periodic data retrieval problem in rings containing a malicious host

In the problems of exploration of faulty graphs, a team of cooperating agents is considered moving in a network containing one or more nodes that can harm the agents. A most notable among these problems is the problem of black hole location, where the network contains one node that destroys any incoming agent, and the task of the agents is to determine the location of this node. The main complexity measure is the number of agents needed to solve the problem. In this paper we begin with a study of malicious hosts with more varied behavior. We study the problem of periodic data retrieval which is equivalent to periodic exploration in fault-free networks, and to black hole location in networks with one black hole. The main result of the paper states that, in case of rings, it is sufficient to protect the internal state of the agent (i.e. the malicious host cannot change or create the content of agents memory), and the periodic data retrieval problem is solvable by a constant number of agents.