60,282 research outputs found
Approximately counting locally-optimal structures
A locally-optimal structure is a combinatorial structure such as a maximal
independent set that cannot be improved by certain (greedy) local moves, even
though it may not be globally optimal. It is trivial to construct an
independent set in a graph. It is easy to (greedily) construct a maximal
independent set. However, it is NP-hard to construct a globally-optimal
(maximum) independent set. In general, constructing a locally-optimal structure
is somewhat more difficult than constructing an arbitrary structure, and
constructing a globally-optimal structure is more difficult than constructing a
locally-optimal structure. The same situation arises with listing. The
differences between the problems become obscured when we move from listing to
counting because nearly everything is #P-complete. However, we highlight an
interesting phenomenon that arises in approximate counting, where the situation
is apparently reversed. Specifically, we show that counting maximal independent
sets is complete for #P with respect to approximation-preserving reductions,
whereas counting all independent sets, or counting maximum independent sets is
complete for an apparently smaller class, which has a
prominent role in the complexity of approximate counting. Motivated by the
difficulty of approximately counting maximal independent sets in bipartite
graphs, we also study the problem of approximately counting other
locally-optimal structures that arise in algorithmic applications, particularly
problems involving minimal separators and minimal edge separators. Minimal
separators have applications via fixed-parameter-tractable algorithms for
constructing triangulations and phylogenetic trees. Although exact
(exponential-time) algorithms exist for listing these structures, we show that
the counting problems are #P-complete with respect to both exact and
approximation-preserving reductions.Comment: Accepted to JCSS, preliminary version accepted to ICALP 2015 (Track
A
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
Counting hypergraph matchings up to uniqueness threshold
We study the problem of approximately counting matchings in hypergraphs of
bounded maximum degree and maximum size of hyperedges. With an activity
parameter , each matching is assigned a weight .
The counting problem is formulated as computing a partition function that gives
the sum of the weights of all matchings in a hypergraph. This problem unifies
two extensively studied statistical physics models in approximate counting: the
hardcore model (graph independent sets) and the monomer-dimer model (graph
matchings).
For this model, the critical activity
is the threshold for the uniqueness of Gibbs measures on the infinite
-uniform -regular hypertree. Consider hypergraphs of maximum
degree at most and maximum size of hyperedges at most . We show that
when , there is an FPTAS for computing the partition
function; and when , there is a PTAS for computing the
log-partition function. These algorithms are based on the decay of correlation
(strong spatial mixing) property of Gibbs distributions. When , there is no PRAS for the partition function or the log-partition
function unless NPRP.
Towards obtaining a sharp transition of computational complexity of
approximate counting, we study the local convergence from a sequence of finite
hypergraphs to the infinite lattice with specified symmetry. We show a
surprising connection between the local convergence and the reversibility of a
natural random walk. This leads us to a barrier for the hardness result: The
non-uniqueness of infinite Gibbs measure is not realizable by any finite
gadgets
Statistical structures for internet-scale data management
Efficient query processing in traditional database management systems relies on statistics on base data. For centralized systems, there is a rich body of research results on such statistics, from simple aggregates to more elaborate synopses such as sketches and histograms. For Internet-scale distributed systems, on the other hand, statistics management still poses major challenges. With the work in this paper we aim to endow peer-to-peer data management over structured overlays with the power associated with such statistical information, with emphasis on meeting the scalability challenge. To this end, we first contribute efficient, accurate, and decentralized algorithms that can compute key aggregates such as Count, CountDistinct, Sum, and Average. We show how to construct several types of histograms, such as simple Equi-Width, Average-Shifted Equi-Width, and Equi-Depth histograms. We present a full-fledged open-source implementation of these tools for distributed statistical synopses, and report on a comprehensive experimental performance evaluation, evaluating our contributions in terms of efficiency, accuracy, and scalability
Estimating and exploiting the degree of independent information in distributed data fusion
Double counting is a major problem in distributed data fusion systems. To maintain flexibility and scalability, distributed data fusion algorithms should just use local information. However globally optimal solutions only exist in highly restricted circumstances. Suboptimal algorithms can be applied in a far wider range of cases, but can be very conservative.
In this paper we present preliminary work to develop
distributed data fusion algorithms that can estimate and
exploit the correlations between the estimates stored in
different nodes in a distributed data fusion network.
We show that partial information can be modelled as
kind of “overweighted” Covariance Intersection algorithm. We motivate the need for an adaptive scheme
by analysing the correlation behaviour of a simple distributed data fusion network and show that it is complicated and counterintuitive. Two simple approaches
to estimate the correlation structure are presented and
their results analysed. We show that significant advantages can be obtained
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