Monika Steinová

Advice complexity of online coloring for paths

In online graph coloring a graph is revealed to an online algorithm one vertex at a time, and the algorithm must color the vertices as they appear. This paper starts to investigate the advice complexity of this problem --- the amount of oracle information an online algorithm needs in order to make optimal choices. We also consider a more general problem --- a trade-off between online and offline graph coloring. In the paper we prove that precisely ⌈n/2 ⌉−1 bits of advice are needed when the vertices on a path are presented for coloring in arbitrary order. The same holds in the more general case when just a subset of the vertices is colored online. However, the problem turns out to be non-trivial for the case where the online algorithm is guaranteed that the vertices it receives form a subset of a path and are presented in the order in which they lie on the path. For this variant we prove that its advice complexity is βn+O(logn) bits, where β≈0.406 is a fixed constant (we give its closed form). This suggests that the generalized problem will be challenging for more complex graph classes.

Metaphors and analogies for teaching algorithms

In this paper we explore the topic of using metaphors and analogies in teaching algorithms. We argue their importance in the teaching process. We present a selection of metaphors we successfully used when teaching algorithms to secondary school students. We also discuss the suitability of several commonly used metaphors, and in several cases we show significant weaknesses of these metaphors.

On the Clique Editing Problem

We study the hardness and approximability of the problem CliqueEditing, where the goal is to edit a given graph G into a graph consisting of a clique and a set of isolated vertices while using a minimum number of editing operations. The problem is interesting from both practical and theoretical points of view, and it belongs to the well-studied family of graph modification problems. We prove that the problem is NP-complete and construct a 3.524-approximation algorithm. Furthermore, we prove an existence of a PTAS for the still NP-complete version of the problem restricted to bipartite graphs, and the existence of a polynomial-time algorithm for the problem restricted to planar graphs.

On the approximability and hardness of minimum topic connected overlay and its special instances

In the context of designing a scalable overlay network to support decentralized topic-based pub/sub communication, the Minimum Topic-Connected Overlay problem (Min-TCO in short) has been investigated: given a set of t topics and a collection of n users together with the lists of topics they are interested in, the aim is to connect these users to a network by a minimum number of edges such that every graph induced by users interested in a common topic is connected. It is known that Min-TCO is NP-hard and approximable within O(logt) in polynomial time. In this paper, we further investigate the problem and some of its special instances. We give various hardness results for instances where the number of topics in which a user is interested in is bounded by a constant, and also for the instances where the number of users interested in a common topic is a constant. For the latter case, we present a first constant approximation algorithm. We also present some polynomial-time algorithms for very restricted instances of Min-TCO.

On the Power of Local Orientations

We consider a network represented by a simple connected undirected graph with Nanonymous nodes that have local orientations, i.e. incident edges of each vertex have locally-unique labels --- port names. We define a pre-processing phase that enables a right-hand rule using agent (RH-agent) to traverse the entire graph. For this phase we design an algorithm for an agent that performs the precomputation. The agent will alter the network by modifying the local orientations using a simple operation of exchanging two local labels in each step. We show a polynomial-time algorithm for this precomputation that needs only one pebble and O(logN) memory in the agent. Furthermore we design a similar algorithm where the memory that the agent uses for the precomputation is decreased to O(1). In this case, the agent is not able to perform some operations by itself due to the lack of memory and needs support from the environment.

On the hardness and approximation of minimum topic-connected overlay

The design of a scalable overlay network to support decentralized topic-based publish/subscribe communication is nowadays a problem of a great importance. We investigate here special instances of one such design problem called Minimum Topic-Connected Overlay. Given a collection of users together with the lists of topics they are interested in, the aim is to connect these users to a network by a minimum number of edges such that every graph induced by users interested in a common topic is connected. We investigate instances where in addition the number of users interested in a particular topic is bounded by a constant d > 2. It is known that the general Topic-Connected Overlay is Ω(log n) hard to approximate and approximable by a logarithmic factor. For our special instances, we design a one-to-one reduction to special instances of the hitting set problem. This allows us to present the first constant approximation algorithm, the first kernelization and the first nontrivial exact algorithm for the special instances discussed.

Improved Approximations for Ordered TSP on Near-Metric Graphs,

The traveling salesman problem with precedence constraints is one of the most important problems in operations research. Here, we consider the well-known variant where a linear order on k special vertices is given that has to be preserved in any feasible Hamiltonian cycle. This problem is called Ordered TSP and we consider it on input instances where the edge-cost function satisfies a β-relaxed triangle inequality, i.e., where the length of a direct edge cannot exceed the cost of any detour via a third vertex by more than a factor of β > 1.

On the approximability of minimum topic connected overlay and its special instances

In the context of designing a scalable overlay network to support decentralized topic-based pub/sub communication, the Minimum Topic-Connected Overlay problem (Min-TCO in short) has been investigated: Given a set of t topics, collection of n users together with the lists of topics they are interested in, the aim is to connect these users to a network by a minimum number of edges such that every graph induced by users interested in a common topic is connected. It is known that Min-TCO is NP-hard and approximable within O(log t) in polynomial time. In this paper, we further investigate the problem and some of its special instances. We give various hardness results for instances where the number of users interested in a common topic is constant, and also for the instances where the number of topics in which an user is interested in is bounded by a constant. Furthermore, we close the gap of hardness of Min-TCO by showing its LOGAPX-completeness. We also present a few polynomial-time algorithms for very restricted instances of Min-TCO.

Didactic Games for Teaching Information Theory

We developed a set of didactic games and activities that can be used to illustrate and teach various concepts from Information Theory. For each of the games and activities we list the topics it covers, give its rules and related information, describe our practical experiences and give an overview of its scientific background. We also discuss the proper ways to integrate these games into the knowledge acquisition process.

Programming Camps: Letting Children Discover the Computer Science

Daily and summer camps for children are certainly well known to every adult. They provide many activities that encourage interests, create opportunities to start new friendships and, last but not least, help parents with the child care when they are unable to take holidays. However, camps designed as a support for contests in the form as we are presenting them here are unique, established in former Czechoslovakia by Vit and Milan Hejný in the late 70s. In this paper, we provide a basic outline of the structure of these programming camps for high school students, we describe the daily program, and introduce the games and activities used to attract and educate participants. Our hope is to provide a comprehensive overview that can serve as a guideline for organizing similar camps in other countries, and in this way to draw the attention of the talented students, to help them to prepare for the contests like the International Olympiad in Informatics (IOI) and to encourage them to choose a career in computing.