9,941 research outputs found
Trace Reconstruction: Generalized and Parameterized
In the beautifully simple-to-state problem of trace reconstruction, the goal is to reconstruct an unknown binary string x given random "traces" of x where each trace is generated by deleting each coordinate of x independently with probability p<1. The problem is well studied both when the unknown string is arbitrary and when it is chosen uniformly at random. For both settings, there is still an exponential gap between upper and lower sample complexity bounds and our understanding of the problem is still surprisingly limited. In this paper, we consider natural parameterizations and generalizations of this problem in an effort to attain a deeper and more comprehensive understanding. Perhaps our most surprising results are:
1) We prove that exp(O(n^(1/4) sqrt{log n})) traces suffice for reconstructing arbitrary matrices. In the matrix version of the problem, each row and column of an unknown sqrt{n} x sqrt{n} matrix is deleted independently with probability p. Our results contrasts with the best known results for sequence reconstruction where the best known upper bound is exp(O(n^(1/3))).
2) An optimal result for random matrix reconstruction: we show that Theta(log n) traces are necessary and sufficient. This is in contrast to the problem for random sequences where there is a super-logarithmic lower bound and the best known upper bound is exp({O}(log^(1/3) n)).
3) We show that exp(O(k^(1/3) log^(2/3) n)) traces suffice to reconstruct k-sparse strings, providing an improvement over the best known sequence reconstruction results when k = o(n/log^2 n).
4) We show that poly(n) traces suffice if x is k-sparse and we additionally have a "separation" promise, specifically that the indices of 1\u27s in x all differ by Omega(k log n)
Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter
An important result in the study of polynomial-time preprocessing shows that
there is an algorithm which given an instance (G,k) of Vertex Cover outputs an
equivalent instance (G',k') in polynomial time with the guarantee that G' has
at most 2k' vertices (and thus O((k')^2) edges) with k' <= k. Using the
terminology of parameterized complexity we say that k-Vertex Cover has a kernel
with 2k vertices. There is complexity-theoretic evidence that both 2k vertices
and Theta(k^2) edges are optimal for the kernel size. In this paper we consider
the Vertex Cover problem with a different parameter, the size fvs(G) of a
minimum feedback vertex set for G. This refined parameter is structurally
smaller than the parameter k associated to the vertex covering number vc(G)
since fvs(G) <= vc(G) and the difference can be arbitrarily large. We give a
kernel for Vertex Cover with a number of vertices that is cubic in fvs(G): an
instance (G,X,k) of Vertex Cover, where X is a feedback vertex set for G, can
be transformed in polynomial time into an equivalent instance (G',X',k') such
that |V(G')| <= 2k and |V(G')| <= O(|X'|^3). A similar result holds when the
feedback vertex set X is not given along with the input. In sharp contrast we
show that the Weighted Vertex Cover problem does not have a polynomial kernel
when parameterized by the cardinality of a given vertex cover of the graph
unless NP is in coNP/poly and the polynomial hierarchy collapses to the third
level.Comment: Published in "Theory of Computing Systems" as an Open Access
publicatio
Kernelization Lower Bounds By Cross-Composition
We introduce the cross-composition framework for proving kernelization lower
bounds. A classical problem L AND/OR-cross-composes into a parameterized
problem Q if it is possible to efficiently construct an instance of Q with
polynomially bounded parameter value that expresses the logical AND or OR of a
sequence of instances of L. Building on work by Bodlaender et al. (ICALP 2008)
and using a result by Fortnow and Santhanam (STOC 2008) with a refinement by
Dell and van Melkebeek (STOC 2010), we show that if an NP-hard problem
OR-cross-composes into a parameterized problem Q then Q does not admit a
polynomial kernel unless NP \subseteq coNP/poly and the polynomial hierarchy
collapses. Similarly, an AND-cross-composition for Q rules out polynomial
kernels for Q under Bodlaender et al.'s AND-distillation conjecture.
Our technique generalizes and strengthens the recent techniques of using
composition algorithms and of transferring the lower bounds via polynomial
parameter transformations. We show its applicability by proving kernelization
lower bounds for a number of important graphs problems with structural
(non-standard) parameterizations, e.g., Clique, Chromatic Number, Weighted
Feedback Vertex Set, and Weighted Odd Cycle Transversal do not admit polynomial
kernels with respect to the vertex cover number of the input graphs unless the
polynomial hierarchy collapses, contrasting the fact that these problems are
trivially fixed-parameter tractable for this parameter.
After learning of our results, several teams of authors have successfully
applied the cross-composition framework to different parameterized problems.
For completeness, our presentation of the framework includes several extensions
based on this follow-up work. For example, we show how a relaxed version of
OR-cross-compositions may be used to give lower bounds on the degree of the
polynomial in the kernel size.Comment: A preliminary version appeared in the proceedings of the 28th
International Symposium on Theoretical Aspects of Computer Science (STACS
2011) under the title "Cross-Composition: A New Technique for Kernelization
Lower Bounds". Several results have been strengthened compared to the
preliminary version (http://arxiv.org/abs/1011.4224). 29 pages, 2 figure
Cross-Composition: A New Technique for Kernelization Lower Bounds
We introduce a new technique for proving kernelization lower bounds, called
cross-composition. A classical problem L cross-composes into a parameterized
problem Q if an instance of Q with polynomially bounded parameter value can
express the logical OR of a sequence of instances of L. Building on work by
Bodlaender et al. (ICALP 2008) and using a result by Fortnow and Santhanam
(STOC 2008) we show that if an NP-complete problem cross-composes into a
parameterized problem Q then Q does not admit a polynomial kernel unless the
polynomial hierarchy collapses. Our technique generalizes and strengthens the
recent techniques of using OR-composition algorithms and of transferring the
lower bounds via polynomial parameter transformations. We show its
applicability by proving kernelization lower bounds for a number of important
graphs problems with structural (non-standard) parameterizations, e.g.,
Chromatic Number, Clique, and Weighted Feedback Vertex Set do not admit
polynomial kernels with respect to the vertex cover number of the input graphs
unless the polynomial hierarchy collapses, contrasting the fact that these
problems are trivially fixed-parameter tractable for this parameter. We have
similar lower bounds for Feedback Vertex Set.Comment: Updated information based on final version submitted to STACS 201
Revisiting the Complexity of Stability of Continuous and Hybrid Systems
We develop a framework to give upper bounds on the "practical" computational
complexity of stability problems for a wide range of nonlinear continuous and
hybrid systems. To do so, we describe stability properties of dynamical systems
using first-order formulas over the real numbers, and reduce stability problems
to the delta-decision problems of these formulas. The framework allows us to
obtain a precise characterization of the complexity of different notions of
stability for nonlinear continuous and hybrid systems. We prove that bounded
versions of the stability problems are generally decidable, and give upper
bounds on their complexity. The unbounded versions are generally undecidable,
for which we give upper bounds on their degrees of unsolvability
On Sparsification for Computing Treewidth
We investigate whether an n-vertex instance (G,k) of Treewidth, asking
whether the graph G has treewidth at most k, can efficiently be made sparse
without changing its answer. By giving a special form of OR-cross-composition,
we prove that this is unlikely: if there is an e > 0 and a polynomial-time
algorithm that reduces n-vertex Treewidth instances to equivalent instances, of
an arbitrary problem, with O(n^{2-e}) bits, then NP is in coNP/poly and the
polynomial hierarchy collapses to its third level.
Our sparsification lower bound has implications for structural
parameterizations of Treewidth: parameterizations by measures that do not
exceed the vertex count, cannot have kernels with O(k^{2-e}) bits for any e >
0, unless NP is in coNP/poly. Motivated by the question of determining the
optimal kernel size for Treewidth parameterized by vertex cover, we improve the
O(k^3)-vertex kernel from Bodlaender et al. (STACS 2011) to a kernel with
O(k^2) vertices. Our improved kernel is based on a novel form of
treewidth-invariant set. We use the q-expansion lemma of Fomin et al. (STACS
2011) to find such sets efficiently in graphs whose vertex count is
superquadratic in their vertex cover number.Comment: 21 pages. Full version of the extended abstract presented at IPEC
201
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