160 research outputs found
Some hard families of parameterised counting problems
We consider parameterised subgraph-counting problems of the following form:
given a graph G, how many k-tuples of its vertices have a given property? A
number of such problems are known to be #W[1]-complete; here we substantially
generalise some of these existing results by proving hardness for two large
families of such problems. We demonstrate that it is #W[1]-hard to count the
number of k-vertex subgraphs having any property where the number of distinct
edge-densities of labelled subgraphs that satisfy the property is o(k^2). In
the special case that the property in question depends only on the number of
edges in the subgraph, we give a strengthening of this result which leads to
our second family of hard problems.Comment: A few more minor changes. This version to appear in the ACM
Transactions on Computation Theor
The parameterised complexity of counting connected subgraphs and graph motifs
We introduce a family of parameterised counting problems on graphs, p-#Induced Subgraph With Property(Φ), which generalises a number of problems which have previously been studied. This paper focuses on the case in which Φ defines a family of graphs whose edge-minimal elements all have bounded treewidth; this includes the special case in which Φ describes the property of being connected. We show that exactly counting the number of connected induced k-vertex subgraphs in an n-vertex graph is #W[1]-hard, but on the other hand there exists an FPTRAS for the problem; more generally, we show that there exists an FPTRAS for p-#Induced Subgraph With Property(Φ) whenever Φ is monotone and all the minimal graphs satisfying Φ have bounded treewidth. We then apply these results to a counting version of the Graph Motif problem
The parameterised complexity of counting even and odd induced subgraphs
We consider the problem of counting, in a given graph, the number of induced k-vertex subgraphs which have an even number of edges, and also the complementary problem of counting the k-vertex induced subgraphs having an odd number of edges. We demonstrate that both problems are #W[1]-hard when parameterised by k, in fact proving a somewhat stronger result about counting subgraphs with a property that only holds for some subset of k-vertex subgraphs which have an even (respectively odd) number of edges. On the other hand, we show that each of the problems admits an FPTRAS. These approximation schemes are based on a surprising structural result, which exploits ideas from Ramsey theory
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)
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
Fundamentals of Partial Rejection Sampling
Partial Rejection Sampling is an algorithmic approach to obtaining a perfect
sample from a specified distribution. The objects to be sampled are assumed to
be represented by a number of random variables. In contrast to classical
rejection sampling, in which all variables are resampled until a feasible
solution is found, partial rejection sampling aims at greater efficiency by
resampling only a subset of variables that `go wrong'. Partial rejection
sampling is closely related to Moser and Tardos' algorithmic version of the
Lov\'asz Local Lemma, but with the additional requirement that a specified
output distribution should be met. This article provides a largely
self-contained account of the basic form of the algorithm and its analysis
- …