arXiv subjects

Prasad Tetali

Publications and source records attributed to Prasad Tetali.

At least 55 records · Page 3Linked to original sources

arXiv2013

Combinatorial approach to the interpolation method and scaling limits in sparse random graphs

We establish the existence of free energy limits for several combinatorial models on Erdös-Rényi graph $\mathbb {G}(N,\lfloor cN\rfloor)$ and random $r$-regular graph $\mathbb {G}(N,r)$. For a variety of models, including independent sets, MAX-CUT, coloring and K-SAT, we prove that the free energy both at a positive and zero temperature, appropriately rescaled, converges to a limit as the size of the underlying graph diverges to infinity. In the zero temperature case, this is interpreted as the existence of the scaling limit for the corresponding combinatorial optimization problem. For example, as a special case we prove that the size of a largest independent set in these graphs, normalized by the number of nodes converges to a limit w.h.p. This resolves an open problem which was proposed by Aldous (Some open problems) as one of his six favorite open problems. It was also mentioned as an open problem in several other places: Conjecture 2.20 in Wormald [In Surveys in Combinatorics, 1999 (Canterbury) (1999) 239-298 Cambridge Univ. Press]; Bollobás and Riordan [Random Structures Algorithms 39 (2011) 1-38]; Janson and Thomason [Combin. Probab. Comput. 17 (2008) 259-264] and Aldous and Steele [In Probability on Discrete Structures (2004) 1-72 Springer].

math.PR↗

arXiv2013

The distribution of second degrees in the Buckley-Osthus random graph model

In this paper we consider a well-known generalization of the Barabási and Albert preferential attachment model - the Buckley-Osthus model. Buckley and Osthus proved that in this model the degree sequence has a power law distribution. As a natural (and arguably more interesting) next step, we study the second degrees of vertices. Roughly speaking, the second degree of a vertex is the number of vertices at distance two from this vertex. The distribution of second degrees is of interest because it is a good approximation of PageRank, where the importance of a vertex is measured by taking into account the popularity of its neighbors. We prove that the second degrees also obey a power law. More precisely, we estimate the expectation of the number of vertices with the second degree greater than or equal to k and prove the concentration of this random variable around its expectation using the now-famous Talagrand's concentration inequality over product spaces. As far as we know this is the only application of Talagrand's inequality to random web graphs, where the (preferential attachment) edges are not defined over a product distribution, making the application nontrivial, and requiring certain novelty.

math.PR↗

arXiv2013

Distributed Random Walks

Performing random walks in networks is a fundamental primitive that has found applications in many areas of computer science, including distributed computing. In this paper, we focus on the problem of sampling random walks efficiently in a distributed network and its applications. Given bandwidth constraints, the goal is to minimize the number of rounds required to obtain random walk samples. All previous algorithms that compute a random walk sample of length $\ell$ as a subroutine always do so naively, i.e., in $O(\ell)$ rounds. The main contribution of this paper is a fast distributed algorithm for performing random walks. We present a sublinear time distributed algorithm for performing random walks whose time complexity is sublinear in the length of the walk. Our algorithm performs a random walk of length $\ell$ in $\tilde{O}(\sqrt{\ell D})$ rounds ($\tilde{O}$ hides $\polylog{n}$ factors where $n$ is the number of nodes in the network) with high probability on an undirected network, where $D$ is the diameter of the network. For small diameter graphs, this is a significant improvement over the naive $O(\ell)$ bound. Furthermore, our algorithm is optimal within a poly-logarithmic factor as there exists a matching lower bound [Nanongkai et al. PODC 2011]. We further extend our algorithms to efficiently perform $k$ independent random walks in $\tilde{O}(\sqrt{k\ell D} + k)$ rounds. We also show that our algorithm can be applied to speedup the more general Metropolis-Hastings sampling. Our random walk algorithms can be used to speed up distributed algorithms in applications that use random walks as a subroutine, such as computing a random spanning tree and estimating mixing time and related parameters. Our algorithm is fully decentralized and can serve as a building block in the design of topologically-aware networks.

cs.DC↗

arXiv2012

Phase Coexistence and Slow Mixing for the Hard-Core Model on Z^2

In the hard-core model on a finite graph we are given a parameter lambda>0, and an independent set I arises with probability proportional to lambda^|I|. On infinite graphs a Gibbs distribution is defined as a suitable limit with the correct conditional probabilities. In the infinite setting we are interested in determining when this limit is unique and when there is phase coexistence, i.e., existence of multiple Gibbs states. On finite graphs we are interested in determining the mixing time of local Markov chains. On Z^2 it is conjectured that these problems are related and that both undergo a phase transition at some critical point lambda_c approx 3.79. For phase coexistence, much of the work to date has focused on the regime of uniqueness, with the best result being recent work of Restrepo et al. showing that there is a unique Gibbs state for all lambda < 2.3882. Here we give the first non-trivial result in the other direction, showing that there are multiple Gibbs states for all lambda > 5.3646. Our proof adds two significant innovations to the standard Peierls argument. First, building on the idea of fault lines introduced by Randall, we construct an event that distinguishes two boundary conditions and always has long contours associated with it, obviating the need to accurately enumerate short contours. Second, we obtain vastly improved bounds on the number of contours by relating them to a new class of self-avoiding walks on an oriented version of Z^2. We extend our characterization of fault lines to show that local Markov chains will mix slowly when lambda > 5.3646 on lattice regions with periodic (toroidal) boundary conditions and when lambda > 7.1031 with non-periodic (free) boundary conditions. The arguments here rely on a careful analysis that relates contours to taxi walks and represent a sevenfold improvement to the previously best known values of λ.

math.CO↗

arXiv2012

Phase transition for the mixing time of the Glauber dynamics for coloring regular trees

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 $Ω(n^{1/C-o_b(1)})$. 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.

math.PR↗

arXiv2012

On Mimicking Networks Representing Minimum Terminal Cuts

Given a capacitated undirected graph $G=(V,E)$ with a set of terminals $K \subset V$, a mimicking network is a smaller graph $H=(V_H,E_H)$ that exactly preserves all the minimum cuts between the terminals. Specifically, the vertex set of the sparsifier $V_H$ contains the set of terminals $K$ and for every bipartition $U, K-U $ of the terminals $K$, the size of the minimum cut separating $U$ from $K-U$ in $G$ is exactly equal to the size of the minimum cut separating $U$ from $K-U$ in $H$. This notion of a mimicking network was introduced by Hagerup, Katajainen, Nishimura and Ragde (1995) who also exhibited a mimicking network of size $2^{2^{k}}$ for every graph with $k$ terminals. The best known lower bound on the size of a mimicking network is linear in the number of terminals. More precisely, the best known lower bound is $k+1$ for graphs with $k$ terminals (Chaudhuri et al. 2000). In this work, we improve both the upper and lower bounds reducing the doubly-exponential gap between them to a single-exponential gap. Specifically, we obtain the following upper and lower bounds on mimicking networks: 1) Given a graph $G$, we exhibit a construction of mimicking network with at most $(|K|-1)$'th Dedekind number ($\approx 2^{{(k-1)} \choose {\lfloor {{(k-1)}/2} \rfloor}}$) of vertices (independent of size of $V$). Furthermore, we show that the construction is optimal among all {\it restricted mimicking networks} -- a natural class of mimicking networks that are obtained by clustering vertices together. 2) There exists graphs with $k$ terminals that have no mimicking network of size smaller than $2^{\frac{k-1}{2}}$. We also exhibit improved constructions of mimicking networks for trees and graphs of bounded tree-width.

cs.DS↗

arXiv2012

Displacement convexity of entropy and related inequalities on graphs

We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path and derive Prekopa-Leindler type inequalities, a Talagrand transport-entropy inequality, certain HWI type as well as log-Sobolev type inequalities in discrete settings. To illustrate through examples, we apply our results to the complete graph and to the hypercube for which our results are optimal -- by passing to the limit, we recover the classical log-Sobolev inequality for the standard Gaussian measure with the optimal constant.

math.PR↗

arXiv2012

On weighted graph homomorphisms

For given graphs $G$ and $H$, let $|Hom(G,H)|$ denote the set of graph homomorphisms from $G$ to $H$. We show that for any finite, $n$-regular, bipartite graph $G$ and any finite graph $H$ (perhaps with loops), $|Hom(G,H)|$ is maximum when $G$ is a disjoint union of $K_{n,n}$'s. This generalizes a result of J. Kahn on the number of independent sets in a regular bipartite graph. We also give the asymptotics of the logarithm of $|Hom(G,H)|$ in terms of a simply expressed parameter of $H$. We also consider weighted versions of these results which may be viewed as statements about the partition functions of certain models of physical systems with hard constraints.

math.CO↗

arXiv2012

Slow mixing of Glauber Dynamics for the hard-core model on regular bipartite graphs

Let $\gS=(V,E)$ be a finite, $d$-regular bipartite graph. For any $λ>0$ let $π_λ$ be the probability measure on the independent sets of $\gS$ in which the set $I$ is chosen with probability proportional to $λ^{|I|}$ ($π_λ$ is the {\em hard-core measure with activity $λ$ on $\gS$}). We study the Glauber dynamics, or single-site update Markov chain, whose stationary distribution is $π_λ$. We show that when $λ$ is large enough (as a function of $d$ and the expansion of subsets of single-parity of $V$) then the convergence to stationarity is exponentially slow in $|V(\gS)|$. In particular, if $\gS$ is the $d$-dimensional hypercube $\{0,1\}^d$ we show that for values of $λ$ tending to 0 as $d$ grows, the convergence to stationarity is exponentially slow in the volume of the cube. The proof combines a conductance argument with combinatorial enumeration methods.

math.CO↗

arXiv2012

Matchings and Independent Sets of a Fixed Size in Regular Graphs

We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size $\ell$ in a $d$-regular graph on $N$ vertices. For $\frac{2\ell}{N}$ bounded away from 0 and 1, the logarithm of the bound we obtain agrees in its leading term with the logarithm of the number of matchings of size $\ell$ in the graph consisting of $\frac{N}{2d}$ disjoint copies of the complete bipartite graph $K_{d,d}$. This provides asymptotic evidence for a conjecture of S. Friedland {\it et al.}. We also obtain an analogous result for independent sets of a fixed size in regular graphs, giving asymptotic evidence for a conjecture of J. Kahn. Our bounds on the number of matchings and independent sets of a fixed size are derived from bounds on the partition function (or generating polynomial) for matchings and independent sets.

math.CO↗

arXiv2012

Matching with Commitments

We consider the following stochastic optimization problem first introduced by Chen et al. in \cite{chen}. We are given a vertex set of a random graph where each possible edge is present with probability p_e. We do not know which edges are actually present unless we scan/probe an edge. However whenever we probe an edge and find it to be present, we are constrained to picking the edge and both its end points are deleted from the graph. We wish to find the maximum matching in this model. We compare our results against the optimal omniscient algorithm that knows the edges of the graph and present a 0.573 factor algorithm using a novel sampling technique. We also prove that no algorithm can attain a factor better than 0.898 in this model.

cs.DS↗

arXiv2012

Algorithms for Sampling 3-Orientations of Planar Triangulations

Given a planar triangulation, a 3-orientation is an orientation of the internal edges so all internal vertices have out-degree three. Each 3-orientation gives rise to a unique edge coloring known as a Schnyder wood that has proven powerful for various computing and combinatorics applications. We consider natural Markov chains for sampling uniformly from the set of 3-orientations. First, we study a "triangle-reversing" chain on the space of 3-orientations of a fixed triangulation that reverses the orientation of the edges around a triangle in each move. It was shown previously that this chain connects the state space and we show that (i) when restricted to planar triangulations of maximum degree six, the Markov chain is rapidly mixing, and (ii) there exists a triangulation with high degree on which this Markov chain mixes slowly. Next, we consider an "edge-flipping" chain on the larger state space consisting of 3-orientations of all planar triangulations on a fixed number of vertices. It was also shown previously that this chain connects the state space and we prove that the chain is always rapidly mixing. The triangle-reversing and edge-flipping Markov chains both arise in the context of sampling other combinatorial structures, such as Eulerian orientations and triangulations of planar point sets, so our results here may shed light on the mixing rate of these related chains as well.

cs.DS↗

arXiv2011

Many Sparse Cuts via Higher Eigenvalues

Cheeger's fundamental inequality states that any edge-weighted graph has a vertex subset $S$ such that its expansion (a.k.a. conductance) is bounded as follows: \[ ϕ(S) \defeq \frac{w(S,\bar{S})}{\min \set{w(S), w(\bar{S})}} \leq 2\sqrt{λ_2} \] where $w$ is the total edge weight of a subset or a cut and $λ_2$ is the second smallest eigenvalue of the normalized Laplacian of the graph. Here we prove the following natural generalization: for any integer $k \in [n]$, there exist $ck$ disjoint subsets $S_1, ..., S_{ck}$, such that \[ \max_i ϕ(S_i) \leq C \sqrt{λ_{k} \log k} \] where $λ_i$ is the $i^{th}$ smallest eigenvalue of the normalized Laplacian and $c<1,C>0$ are suitable absolute constants. Our proof is via a polynomial-time algorithm to find such subsets, consisting of a spectral projection and a randomized rounding. As a consequence, we get the same upper bound for the small set expansion problem, namely for any $k$, there is a subset $S$ whose weight is at most a $\bigO(1/k)$ fraction of the total weight and $ϕ(S) \le C \sqrt{λ_k \log k}$. Both results are the best possible up to constant factors. The underlying algorithmic problem, namely finding $k$ subsets such that the maximum expansion is minimized, besides extending sparse cuts to more than one subset, appears to be a natural clustering problem in its own right.

cs.DS↗

arXiv2011

Improved Mixing Condition on the Grid for Counting and Sampling Independent Sets

We study the hard-core model defined on independent sets, where each independent set I in a graph G is weighted proportionally to $λ^{|I|}$, for a positive real parameter $λ$. For large $λ$, 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 $λ_c(T_D)$ 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 $λ_c(T_D)$ 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 $λ_c(T_4)$. 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 $λ<2.3882$, improving upon the work of Weitz that held for $λ<27/16=1.6875$. 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.

math.PR↗

arXiv2011

Entropy and set cardinality inequalities for partition-determined functions

A new notion of partition-determined functions is introduced, and several basic inequalities are developed for the entropy of such functions of independent random variables, as well as for cardinalities of compound sets obtained using these functions. Here a compound set means a set obtained by varying each argument of a function of several variables over a set associated with that argument, where all the sets are subsets of an appropriate algebraic structure so that the function is well defined. On the one hand, the entropy inequalities developed for partition-determined functions imply entropic analogues of general inequalities of Plünnecke-Ruzsa type. On the other hand, the cardinality inequalities developed for compound sets imply several inequalities for sumsets, including for instance a generalization of inequalities proved by Gyarmati, Matolcsi and Ruzsa (2010). We also provide partial progress towards a conjecture of Ruzsa (2007) for sumsets in nonabelian groups. All proofs are elementary and rely on properly developing certain information-theoretic inequalities.

cs.IT↗

arXiv2011

Efficient Distributed Medium Access

Consider a wireless network of n nodes represented by a graph G=(V, E) where an edge (i,j) models the fact that transmissions of i and j interfere with each other, i.e. simultaneous transmissions of i and j become unsuccessful. Hence it is required that at each time instance a set of non-interfering nodes (corresponding to an independent set in G) access the wireless medium. To utilize wireless resources efficiently, it is required to arbitrate the access of medium among interfering nodes properly. Moreover, to be of practical use, such a mechanism is required to be totally distributed as well as simple. As the main result of this paper, we provide such a medium access algorithm. It is randomized, totally distributed and simple: each node attempts to access medium at each time with probability that is a function of its local information. We establish efficiency of the algorithm by showing that the corresponding network Markov chain is positive recurrent as long as the demand imposed on the network can be supported by the wireless network (using any algorithm). In that sense, the proposed algorithm is optimal in terms of utilizing wireless resources. The algorithm is oblivious to the network graph structure, in contrast with the so-called `polynomial back-off' algorithm by Hastad-Leighton-Rogoff (STOC '87, SICOMP '96) that is established to be optimal for the complete graph and bipartite graphs (by Goldberg-MacKenzie (SODA '96, JCSS '99)).

cs.NI↗

arXiv2010

Tight Bounds for Mixing of the Swendsen-Wang Algorithm at the Potts Transition Point

We study two widely used algorithms for the Potts model on rectangular subsets of the hypercubic lattice Z^d - heat bath dynamics and the Swendsen-Wang algorithm - and prove that, under certain circumstances, the mixing in these algorithms is torpid or slow. In particular, we show that for heat bath dynamics throughout the region of phase coexistence, and for the Swendsen-Wang algorithm at the transition point, the mixing time in a box of side length L with periodic boundary conditions has upper and lower bounds which are exponential in L^{d-1}. This work provides the first upper bound of this form for the Swendsen-Wang algorithm, and gives lower bounds for both algorithms which significantly improve the previous lower bounds that were exponential in L/(log L)^2.

math.PR↗

arXiv2010

How long does it take to catch a wild kangaroo?

We develop probabilistic tools for upper and lower bounding the expected time until two independent random walks on $\ZZ$ intersect each other. This leads to the first sharp analysis of a non-trivial Birthday attack, proving that Pollard's Kangaroo method solves the discrete logarithm problem $g^x=h$ on a cyclic group in expected time $(2+o(1))\sqrt{b-a}$ for an average $x\in_{uar}[a,b]$. Our methods also resolve a conjecture of Pollard's, by showing that the same bound holds when step sizes are generalized from powers of 2 to powers of any fixed $n$.

math.PR↗