7 research outputs found
A complexity dichotomy for hypergraph partition functions
We consider the complexity of counting homomorphisms from an -uniform
hypergraph to a symmetric -ary relation . We give a dichotomy theorem
for , showing for which this problem is in FP and for which it is
#P-complete. This generalises a theorem of Dyer and Greenhill (2000) for the
case , which corresponds to counting graph homomorphisms. Our dichotomy
theorem extends to the case in which the relation is weighted, and the goal
is to compute the \emph{partition function}, which is the sum of weights of the
homomorphisms. This problem is motivated by statistical physics, where it
arises as computing the partition function for particle models in which certain
combinations of sites interact symmetrically. In the weighted case, our
dichotomy theorem generalises a result of Bulatov and Grohe (2005) for graphs,
where . When , the polynomial time cases of the dichotomy correspond
simply to rank-1 weights. Surprisingly, for all the polynomial time cases
of the dichotomy have rather more structure. It turns out that the weights must
be superimposed on a combinatorial structure defined by solutions of an
equation over an Abelian group. Our result also gives a dichotomy for a closely
related constraint satisfaction problem.Comment: 21 page
The complexity of weighted and unweighted #CSP
We give some reductions among problems in (nonnegative) weighted #CSP which
restrict the class of functions that needs to be considered in computational
complexity studies. Our reductions can be applied to both exact and approximate
computation. In particular, we show that a recent dichotomy for unweighted #CSP
can be extended to rational-weighted #CSP.Comment: 11 page
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
A complexity dichotomy for partition functions with mixed signs
Partition functions, also known as homomorphism functions, form a rich family
of graph invariants that contain combinatorial invariants such as the number of
k-colourings or the number of independent sets of a graph and also the
partition functions of certain "spin glass" models of statistical physics such
as the Ising model.
Building on earlier work by Dyer, Greenhill and Bulatov, Grohe, we completely
classify the computational complexity of partition functions. Our main result
is a dichotomy theorem stating that every partition function is either
computable in polynomial time or #P-complete. Partition functions are described
by symmetric matrices with real entries, and we prove that it is decidable in
polynomial time in terms of the matrix whether a given partition function is in
polynomial time or #P-complete.
While in general it is very complicated to give an explicit algebraic or
combinatorial description of the tractable cases, for partition functions
described by a Hadamard matrices -- these turn out to be central in our proofs
-- we obtain a simple algebraic tractability criterion, which says that the
tractable cases are those "representable" by a quadratic polynomial over the
field GF(2)