A vertex i of a graph G = (V,E) is said to be controlled by $M \subseteq V$ if the majority of the elements of the neighborhood of i (including itself) belong to M. The set M is a monopoly in G if every vertex $i\in V$ is controlled by M. Given a set $M \subseteq V$ and two graphs G1 = ($V,E_1$) and G2 = ($V,E_2$) where $E_1\subseteq E_2$, the monopoly verification problem (mvp) consists of deciding whether there exists a sandwich graph G = (V,E) (i.e., a graph where $E_1\subseteq E\subseteq E_2$) such that M is a monopoly in G = (V,E). If the answer to the mvp is No, we then consider the max-controlled set problem (mcsp), whose objective is to find a sandwich graph G = (V,E) such that the number of vertices of G controlled by M is maximized. The mvp can be solved in polynomial time; the mcsp, however, is NP-hard. In this work, we present a deterministic polynomial time approximation algorithm for the mcsp with ratio $\frac{1}{2}$ + $\frac{1+\sqrt{n}}{2n-2}$, where n=|V|>4. (The case $n\leq4$ is solved exactly by considering the parameterized version of the mcsp.) The algorithm is obtained through the use of randomized rounding and derandomization techniques based on the method of conditional expectations. Additionally, we show how to improve this ratio if good estimates of expectation are obtained in advance.

In this paper we study the parameterized complexity of approximating the parameterized counting problems contained in the class $\#W[P]$, the parameterized analogue of $\#P$. We prove a parameterized analogue of a famous theorem of Stockmeyer claiming that approximate counting belongs to the second level of the polynomial hierarchy.

Sets of integers form a monoid, where the product of two sets A and B is defined as the set containing a+b for all $a\in A$ and $b\in B$. We give a characterization of when a family of finite sets is a code in this monoid, that is when the sets do not satisfy any nontrivial relation. We also extend this result for some infinite sets, including all infinite rational sets.

It often occurs that local copies of a text are modified by users but that the local modifications are not synchronized (thus allowing the merged text to become the source for later editions) until later when, for instance the network connection is reestablished. Since text editions usually affect a small fraction of the whole content, the history of edit operations provides a compact representation of the modified file. In this paper, we define a normal form for these records which will permit for the comparison of all text files that have been obtained by editing a common source S when the difference between each output file Oi and the source file is given as a sequence Li of edit operations. We show that the normalized sequence is unique for all the equivalent text editions and provide efficient procedures with which to compute this normal form and to obtain the edit sequence LM transforming S into a merged file M which integrates all the local modifications. We also discuss how these normalization can be integrated into the operational transformation paradigm for optimistic replication.

Recently, a new measurement – the advice complexity – was introduced for measuring the information content of online problems. The aim is to measure the bitwise information that online algorithms lack, causing them to perform worse than offline algorithms. Among a large number of problems, a well-known scheduling problem, job shop scheduling with unit length tasks, and the paging problem were analyzed within this model. We observe some connections between advice complexity and randomization. Our special focus goes to barely random algorithms, i.e., randomized algorithms that use only a constant number of random bits, regardless of the input size. We adapt the results on advice complexity to obtain efficient barely random algorithms for both the job shop scheduling and the paging problem. Furthermore, so far, it has not yet been investigated for job shop scheduling how good an online algorithm may perform when only using a very small (e.g., constant) number of advice bits. In this paper, we answer this question by giving both lower and upper bounds, and also improve the best known upper bound for optimal algorithms.