1,278 research outputs found
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
Lossy Kernelization
In this paper we propose a new framework for analyzing the performance of
preprocessing algorithms. Our framework builds on the notion of kernelization
from parameterized complexity. However, as opposed to the original notion of
kernelization, our definitions combine well with approximation algorithms and
heuristics. The key new definition is that of a polynomial size
-approximate kernel. Loosely speaking, a polynomial size
-approximate kernel is a polynomial time pre-processing algorithm that
takes as input an instance to a parameterized problem, and outputs
another instance to the same problem, such that . Additionally, for every , a -approximate solution
to the pre-processed instance can be turned in polynomial time into a
-approximate solution to the original instance .
Our main technical contribution are -approximate kernels of
polynomial size for three problems, namely Connected Vertex Cover, Disjoint
Cycle Packing and Disjoint Factors. These problems are known not to admit any
polynomial size kernels unless . Our approximate
kernels simultaneously beat both the lower bounds on the (normal) kernel size,
and the hardness of approximation lower bounds for all three problems. On the
negative side we prove that Longest Path parameterized by the length of the
path and Set Cover parameterized by the universe size do not admit even an
-approximate kernel of polynomial size, for any , unless
. In order to prove this lower bound we need to combine
in a non-trivial way the techniques used for showing kernelization lower bounds
with the methods for showing hardness of approximationComment: 58 pages. Version 2 contain new results: PSAKS for Cycle Packing and
approximate kernel lower bounds for Set Cover and Hitting Set parameterized
by universe siz
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
Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal
The Odd Cycle Transversal problem (OCT) asks whether a given graph can be
made bipartite by deleting at most of its vertices. In a breakthrough
result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a
\BigOh(4^kkmn) time algorithm for it, the first algorithm with polynomial
runtime of uniform degree for every fixed . It is known that this implies a
polynomial-time compression algorithm that turns OCT instances into equivalent
instances of size at most \BigOh(4^k), a so-called kernelization. Since then
the existence of a polynomial kernel for OCT, i.e., a kernelization with size
bounded polynomially in , has turned into one of the main open questions in
the study of kernelization.
This work provides the first (randomized) polynomial kernelization for OCT.
We introduce a novel kernelization approach based on matroid theory, where we
encode all relevant information about a problem instance into a matroid with a
representation of size polynomial in . For OCT, the matroid is built to
allow us to simulate the computation of the iterative compression step of the
algorithm of Reed, Smith, and Vetta, applied (for only one round) to an
approximate odd cycle transversal which it is aiming to shrink to size . The
process is randomized with one-sided error exponentially small in , where
the result can contain false positives but no false negatives, and the size
guarantee is cubic in the size of the approximate solution. Combined with an
\BigOh(\sqrt{\log n})-approximation (Agarwal et al., STOC 2005), we get a
reduction of the instance to size \BigOh(k^{4.5}), implying a randomized
polynomial kernelization.Comment: Minor changes to agree with SODA 2012 version of the pape
Streaming Kernelization
Kernelization is a formalization of preprocessing for combinatorially hard
problems. We modify the standard definition for kernelization, which allows any
polynomial-time algorithm for the preprocessing, by requiring instead that the
preprocessing runs in a streaming setting and uses
bits of memory on instances . We obtain
several results in this new setting, depending on the number of passes over the
input that such a streaming kernelization is allowed to make. Edge Dominating
Set turns out as an interesting example because it has no single-pass
kernelization but two passes over the input suffice to match the bounds of the
best standard kernelization
Crossing Paths with Hans Bodlaender:A Personal View on Cross-Composition for Sparsification Lower Bounds
On the occasion of Hans Bodlaender’s 60th birthday, I give a personal account of our history and work together on the technique of cross-composition for kernelization lower bounds. I present several simple new proofs for polynomial kernelization lower bounds using cross-composition, interlaced with personal anecdotes about my time as Hans’ PhD student at Utrecht University. Concretely, I will prove that Vertex Cover, Feedback Vertex Set, and the H-Factor problem for every graph H that has a connected component of at least three vertices, do not admit kernels of (formula presented) bits when parameterized by the number of vertices n for any (formula presented), unless (formula presented). These lower bounds are obtained by elementary gadget constructions, in particular avoiding the use of the Packing Lemma by Dell and van Melkebeek.</p
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
Polynomial Kernels for Weighted Problems
Kernelization is a formalization of efficient preprocessing for NP-hard
problems using the framework of parameterized complexity. Among open problems
in kernelization it has been asked many times whether there are deterministic
polynomial kernelizations for Subset Sum and Knapsack when parameterized by the
number of items.
We answer both questions affirmatively by using an algorithm for compressing
numbers due to Frank and Tardos (Combinatorica 1987). This result had been
first used by Marx and V\'egh (ICALP 2013) in the context of kernelization. We
further illustrate its applicability by giving polynomial kernels also for
weighted versions of several well-studied parameterized problems. Furthermore,
when parameterized by the different item sizes we obtain a polynomial
kernelization for Subset Sum and an exponential kernelization for Knapsack.
Finally, we also obtain kernelization results for polynomial integer programs
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