2,241 research outputs found
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
Approximation for Maximum Surjective Constraint Satisfaction Problems
Maximum surjective constraint satisfaction problems (Max-Sur-CSPs) are
computational problems where we are given a set of variables denoting values
from a finite domain B and a set of constraints on the variables. A solution to
such a problem is a surjective mapping from the set of variables to B such that
the number of satisfied constraints is maximized. We study the approximation
performance that can be acccchieved by algorithms for these problems, mainly by
investigating their relation with Max-CSPs (which are the corresponding
problems without the surjectivity requirement). Our work gives a complexity
dichotomy for Max-Sur-CSP(B) between PTAS and APX-complete, under the
assumption that there is a complexity dichotomy for Max-CSP(B) between PO and
APX-complete, which has already been proved on the Boolean domain and 3-element
domains
The complexity of approximating conservative counting CSPs
We study the complexity of approximately solving the weighted counting
constraint satisfaction problem #CSP(F). In the conservative case, where F
contains all unary functions, there is a classification known for the case in
which the domain of functions in F is Boolean. In this paper, we give a
classification for the more general problem where functions in F have an
arbitrary finite domain. We define the notions of weak log-modularity and weak
log-supermodularity. We show that if F is weakly log-modular, then #CSP(F)is in
FP. Otherwise, it is at least as difficult to approximate as #BIS, the problem
of counting independent sets in bipartite graphs. #BIS is complete with respect
to approximation-preserving reductions for a logically-defined complexity class
#RHPi1, and is believed to be intractable. We further sub-divide the #BIS-hard
case. If F is weakly log-supermodular, then we show that #CSP(F) is as easy as
a (Boolean) log-supermodular weighted #CSP. Otherwise, we show that it is
NP-hard to approximate. Finally, we give a full trichotomy for the arity-2
case, where #CSP(F) is in FP, or is #BIS-equivalent, or is equivalent in
difficulty to #SAT, the problem of approximately counting the satisfying
assignments of a Boolean formula in conjunctive normal form. We also discuss
the algorithmic aspects of our classification.Comment: Minor revisio
Near-Optimal UGC-hardness of Approximating Max k-CSP_R
In this paper, we prove an almost-optimal hardness for Max -CSP based
on Khot's Unique Games Conjecture (UGC). In Max -CSP, we are given a set
of predicates each of which depends on exactly variables. Each variable can
take any value from . The goal is to find an assignment to
variables that maximizes the number of satisfied predicates.
Assuming the Unique Games Conjecture, we show that it is NP-hard to
approximate Max -CSP to within factor for any . To the best of our knowledge, this result
improves on all the known hardness of approximation results when . In this case, the previous best hardness result was
NP-hardness of approximating within a factor by Chan. When , our result matches the best known UGC-hardness result of Khot, Kindler,
Mossel and O'Donnell.
In addition, by extending an algorithm for Max 2-CSP by Kindler, Kolla
and Trevisan, we provide an -approximation algorithm
for Max -CSP. This algorithm implies that our inapproximability result
is tight up to a factor of . In comparison,
when is a constant, the previously known gap was , which is
significantly larger than our gap of .
Finally, we show that we can replace the Unique Games Conjecture assumption
with Khot's -to-1 Conjecture and still get asymptotically the same hardness
of approximation
Approximation Resistant Predicates From Pairwise Independence
We study the approximability of predicates on variables from a domain
, and give a new sufficient condition for such predicates to be
approximation resistant under the Unique Games Conjecture. Specifically, we
show that a predicate is approximation resistant if there exists a balanced
pairwise independent distribution over whose support is contained in
the set of satisfying assignments to
A Characterization of Approximation Resistance for Even -Partite CSPs
A constraint satisfaction problem (CSP) is said to be \emph{approximation
resistant} if it is hard to approximate better than the trivial algorithm which
picks a uniformly random assignment. Assuming the Unique Games Conjecture, we
give a characterization of approximation resistance for -partite CSPs
defined by an even predicate
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