69,057 research outputs found
Approximating the minimum directed tree cover
Given a directed graph with non negative cost on the arcs, a directed
tree cover of is a rooted directed tree such that either head or tail (or
both of them) of every arc in is touched by . The minimum directed tree
cover problem (DTCP) is to find a directed tree cover of minimum cost. The
problem is known to be -hard. In this paper, we show that the weighted Set
Cover Problem (SCP) is a special case of DTCP. Hence, one can expect at best to
approximate DTCP with the same ratio as for SCP. We show that this expectation
can be satisfied in some way by designing a purely combinatorial approximation
algorithm for the DTCP and proving that the approximation ratio of the
algorithm is with is the maximum outgoing degree of
the nodes in .Comment: 13 page
Discovering a junction tree behind a Markov network by a greedy algorithm
In an earlier paper we introduced a special kind of k-width junction tree,
called k-th order t-cherry junction tree in order to approximate a joint
probability distribution. The approximation is the best if the Kullback-Leibler
divergence between the true joint probability distribution and the
approximating one is minimal. Finding the best approximating k-width junction
tree is NP-complete if k>2. In our earlier paper we also proved that the best
approximating k-width junction tree can be embedded into a k-th order t-cherry
junction tree. We introduce a greedy algorithm resulting very good
approximations in reasonable computing time.
In this paper we prove that if the Markov network underlying fullfills some
requirements then our greedy algorithm is able to find the true probability
distribution or its best approximation in the family of the k-th order t-cherry
tree probability distributions. Our algorithm uses just the k-th order marginal
probability distributions as input.
We compare the results of the greedy algorithm proposed in this paper with
the greedy algorithm proposed by Malvestuto in 1991.Comment: The paper was presented at VOCAL 2010 in Veszprem, Hungar
Inapproximability of the independent set polynomial in the complex plane
We study the complexity of approximating the independent set polynomial
of a graph with maximum degree when the activity
is a complex number.
This problem is already well understood when is real using
connections to the -regular tree . The key concept in that case is
the "occupation ratio" of the tree . This ratio is the contribution to
from independent sets containing the root of the tree, divided
by itself. If is such that the occupation ratio
converges to a limit, as the height of grows, then there is an FPTAS for
approximating on a graph with maximum degree .
Otherwise, the approximation problem is NP-hard.
Unsurprisingly, the case where is complex is more challenging.
Peters and Regts identified the complex values of for which the
occupation ratio of the -regular tree converges. These values carve a
cardioid-shaped region in the complex plane. Motivated by the
picture in the real case, they asked whether marks the true
approximability threshold for general complex values .
Our main result shows that for every outside of ,
the problem of approximating on graphs with maximum degree
at most is indeed NP-hard. In fact, when is outside of
and is not a positive real number, we give the stronger result
that approximating is actually #P-hard. If is a
negative real number outside of , we show that it is #P-hard to
even decide whether , resolving in the affirmative a conjecture
of Harvey, Srivastava and Vondrak.
Our proof techniques are based around tools from complex analysis -
specifically the study of iterative multivariate rational maps
Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models
Recent inapproximability results of Sly (2010), together with an
approximation algorithm presented by Weitz (2006) establish a beautiful picture
for the computational complexity of approximating the partition function of the
hard-core model. Let denote the critical activity for the
hard-model on the infinite -regular tree. Weitz presented an FPTAS for
the partition function when for graphs with
constant maximum degree . In contrast, Sly showed that for all
, there exists such that (unless RP=NP) there
is no FPRAS for approximating the partition function on graphs of maximum
degree for activities satisfying
.
We prove that a similar phenomenon holds for the antiferromagnetic Ising
model. Recent results of Li et al. and Sinclair et al. extend Weitz's approach
to any 2-spin model, which includes the antiferromagnetic Ising model, to yield
an FPTAS for the partition function for all graphs of constant maximum degree
when the parameters of the model lie in the uniqueness regime of the
infinite tree . We prove the complementary result that for the
antiferrogmanetic Ising model without external field that, unless RP=NP, for
all , there is no FPRAS for approximating the partition function
on graphs of maximum degree when the inverse temperature lies in the
non-uniqueness regime of the infinite tree . Our results extend to a
region of the parameter space for general 2-spin models. Our proof works by
relating certain second moment calculations for random -regular
bipartite graphs to the tree recursions used to establish the critical points
on the infinite tree.Comment: Journal version (no changes
Causal Dependence Tree Approximations of Joint Distributions for Multiple Random Processes
We investigate approximating joint distributions of random processes with
causal dependence tree distributions. Such distributions are particularly
useful in providing parsimonious representation when there exists causal
dynamics among processes. By extending the results by Chow and Liu on
dependence tree approximations, we show that the best causal dependence tree
approximation is the one which maximizes the sum of directed informations on
its edges, where best is defined in terms of minimizing the KL-divergence
between the original and the approximate distribution. Moreover, we describe a
low-complexity algorithm to efficiently pick this approximate distribution.Comment: 9 pages, 15 figure
Optimising the Solovay-Kitaev algorithm
The Solovay-Kitaev algorithm is the standard method used for approximating
arbitrary single-qubit gates for fault-tolerant quantum computation. In this
paper we introduce a technique called "search space expansion", which modifies
the initial stage of the Solovay-Kitaev algorithm, increasing the length of the
possible approximating sequences but without requiring an exhaustive search
over all possible sequences. We show that our technique, combined with a GNAT
geometric tree search outputs gate sequences that are almost an order of
magnitude smaller for the same level of accuracy. This therefore significantly
reduces the error correction requirements for quantum algorithms on encoded
fault-tolerant hardware.Comment: 9 page
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