87 research outputs found
An approximation trichotomy for Boolean #CSP
We give a trichotomy theorem for the complexity of approximately counting the
number of satisfying assignments of a Boolean CSP instance. Such problems are
parameterised by a constraint language specifying the relations that may be
used in constraints. If every relation in the constraint language is affine
then the number of satisfying assignments can be exactly counted in polynomial
time. Otherwise, if every relation in the constraint language is in the
co-clone IM_2 from Post's lattice, then the problem of counting satisfying
assignments is complete with respect to approximation-preserving reductions in
the complexity class #RH\Pi_1. This means that the problem of approximately
counting satisfying assignments of such a CSP instance is equivalent in
complexity to several other known counting problems, including the problem of
approximately counting the number of independent sets in a bipartite graph. For
every other fixed constraint language, the problem is complete for #P with
respect to approximation-preserving reductions, meaning that there is no fully
polynomial randomised approximation scheme for counting satisfying assignments
unless NP=RP
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
The complexity of counting locally maximal satisfying assignments of Boolean CSPs
We investigate the computational complexity of the problem of counting the
maximal satisfying assignments of a Constraint Satisfaction Problem (CSP) over
the Boolean domain {0,1}. A satisfying assignment is maximal if any new
assignment which is obtained from it by changing a 0 to a 1 is unsatisfying.
For each constraint language Gamma, #MaximalCSP(Gamma) denotes the problem of
counting the maximal satisfying assignments, given an input CSP with
constraints in Gamma. We give a complexity dichotomy for the problem of exactly
counting the maximal satisfying assignments and a complexity trichotomy for the
problem of approximately counting them. Relative to the problem #CSP(Gamma),
which is the problem of counting all satisfying assignments, the maximal
version can sometimes be easier but never harder. This finding contrasts with
the recent discovery that approximately counting maximal independent sets in a
bipartite graph is harder (under the usual complexity-theoretic assumptions)
than counting all independent sets.Comment: V2 adds contextual material relating the results obtained here to
earlier work in a different but related setting. The technical content is
unchanged. V3 (this version) incorporates minor revisions. The title has been
changed to better reflect what is novel in this work. This version has been
accepted for publication in Theoretical Computer Science. 19 page
The Complexity of Approximately Counting Tree Homomorphisms
We study two computational problems, parameterised by a fixed tree H.
#HomsTo(H) is the problem of counting homomorphisms from an input graph G to H.
#WHomsTo(H) is the problem of counting weighted homomorphisms to H, given an
input graph G and a weight function for each vertex v of G. Even though H is a
tree, these problems turn out to be sufficiently rich to capture all of the
known approximation behaviour in #P. We give a complete trichotomy for
#WHomsTo(H). If H is a star then #WHomsTo(H) is in FP. If H is not a star but
it does not contain a certain induced subgraph J_3 then #WHomsTo(H) is
equivalent under approximation-preserving (AP) reductions to #BIS, the problem
of counting independent sets in a bipartite graph. This problem is complete for
the class #RHPi_1 under AP-reductions. Finally, if H contains an induced J_3
then #WHomsTo(H) is equivalent under AP-reductions to #SAT, the problem of
counting satisfying assignments to a CNF Boolean formula. Thus, #WHomsTo(H) is
complete for #P under AP-reductions. The results are similar for #HomsTo(H)
except that a rich structure emerges if H contains an induced J_3. We show that
there are trees H for which #HomsTo(H) is #SAT-equivalent (disproving a
plausible conjecture of Kelk). There is an interesting connection between these
homomorphism-counting problems and the problem of approximating the partition
function of the ferromagnetic Potts model. In particular, we show that for a
family of graphs J_q, parameterised by a positive integer q, the problem
#HomsTo(H) is AP-interreducible with the problem of approximating the partition
function of the q-state Potts model. It was not previously known that the Potts
model had a homomorphism-counting interpretation. We use this connection to
obtain some additional upper bounds for the approximation complexity of
#HomsTo(J_q)
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 Weighted Boolean #CSP with Mixed Signs
We give a complexity dichotomy for the problem of computing the partition
function of a weighted Boolean constraint satisfaction problem. Such a problem
is parameterized by a set of rational-valued functions, which generalize
constraints. Each function assigns a weight to every assignment to a set of
Boolean variables. Our dichotomy extends previous work in which the weight
functions were restricted to being non-negative. We represent a weight function
as a product of the form (-1)^s g, where the polynomial s determines the sign
of the weight and the non-negative function g determines its magnitude. We show
that the problem of computing the partition function (the sum of the weights of
all possible variable assignments) is in polynomial time if either every weight
function can be defined by a "pure affine" magnitude with a quadratic sign
polynomial or every function can be defined by a magnitude of "product type"
with a linear sign polynomial. In all other cases, computing the partition
function is FP^#P-complete.Comment: 24 page
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