90 research outputs found
Optimal Sparsification for Some Binary CSPs Using Low-degree Polynomials
This paper analyzes to what extent it is possible to efficiently reduce the
number of clauses in NP-hard satisfiability problems, without changing the
answer. Upper and lower bounds are established using the concept of
kernelization. Existing results show that if NP is not contained in coNP/poly,
no efficient preprocessing algorithm can reduce n-variable instances of CNF-SAT
with d literals per clause, to equivalent instances with bits for
any e > 0. For the Not-All-Equal SAT problem, a compression to size
exists. We put these results in a common framework by analyzing
the compressibility of binary CSPs. We characterize constraint types based on
the minimum degree of multivariate polynomials whose roots correspond to the
satisfying assignments, obtaining (nearly) matching upper and lower bounds in
several settings. Our lower bounds show that not just the number of
constraints, but also the encoding size of individual constraints plays an
important role. For example, for Exact Satisfiability with unbounded clause
length it is possible to efficiently reduce the number of constraints to n+1,
yet no polynomial-time algorithm can reduce to an equivalent instance with
bits for any e > 0, unless NP is a subset of coNP/poly.Comment: Updated the cross-composition in lemma 18 (minor update), since the
previous version did NOT satisfy requirement 4 of lemma 18 (the proof of
Claim 20 was incorrect
Improved Parameterized Algorithms for Constraint Satisfaction
For many constraint satisfaction problems, the algorithm which chooses a
random assignment achieves the best possible approximation ratio. For instance,
a simple random assignment for {\sc Max-E3-Sat} allows 7/8-approximation and
for every \eps >0 there is no polynomial-time (7/8+\eps)-approximation
unless P=NP. Another example is the {\sc Permutation CSP} of bounded arity.
Given the expected fraction of the constraints satisfied by a random
assignment (i.e. permutation), there is no (\rho+\eps)-approximation
algorithm for every \eps >0, assuming the Unique Games Conjecture (UGC).
In this work, we consider the following parameterization of constraint
satisfaction problems. Given a set of constraints of constant arity, can we
satisfy at least constraint, where is the expected fraction
of constraints satisfied by a random assignment? {\sc Constraint Satisfaction
Problems above Average} have been posed in different forms in the literature
\cite{Niedermeier2006,MahajanRamanSikdar09}. We present a faster parameterized
algorithm for deciding whether equations can be simultaneously
satisfied over . As a consequence, we obtain -variable
bikernels for {\sc boolean CSPs} of arity for every fixed , and for {\sc
permutation CSPs} of arity 3. This implies linear bikernels for many problems
under the "above average" parameterization, such as {\sc Max--Sat}, {\sc
Set-Splitting}, {\sc Betweenness} and {\sc Max Acyclic Subgraph}. As a result,
all the parameterized problems we consider in this paper admit -time
algorithms.
We also obtain non-trivial hybrid algorithms for every Max -CSP: for every
instance , we can either approximate beyond the random assignment
threshold in polynomial time, or we can find an optimal solution to in
subexponential time.Comment: A preliminary version of this paper has been accepted for IPEC 201
On the Parameterized Complexity and Kernelization of the Workflow Satisfiability Problem
A workflow specification defines a set of steps and the order in which those
steps must be executed. Security requirements may impose constraints on which
groups of users are permitted to perform subsets of those steps. A workflow
specification is said to be satisfiable if there exists an assignment of users
to workflow steps that satisfies all the constraints. An algorithm for
determining whether such an assignment exists is important, both as a static
analysis tool for workflow specifications, and for the construction of run-time
reference monitors for workflow management systems. Finding such an assignment
is a hard problem in general, but work by Wang and Li in 2010 using the theory
of parameterized complexity suggests that efficient algorithms exist under
reasonable assumptions about workflow specifications. In this paper, we improve
the complexity bounds for the workflow satisfiability problem. We also
generalize and extend the types of constraints that may be defined in a
workflow specification and prove that the satisfiability problem remains
fixed-parameter tractable for such constraints. Finally, we consider
preprocessing for the problem and prove that in an important special case, in
polynomial time, we can reduce the given input into an equivalent one, where
the number of users is at most the number of steps. We also show that no such
reduction exists for two natural extensions of this case, which bounds the
number of users by a polynomial in the number of steps, provided a
widely-accepted complexity-theoretical assumption holds
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
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
09441 Abstracts Collection -- The Constraint Satisfaction Problem: Complexity and Approximability
From 25th to 30th October 2009, the Dagstuhl Seminar 09441 ``The Constraint Satisfaction Problem: Complexity and Approximability\u27\u27 was held
in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
- …