1,424 research outputs found
Parameterized bounded-depth Frege is not optimal
A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [9]. There the authors concentrate on tree-like Parameterized Resolution-a parameterized version of classical Resolution-and their gap complexity theorem implies lower bounds for that system. The main result of the present paper significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size n in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in [9]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNF's
On the Complexity of -Closeness Anonymization and Related Problems
An important issue in releasing individual data is to protect the sensitive
information from being leaked and maliciously utilized. Famous privacy
preserving principles that aim to ensure both data privacy and data integrity,
such as -anonymity and -diversity, have been extensively studied both
theoretically and empirically. Nonetheless, these widely-adopted principles are
still insufficient to prevent attribute disclosure if the attacker has partial
knowledge about the overall sensitive data distribution. The -closeness
principle has been proposed to fix this, which also has the benefit of
supporting numerical sensitive attributes. However, in contrast to
-anonymity and -diversity, the theoretical aspect of -closeness has
not been well investigated.
We initiate the first systematic theoretical study on the -closeness
principle under the commonly-used attribute suppression model. We prove that
for every constant such that , it is NP-hard to find an optimal
-closeness generalization of a given table. The proof consists of several
reductions each of which works for different values of , which together
cover the full range. To complement this negative result, we also provide exact
and fixed-parameter algorithms. Finally, we answer some open questions
regarding the complexity of -anonymity and -diversity left in the
literature.Comment: An extended abstract to appear in DASFAA 201
On Covering Segments with Unit Intervals
We study the problem of covering a set of segments on a line with the minimum number of unit-length intervals, where an interval covers a segment if at least one of the two endpoints of the segment falls in the unit interval. We also study several variants of this problem.
We show that the restrictions of the aforementioned problems to the set of instances in which all the segments have the same length are NP-hard. This result implies several NP-hardness results in the literature for variants and generalizations of the problems under consideration.
We then study the parameterized complexity of the aforementioned problems. We provide tight results for most of them by showing that they are fixed-parameter tractable for the restrictions in which all the segments have the same length, and are W[1]-complete otherwise
Lower Bounds for the Graph Homomorphism Problem
The graph homomorphism problem (HOM) asks whether the vertices of a given
-vertex graph can be mapped to the vertices of a given -vertex graph
such that each edge of is mapped to an edge of . The problem
generalizes the graph coloring problem and at the same time can be viewed as a
special case of the -CSP problem. In this paper, we prove several lower
bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main
result is a lower bound .
This rules out the existence of a single-exponential algorithm and shows that
the trivial upper bound is almost asymptotically
tight.
We also investigate what properties of graphs and make it difficult
to solve HOM. An easy observation is that an upper
bound can be improved to where
is the minimum size of a vertex cover of . The second
lower bound shows that the upper bound is
asymptotically tight. As to the properties of the "right-hand side" graph ,
it is known that HOM can be solved in time and
where is the maximum degree of
and is the treewidth of . This gives
single-exponential algorithms for graphs of bounded maximum degree or bounded
treewidth. Since the chromatic number does not exceed
and , it is natural to ask whether similar
upper bounds with respect to can be obtained. We provide a negative
answer to this question by establishing a lower bound for any
function . We also observe that similar lower bounds can be obtained for
locally injective homomorphisms.Comment: 19 page
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