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ABSTRACT: A remarkable connection has been established for 2-spin systems, including
the Ising and hard-core models, showing that the computational complexity of
approximating the partition function for graphs with maximum degree D undergoes
a phase transition that coincides with the statistical physics
uniqueness/non-uniqueness phase transition on the infinite D-regular tree.
Despite this clear picture for 2-spin systems, there is little known for
multi-spin systems. We present an analog of the above inapproximability results
for multi-spin systems. We prove that, unless NP=RP, for any antiferromagnetic
spin system, there is no FPRAS for the partition function of D-regular graphs
when the dominant semi-translation invariant Gibbs measures on the infinite
D-regular tree are not translation invariant and are permutation symmetric of
each other. Our results apply to the antiferromagnetic Potts model (even q) and
colorings problem (even k), which are the multi-spin systems of particular
interest. Our proof relies on a simple and generic analysis of the second
moment for any spin system. As a consequence we get concentration results for
any spin system in which one can analyze the first moment. We also present a
tool for simplifying the associated first moment calculations by relating it to
the stable fixed points for the tree recursions.
05/2013;
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ABSTRACT: 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 $\lambda_c(T_\Delta)$ denote the critical activity for the
hard-model on the infinite $\Delta$-regular tree. Weitz presented an FPTAS for
the partition function when $\lambda<\lambda_c(T_\Delta)$ for graphs with
constant maximum degree $\Delta$. In contrast, Sly showed that for all
$\Delta\geq 3$, there exists $\epsilon_\Delta>0$ such that (unless RP=NP) there
is no FPRAS for approximating the partition function on graphs of maximum
degree $\Delta$ for activities $\lambda$ satisfying
$\lambda_c(T_\Delta)<\lambda<\lambda_c(T_\Delta)+\epsilon_\Delta$.
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
$\Delta$ when the parameters of the model lie in the uniqueness regime of the
infinite tree $T_\Delta$. We prove the complementary result that for the
antiferrogmanetic Ising model without external field that, unless RP=NP, for
all $\Delta\geq 3$, there is no FPRAS for approximating the partition function
on graphs of maximum degree $\Delta$ when the inverse temperature lies in the
non-uniqueness regime of the infinite tree $T_\Delta$. 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 $\Delta$-regular
bipartite graphs to the tree recursions used to establish the critical points
on the infinite tree.
03/2012;
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ABSTRACT: We study the computational complexity of approximately counting the number of
independent sets of a graph with maximum degree Delta. More generally, for an
input graph G=(V,E) and an activity lambda>0, we are interested in the quantity
Z_G(lambda) defined as the sum over independent sets I weighted as w(I) =
lambda^|I|. In statistical physics, Z_G(lambda) is the partition function for
the hard-core model, which is an idealized model of a gas where the particles
have non-negibile size.
Recently, an interesting phase transition was shown to occur for the
complexity of approximating the partition function. Weitz showed an FPAS for
the partition function for any graph of maximum degree Delta when Delta is
constant and lambda< lambda_c(Tree_Delta):=(Delta-1)^(Delta-1)/(Delta-2)^Delta.
The quantity lambda_c(Tree_Delta) is the critical point for the so-called
uniqueness threshold on the infinite, regular tree of degree Delta. On the
other side, Sly proved that there does not exist efficient (randomized)
approximation algorithms for lambda_c(Tree_Delta) < lambda <
lambda_c(Tree_Delta)+epsilon(Delta), unless NP=RP, for some function
epsilon(Delta)>0. We remove the upper bound in the assumptions of Sly's result
for Delta not equal to 4 and 5, that is, we show that there does not exist
efficient randomized approximation algorithms for all
lambda>lambda_c(Tree_Delta) for Delta=3 and Delta>= 6. Sly's inapproximability
result uses a clever reduction, combined with a second-moment analysis of
Mossel, Weitz and Wormald which prove torpid mixing of the Glauber dynamics for
sampling from the associated Gibbs distribution on almost every regular graph
of degree Delta for the same range of lambda as in Sly's result. We extend
Sly's result by improving upon the technical work of Mossel et al., via a more
detailed analysis of independent sets in random regular graphs.
05/2011;
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ABSTRACT: We study the hard-core model defined on independent sets, where each
independent set I in a graph G is weighted proportionally to $\lambda^{|I|}$,
for a positive real parameter $\lambda$. For large $\lambda$, computing the
partition function (namely, the normalizing constant which makes the weighting
a probability distribution on a finite graph) on graphs of maximum degree $D\ge
3$, is a well known computationally challenging problem. More concretely, let
$\lambda_c(T_D)$ denote the critical value for the so-called uniqueness
threshold of the hard-core model on the infinite D-regular tree; recent
breakthrough results of Dror Weitz (2006) and Allan Sly (2010) have identified
$\lambda_c(T_D)$ as a threshold where the hardness of estimating the above
partition function undergoes a computational transition.
We focus on the well-studied particular case of the square lattice
$\integers^2$, and provide a new lower bound for the uniqueness threshold, in
particular taking it well above $\lambda_c(T_4)$. Our technique refines and
builds on the tree of self-avoiding walks approach of Weitz, resulting in a new
technical sufficient criterion (of wider applicability) for establishing strong
spatial mixing (and hence uniqueness) for the hard-core model. Our new
criterion achieves better bounds on strong spatial mixing when the graph has
extra structure, improving upon what can be achieved by just using the maximum
degree. Applying our technique to $\integers^2$ we prove that strong spatial
mixing holds for all $\lambda<2.3882$, improving upon the work of Weitz that
held for $\lambda<27/16=1.6875$. Our results imply a fully-polynomial
deterministic approximation algorithm for estimating the partition function, as
well as rapid mixing of the associated Glauber dynamics to sample from the
hard-core distribution.
05/2011;
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SIAM J. Discrete Math. 01/2011; 25:1194-1211.
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Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011; 01/2011
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IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011; 01/2011
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ABSTRACT: Given n elements with nonnegative integer weights w1,..., wn and an integer capacity C, we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most the given capacity. We give a deterministic algorithm that estimates the number of solutions to within relative error 1+-eps in time polynomial in n and 1/eps (fully polynomial approximation scheme). More precisely, our algorithm takes time O(n^3 (1/eps) log (n/eps)). Our algorithm is based on dynamic programming. Previously, randomized polynomial time approximation schemes were known first by Morris and Sinclair via Markov chain Monte Carlo techniques, and subsequently by Dyer via dynamic programming and rejection sampling. Comment: 11 pages
08/2010;
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ABSTRACT: We study the effect of boundary conditions on the relaxation time of the Glauber dynamics for the hard-core model on the tree. The hard-core model is defined on the set of independent sets weighted by a parameter $\lambda$, called the activity. The Glauber dynamics is the Markov chain that updates a randomly chosen vertex in each step. On the infinite tree with branching factor $b$, the hard-core model can be equivalently defined as a broadcasting process with a parameter $\omega$ which is the positive solution to $\lambda=\omega(1+\omega)^b$, and vertices are occupied with probability $\omega/(1+\omega)$ when their parent is unoccupied. This broadcasting process undergoes a phase transition between the so-called reconstruction and non-reconstruction regions at $\omega_r\approx \ln{b}/b$. Reconstruction has been of considerable interest recently since it appears to be intimately connected to the efficiency of local algorithms on locally tree-like graphs, such as sparse random graphs. In this paper we show that the relaxation time of the Glauber dynamics on regular $b$-ary trees $T_h$ of height $h$ and $n$ vertices, undergoes a phase transition around the reconstruction threshold. In particular, we construct a boundary condition for which the relaxation time slows down at the reconstruction threshold. More precisely, for any $\omega \le \ln{b}/b$, for $T_h$ with any boundary condition, the relaxation time is $\Omega(n)$ and $O(n^{1+o_b(1)})$. In contrast, above the reconstruction threshold we show that for every $\delta>0$, for $\omega=(1+\delta)\ln{b}/b$, the relaxation time on $T_h$ with any boundary condition is $O(n^{1+\delta + o_b(1)})$, and we construct a boundary condition where the relaxation time is $\Omega(n^{1+\delta/2 - o_b(1)})$.
07/2010;
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ABSTRACT: This paper studies a Markov chain for phylogenetic reconstruction which uses
a popular transition between tree topologies known as subtree
pruning-and-regrafting (SPR). We analyze the Markov chain in the simpler
setting that the generating tree consists of very short edge lengths, short
enough so that each sample from the generating tree (or character in
phylogenetic terminology) is likely to have only one mutation, and that there
enough samples so that the data looks like the generating distribution. We
prove in this setting that the Markov chain is rapidly mixing, i.e., it quickly
converges to its stationary distribution, which is the posterior distribution
over tree topologies. Our proofs use that the leading term of the maximum
likelihood function of a tree T is the maximum parsimony score, which is the
size of the minimum cut in T needed to realize single edge cuts of the
generating tree. Our main contribution is a combinatorial proof that in our
simplified setting, SPR moves are guaranteed to converge quickly to the maximum
parsimony tree. Our results are in contrast to recent works showing examples
with heterogeneous data (namely, the data is generated from a mixture
distribution) where many natural Markov chains are exponentially slow to
converge to the stationary distribution.
03/2010;
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Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010; 01/2010
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ABSTRACT: We prove that the mixing time of the Glauber dynamics for random k-colorings
of the complete tree with branching factor b undergoes a phase transition at
$k=b(1+o_b(1))/\ln{b}$. Our main result shows nearly sharp bounds on the mixing
time of the dynamics on the complete tree with n vertices for $k=Cb/\ln{b}$
colors with constant C. For $C\geq1$ we prove the mixing time is
$O(n^{1+o_b(1)}\ln{n})$. On the other side, for $C<1$ the mixing time
experiences a slowing down; in particular, we prove it is
$O(n^{1/C+o_b(1)}\ln{n})$ and $\Omega(n^{1/C-o_b(1)})$. The critical point C=1
is interesting since it coincides (at least up to first order) with the
so-called reconstruction threshold which was recently established by Sly. The
reconstruction threshold has been of considerable interest recently since it
appears to have close connections to the efficiency of certain local
algorithms, and this work was inspired by our attempt to understand these
connections in this particular setting.
08/2009;
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ABSTRACT: Given a graph with edges colored Red and Blue, we study the problem of sampling and approximately counting the number of matchings with exactly k
Red edges. We solve the problem of estimating the number of perfect matchings with exactly k
Red edges for dense graphs. We study a Markov chain on the space of all matchings of a graph that favors matchings with k
Red edges. We show that it is rapidly mixing using non-traditional canonical paths that can backtrack. We show that this chain
can be used to sample matchings in the 2-dimensional toroidal lattice of any fixed size ℓ with k
Red edges, where the horizontal edges are Red and the vertical edges are Blue.
Algorithmica 03/2008; 50(4):418-445. · 0.60 Impact Factor
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SIAM J. Comput. 01/2008; 37:1429-1454.
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ABSTRACT: Consider $k$-colorings of the complete tree of depth $\ell$ and branching
factor $\Delta$. If we fix the coloring of the leaves, as $\ell$ tends to
$\infty$, for what range of $k$ is the root uniformly distributed over all $k$
colors? This corresponds to the threshold for uniqueness of the infinite-volume
Gibbs measure. It is straightforward to show the existence of colorings of the
leaves which ``freeze'' the entire tree when $k\le\Delta+1$. For
$k\geq\Delta+2$, Jonasson proved the root is ``unbiased'' for any fixed
coloring of the leaves and thus the Gibbs measure is unique. What happens for a
{\em typical} coloring of the leaves? When the leaves have a non-vanishing
influence on the root in expectation, over random colorings of the leaves,
reconstruction is said to hold. Non-reconstruction is equivalent to extremality
of the free-boundary Gibbs measure. When $k<\Delta/\ln{\Delta}$, it is
straightforward to show that reconstruction is possible and hence the measure
is not extremal.
We prove that for $C>1$ and $k =C\Delta/\ln{\Delta}$, that the Gibbs measure
is extremal in a strong sense: with high probability over the colorings of the
leaves the influence at the root decays exponentially fast with the depth of
the tree. Closely related results were also proven recently by Sly. The above
strong form of extremality implies that a local Markov chain that updates
constant sized blocks has inverse linear entropy constant and hence $O(N\log
N)$ mixing time where $N$ is the number of vertices of the tree.
11/2007;
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ABSTRACT: We present a new technique for constructing and analyzing couplings to bound the convergence rate of finite Markov chains. Our main theorem is a generalization of the path coupling theorem of Bubley and Dyer, allowing the defining partial couplings to have length determined by a random stopping time. Unlike the original path coupling theorem, our version can produce multistep (non-Markovian) couplings. Using our variable length path coupling theorem, we improve the upper bound on the mixing time of the Glauber dynamics for randomly sampling colorings. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007
Random Structures and Algorithms 09/2007; 31(3):251 - 272. · 1.03 Impact Factor
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ABSTRACT: We study Markov chains for randomly sampling $k$-colorings of a graph with
maximum degree $\Delta$. Our main result is a polynomial upper bound on the
mixing time of the single-site update chain known as the Glauber dynamics for
planar graphs when $k=\Omega(\Delta/\log{\Delta})$. Our results can be
partially extended to the more general case where the maximum eigenvalue of the
adjacency matrix of the graph is at most $\Delta^{1-\eps}$, for fixed $\eps >
0$.
The main challenge when $k \le \Delta + 1$ is the possibility of "frozen"
vertices, that is, vertices for which only one color is possible, conditioned
on the colors of its neighbors. Indeed, when $\Delta = O(1)$, even a typical
coloring can have a constant fraction of the vertices frozen. Our proofs rely
on recent advances in techniques for bounding mixing time using "local
uniformity" properties.
06/2007;
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ABSTRACT: We address phylogenetic reconstruction when the data is generated from a mixture distribution. Such topics have gained considerable attention in the biological community with the clear evidence of heterogeneity of mutation rates. In our work we consider data coming from a mixture of trees which share a common topology, but differ in their edge weights (i.e., branch lengths). We first show the pitfalls of popular methods, including maximum likelihood and Markov chain Monte Carlo algorithms. We then determine in which evolutionary models, reconstructing the tree topology, under a mixture distribution, is (im)possible. We prove that every model whose transition matrices can be parameterized by an open set of multilinear polynomials, either has non-identifiable mixture distributions, in which case reconstruction is impossible in general, or there exist linear tests which identify the topology. This duality theorem, relies on our notion of linear tests and uses ideas from convex programming duality. Linear tests are closely related to linear invariants, which were first introduced by Lake, and are natural from an algebraic geometry perspective.
Journal of Computational Biology 04/2007; 14(2):156-89. · 1.55 Impact Factor
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ABSTRACT: Different genes often have different phylogenetic histories. Even within regions having the same phylogenetic history, the mutation rates often vary. We investigate the prospects of phylogenetic reconstruction when all the characters are generated from the same tree topology, but the branch lengths vary (with possibly different tree shapes). Furthering work of Kolaczkowski and Thornton (2004, Nature 431: 980-984) and Chang (1996, Math. Biosci. 134: 189-216), we show examples where maximum likelihood (under a homogeneous model) is an inconsistent estimator of the tree. We then explore the prospects of phylogenetic inference under a heterogeneous model. In some models, there are examples where phylogenetic inference under any method is impossible - despite the fact that there is a common tree topology. In particular, there are nonidentifiable mixture distributions, i.e., multiple topologies generate identical mixture distributions. We address which evolutionary models have nonidentifiable mixture distributions and prove that the following duality theorem holds for most DNA substitution models. The model has either: (i) nonidentifiability - two different tree topologies can produce identical mixture distributions, and hence distinguishing between the two topologies is impossible; or (ii) linear tests - there exist linear tests which identify the common tree topology for character data generated by a mixture distribution. The theorem holds for models whose transition matrices can be parameterized by open sets, which includes most of the popular models, such as Tamura-Nei and Kimura's 2-parameter model. The duality theorem relies on our notion of linear tests, which are related to Lake's linear invariants.
Systematic Biology 03/2007; 56(1):113-24. · 10.23 Impact Factor
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ABSTRACT: We present a near-optimal reduction from approximately counting the cardinality of a discrete set to approximately sampling elements of the set. An important application of our work is to approximating the partition function $Z$ of a discrete system, such as the Ising model, matchings or colorings of a graph. The typical approach to estimating the partition function $Z(\beta^*)$ at some desired inverse temperature $\beta^*$ is to define a sequence, which we call a {\em cooling schedule}, $\beta_0=0<\beta_1<...<\beta_\ell=\beta^*$ where Z(0) is trivial to compute and the ratios $Z(\beta_{i+1})/Z(\beta_i)$ are easy to estimate by sampling from the distribution corresponding to $Z(\beta_i)$. Previous approaches required a cooling schedule of length $O^*(\ln{A})$ where $A=Z(0)$, thereby ensuring that each ratio $Z(\beta_{i+1})/Z(\beta_i)$ is bounded. We present a cooling schedule of length $\ell=O^*(\sqrt{\ln{A}})$. For well-studied problems such as estimating the partition function of the Ising model, or approximating the number of colorings or matchings of a graph, our cooling schedule is of length $O^*(\sqrt{n})$, which implies an overall savings of $O^*(n)$ in the running time of the approximate counting algorithm (since roughly $\ell$ samples are needed to estimate each ratio).
01/2007;
