184 research outputs found

### 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

### Fully Proportional Representation as Resource Allocation: Approximability Results

We model Monroe's and Chamberlin and Courant's multiwinner voting systems as
a certain resource allocation problem. We show that for many restricted
variants of this problem, under standard complexity-theoretic assumptions,
there are no constant-factor approximation algorithms. Yet, we also show cases
where good approximation algorithms exist (briefly put, these variants
correspond to optimizing total voter satisfaction under Borda scores, within
Monroe's and Chamberlin and Courant's voting systems).Comment: 26 pages, 1 figur

### Finding a Collective Set of Items: From Proportional Multirepresentation to Group Recommendation

We consider the following problem: There is a set of items (e.g., movies) and
a group of agents (e.g., passengers on a plane); each agent has some intrinsic
utility for each of the items. Our goal is to pick a set of $K$ items that
maximize the total derived utility of all the agents (i.e., in our example we
are to pick $K$ movies that we put on the plane's entertainment system).
However, the actual utility that an agent derives from a given item is only a
fraction of its intrinsic one, and this fraction depends on how the agent ranks
the item among the chosen, available, ones. We provide a formal specification
of the model and provide concrete examples and settings where it is applicable.
We show that the problem is hard in general, but we show a number of
tractability results for its natural special cases

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