Approximating the MaxCover Problem with Bounded Frequencies in FPT Time

We study approximation algorithms for several variants of the MaxCover problem, with the focus on algorithms that run in FPT time. In the MaxCover problem we are given a set N of elements, a family S of subsets of N, and an integer K. The goal is to find up to K sets from S that jointly cover (i.e., include) as many elements as possible. This problem is well-known to be NP-hard and, under standard complexity-theoretic assumptions, the best possible polynomial-time approximation algorithm has approximation ratio (1 - 1/e). We first consider a variant of MaxCover with bounded element frequencies, i.e., a variant where there is a constant p such that each element belongs to at most p sets in S. For this case we show that there is an FPT approximation scheme (i.e., for each B there is a B-approximation algorithm running in FPT time) for the problem of maximizing the number of covered elements, and a randomized FPT approximation scheme for the problem of minimizing the number of elements left uncovered (we take K to be the parameter). Then, for the case where there is a constant p such that each element belongs to at least p sets from S, we show that the standard greedy approximation algorithm achieves approximation ratio exactly (1-e^{-max(pK/|S|, 1)}). We conclude by considering an unrestricted variant of MaxCover, and show approximation algorithms that run in exponential time and combine an exact algorithm with a greedy approximation. Some of our results improve currently known results for MaxVertexCover

Iterative Voting, Control and Sentiment Analysis

In multi-agent systems agents often need to take a collective decision based on the preferences of individuals. A voting rule is used to decide which decision to take, mapping the agents' preferences over the possible candidate decisions into a winning decision for the collection of agents. In these kind of scenarios acting strategically can be seen in two opposite way. On one hand it may be desirable that agents do not have any incentive to act strategically. That is, to misreport their preferences in order to influence the result of the voting rule in their favor or acting on the structure of the election to change the outcome. On the other hand manipulation can be used to improve the quality of the outcome by enlarging the consensus of the winner. These two different scenarios are studied in this thesis. The first one by modeling and describing a natural form of control named ``replacement control'' and characterizing for several voting rules its computational complexity. The second scenario is studied in the form of iterative voting frameworks where individuals are allowed to change their preferences to change the outcome of the election. Computational social choice techniques can be used in very different scenarios. This work reports a first attempt to introduce the use of voting procedures in the field of sentiment analysis. In this area computer scientists extract the opinion of the community about a specific item. This opinion is extracted aggregating the opinion expressed by each individual which leaves a text in a blog or social network about the given item. We studied and proposed a new aggregation method which can improve performances of sentiment analysis, this new technique is a new variance of a well-known voting rule called Borda