802 research outputs found
Provenance Circuits for Trees and Treelike Instances (Extended Version)
Query evaluation in monadic second-order logic (MSO) is tractable on trees
and treelike instances, even though it is hard for arbitrary instances. This
tractability result has been extended to several tasks related to query
evaluation, such as counting query results [3] or performing query evaluation
on probabilistic trees [10]. These are two examples of the more general problem
of computing augmented query output, that is referred to as provenance. This
article presents a provenance framework for trees and treelike instances, by
describing a linear-time construction of a circuit provenance representation
for MSO queries. We show how this provenance can be connected to the usual
definitions of semiring provenance on relational instances [20], even though we
compute it in an unusual way, using tree automata; we do so via intrinsic
definitions of provenance for general semirings, independent of the operational
details of query evaluation. We show applications of this provenance to capture
existing counting and probabilistic results on trees and treelike instances,
and give novel consequences for probability evaluation.Comment: 48 pages. Presented at ICALP'1
The Bane of Low-Dimensionality Clustering
In this paper, we give a conditional lower bound of on
running time for the classic k-median and k-means clustering objectives (where
n is the size of the input), even in low-dimensional Euclidean space of
dimension four, assuming the Exponential Time Hypothesis (ETH). We also
consider k-median (and k-means) with penalties where each point need not be
assigned to a center, in which case it must pay a penalty, and extend our lower
bound to at least three-dimensional Euclidean space.
This stands in stark contrast to many other geometric problems such as the
traveling salesman problem, or computing an independent set of unit spheres.
While these problems benefit from the so-called (limited) blessing of
dimensionality, as they can be solved in time or
in d dimensions, our work shows that widely-used clustering
objectives have a lower bound of , even in dimension four.
We complete the picture by considering the two-dimensional case: we show that
there is no algorithm that solves the penalized version in time less than
, and provide a matching upper bound of .
The main tool we use to establish these lower bounds is the placement of
points on the moment curve, which takes its inspiration from constructions of
point sets yielding Delaunay complexes of high complexity
Fractional covers of hypergraphs with bounded multi-intersection
Fractional (hyper-)graph theory is concerned with the specific problems that arise when fractional analogues of otherwise integer-valued (hyper-)graph invariants are considered. The focus of this paper is on fractional edge covers of hypergraphs. Our main technical result generalizes and unifies previous conditions under which the size of the support of fractional edge covers is bounded independently of the size of the hypergraph itself. We show how this combinatorial result can be used to extend previous tractability results for checking if the fractional hypertree width of a given hypergraph is ≤k for some constant k. Moreover, we show a dual version of our main result for fractional hitting sets
Multivariate Analyis of Swap Bribery
We consider the computational complexity of a problem modeling bribery in the
context of voting systems. In the scenario of Swap Bribery, each voter assigns
a certain price for swapping the positions of two consecutive candidates in his
preference ranking. The question is whether it is possible, without exceeding a
given budget, to bribe the voters in a way that the preferred candidate wins in
the election. We initiate a parameterized and multivariate complexity analysis
of Swap Bribery, focusing on the case of k-approval. We investigate how
different cost functions affect the computational complexity of the problem. We
identify a special case of k-approval for which the problem can be solved in
polynomial time, whereas we prove NP-hardness for a slightly more general
scenario. We obtain fixed-parameter tractability as well as W[1]-hardness
results for certain natural parameters.Comment: 20 pages. Conference version published at IPEC 201
The Mean-Median Map
PhD ThesisThe mean-median map enlarges a nite (multi)set of real numbers by adjoining to it a new number such that the mean of the enlarged set is equal to the median of the original set. An open conjecture states that, starting with any nite set, the sequence of the new numbers generated by iterating this map stabilises, i.e., is eventually constant. We approach this problem from a new perspective, that of a dynamical system over the space of nite sets of piecewise-a ne continuous functions with rational coe cients, de ning the map pointwise. We develop a theory for the dynamics in the neighbourhoods of the local minima of the limit function |the limit of the generated sequence| establishing its local shapes and symmetries. We also show that the conjecture can be veri ed in exponentially many neighbourhoods simultaneously by computing a single dyadic rational orbit of a variant of the map. We then study a common pre-stabilisation behaviour of rational orbits, and construct a family of initial sets for which stabilisation can be delayed arbitrarily. Finally, using our theory, we extend the existing computational results by over two orders of magnitude. The results reveal that the total measure of the regular neighbourhoods is far from full, suggesting the existence of a region with a new, presently unexplained, dynamical behaviour
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