785 research outputs found
The complexity of approximating bounded-degree Boolean #CSP
AbstractThe degree of a CSP instance is the maximum number of times that any variable appears in the scopes of constraints. We consider the approximate counting problem for Boolean CSP with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum allowed degree is large enough (at least 6) we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of {0}, {1} and binary implication. Otherwise, there is no FPRAS unless NP=RP. For lower degree bounds, additional cases arise, where the complexity is related to the complexity of approximately counting independent sets in hypergraphs
The complexity of approximately counting in 2-spin systems on -uniform bounded-degree hypergraphs
One of the most important recent developments in the complexity of
approximate counting is the classification of the complexity of approximating
the partition functions of antiferromagnetic 2-spin systems on bounded-degree
graphs. This classification is based on a beautiful connection to the so-called
uniqueness phase transition from statistical physics on the infinite
-regular tree. Our objective is to study the impact of this
classification on unweighted 2-spin models on -uniform hypergraphs. As has
already been indicated by Yin and Zhao, the connection between the uniqueness
phase transition and the complexity of approximate counting breaks down in the
hypergraph setting. Nevertheless, we show that for every non-trivial symmetric
-ary Boolean function there exists a degree bound so that for
all the following problem is NP-hard: given a
-uniform hypergraph with maximum degree at most , approximate the
partition function of the hypergraph 2-spin model associated with . It is
NP-hard to approximate this partition function even within an exponential
factor. By contrast, if is a trivial symmetric Boolean function (e.g., any
function that is excluded from our result), then the partition function of
the corresponding hypergraph 2-spin model can be computed exactly in polynomial
time
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
A Dichotomy Theorem for the Approximate Counting of Complex-Weighted Bounded-Degree Boolean CSPs
We determine the computational complexity of approximately counting the total
weight of variable assignments for every complex-weighted Boolean constraint
satisfaction problem (or CSP) with any number of additional unary (i.e., arity
1) constraints, particularly, when degrees of input instances are bounded from
above by a fixed constant. All degree-1 counting CSPs are obviously solvable in
polynomial time. When the instance's degree is more than two, we present a
dichotomy theorem that classifies all counting CSPs admitting free unary
constraints into exactly two categories. This classification theorem extends,
to complex-weighted problems, an earlier result on the approximation complexity
of unweighted counting Boolean CSPs of bounded degree. The framework of the
proof of our theorem is based on a theory of signature developed from Valiant's
holographic algorithms that can efficiently solve seemingly intractable
counting CSPs. Despite the use of arbitrary complex weight, our proof of the
classification theorem is rather elementary and intuitive due to an extensive
use of a novel notion of limited T-constructibility. For the remaining degree-2
problems, in contrast, they are as hard to approximate as Holant problems,
which are a generalization of counting CSPs.Comment: A4, 10pt, 20 pages. This revised version improves its preliminary
version published under a slightly different title in the Proceedings of the
4th International Conference on Combinatorial Optimization and Applications
(COCOA 2010), Lecture Notes in Computer Science, Springer, Vol.6508 (Part I),
pp.285--299, Kailua-Kona, Hawaii, USA, December 18--20, 201
FPTAS for Counting Monotone CNF
A monotone CNF formula is a Boolean formula in conjunctive normal form where
each variable appears positively. We design a deterministic fully
polynomial-time approximation scheme (FPTAS) for counting the number of
satisfying assignments for a given monotone CNF formula when each variable
appears in at most clauses. Equivalently, this is also an FPTAS for
counting set covers where each set contains at most elements. If we allow
variables to appear in a maximum of clauses (or sets to contain
elements), it is NP-hard to approximate it. Thus, this gives a complete
understanding of the approximability of counting for monotone CNF formulas. It
is also an important step towards a complete characterization of the
approximability for all bounded degree Boolean #CSP problems. In addition, we
study the hypergraph matching problem, which arises naturally towards a
complete classification of bounded degree Boolean #CSP problems, and show an
FPTAS for counting 3D matchings of hypergraphs with maximum degree .
Our main technique is correlation decay, a powerful tool to design
deterministic FPTAS for counting problems defined by local constraints among a
number of variables. All previous uses of this design technique fall into two
categories: each constraint involves at most two variables, such as independent
set, coloring, and spin systems in general; or each variable appears in at most
two constraints, such as matching, edge cover, and holant problem in general.
The CNF problems studied here have more complicated structures than these
problems and require new design and proof techniques. As it turns out, the
technique we developed for the CNF problem also works for the hypergraph
matching problem. We believe that it may also find applications in other CSP or
more general counting problems.Comment: 24 pages, 2 figures. version 1=>2: minor edits, highlighted the
picture of set cover/packing, and an implication of our previous result in 3D
matchin
Lower Bounds on Query Complexity for Testing Bounded-Degree CSPs
In this paper, we consider lower bounds on the query complexity for testing
CSPs in the bounded-degree model.
First, for any ``symmetric'' predicate except \equ
where , we show that every (randomized) algorithm that distinguishes
satisfiable instances of CSP(P) from instances -far
from satisfiability requires queries where is the
number of variables and is a constant that depends on and
. This breaks a natural lower bound , which is
obtained by the birthday paradox. We also show that every one-sided error
tester requires queries for such . These results are hereditary
in the sense that the same results hold for any predicate such that
. For EQU, we give a one-sided error tester
whose query complexity is . Also, for 2-XOR (or,
equivalently E2LIN2), we show an lower bound for
distinguishing instances between -close to and -far
from satisfiability.
Next, for the general k-CSP over the binary domain, we show that every
algorithm that distinguishes satisfiable instances from instances
-far from satisfiability requires queries. The
matching NP-hardness is not known, even assuming the Unique Games Conjecture or
the -to- Conjecture. As a corollary, for Maximum Independent Set on
graphs with vertices and a degree bound , we show that every
approximation algorithm within a factor d/\poly\log d and an additive error
of requires queries. Previously, only super-constant
lower bounds were known
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