arXiv subjects

David Gamarnik

Publications and source records attributed to David Gamarnik.

At least 91 records · Page 5Linked to original sources

arXiv2014

Giant Component in Random Multipartite Graphs with Given Degree Sequences

We study the problem of the existence of a giant component in a random multipartite graph. We consider a random multipartite graph with $p$ parts generated according to a given degree sequence $n_i^{\mathbf{d}}(n)$ which denotes the number of vertices in part $i$ of the multipartite graph with degree given by the vector $\mathbf{d}$. We assume that the empirical distribution of the degree sequence converges to a limiting probability distribution. Under certain mild regularity assumptions, we characterize the conditions under which, with high probability, there exists a component of linear size. The characterization involves checking whether the Perron-Frobenius norm of the matrix of means of a certain associated edge-biased distribution is greater than unity. We also specify the size of the giant component when it exists. We use the exploration process of Molloy and Reed combined with techniques from the theory of multidimensional Galton-Watson processes to establish this result.

math.PR↗

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

Steady-state $\mathit{GI}/\mathit{GI}/\mathit{n}$ queue in the Halfin-Whitt regime

We consider the FCFS $\mathit{GI}/\mathit{GI}/n$ queue in the so-called Halfin-Whitt heavy traffic regime. We prove that under minor technical conditions the associated sequence of steady-state queue length distributions, normalized by $n^{1/2}$, is tight. We derive an upper bound on the large deviation exponent of the limiting steady-state queue length matching that conjectured by Gamarnik and Momcilovic [Adv. in Appl. Probab. 40 (2008) 548-577]. We also prove a matching lower bound when the arrival process is Poisson. Our main proof technique is the derivation of new and simple bounds for the FCFS $\mathit{GI}/\mathit{GI}/n$ queue. Our bounds are of a structural nature, hold for all $n$ and all times $t\geq0$, and have intuitive closed-form representations as the suprema of certain natural processes which converge weakly to Gaussian processes. We further illustrate the utility of this methodology by deriving the first nontrivial bounds for the weak limit process studied in [Ann. Appl. Probab. 19 (2009) 2211-2269].

math.PR↗

arXiv2013

On the rate of convergence to stationarity of the M/M/N queue in the Halfin-Whitt regime

We prove several results about the rate of convergence to stationarity, that is, the spectral gap, for the M/M/n queue in the Halfin-Whitt regime. We identify the limiting rate of convergence to steady-state, and discover an asymptotic phase transition that occurs w.r.t. this rate. In particular, we demonstrate the existence of a constant $B^*\approx1.85772$ s.t. when a certain excess parameter $B\in(0,B^*]$, the error in the steady-state approximation converges exponentially fast to zero at rate $\frac{B^2}{4}$. For $B>B^*$, the error in the steady-state approximation converges exponentially fast to zero at a different rate, which is the solution to an explicit equation given in terms of special functions. This result may be interpreted as an asymptotic version of a phase transition proven to occur for any fixed n by van Doorn [Stochastic Monotonicity and Queueing Applications of Birth-death Processes (1981) Springer]. We also prove explicit bounds on the distance to stationarity for the M/M/n queue in the Halfin-Whitt regime, when $B<B^*$. Our bounds scale independently of $n$ in the Halfin-Whitt regime, and do not follow from the weak-convergence theory.

math.PR↗

arXiv2013

Limits of local algorithms over sparse random graphs

Local algorithms on graphs are algorithms that run in parallel on the nodes of a graph to compute some global structural feature of the graph. Such algorithms use only local information available at nodes to determine local aspects of the global structure, while also potentially using some randomness. Recent research has shown that such algorithms show significant promise in computing structures like large independent sets in graphs locally. Indeed the promise led to a conjecture by Hatami, \Lovasz and Szegedy \cite{HatamiLovaszSzegedy} that local algorithms may be able to compute maximum independent sets in (sparse) random $d$-regular graphs. In this paper we refute this conjecture and show that every independent set produced by local algorithms is multiplicative factor $1/2+1/(2\sqrt{2})$ smaller than the largest, asymptotically as $d\rightarrow\infty$. Our result is based on an important clustering phenomena predicted first in the literature on spin glasses, and recently proved rigorously for a variety of constraint satisfaction problems on random graphs. Such properties suggest that the geometry of the solution space can be quite intricate. The specific clustering property, that we prove and apply in this paper shows that typically every two large independent sets in a random graph either have a significant intersection, or have a nearly empty intersection. As a result, large independent sets are clustered according to the proximity to each other. While the clustering property was postulated earlier as an obstruction for the success of local algorithms, such as for example, the Belief Propagation algorithm, our result is the first one where the clustering property is used to formally prove limits on local algorithms.

math.PR↗

arXiv2013

Convergent sequences of sparse graphs: A large deviations approach

In this paper we introduce a new notion of convergence of sparse graphs which we call Large Deviations or LD-convergence and which is based on the theory of large deviations. The notion is introduced by "decorating" the nodes of the graph with random uniform i.i.d. weights and constructing random measures on $[0,1]$ and $[0,1]^2$ based on the decoration of nodes and edges. A graph sequence is defined to be converging if the corresponding sequence of random measures satisfies the Large Deviations Principle with respect to the topology of weak convergence on bounded measures on $[0,1]^d, d=1,2$. We then establish that LD-convergence implies several previous notions of convergence, namely so-called right-convergence, left-convergence, and partition-convergence. The corresponding large deviation rate function can be interpreted as the limit object of the sparse graph sequence. In particular, we can express the limiting free energies in terms of this limit object.

math.PR↗

arXiv2012

Belief Propagation for Min-cost Network Flow: Convergence and Correctness

Message passing type algorithms such as the so-called Belief Propagation algorithm have recently gained a lot of attention in the statistics, signal processing and machine learning communities as attractive algorithms for solving a variety of optimization and inference problems. As a decentralized, easy to implement and empirically successful algorithm, BP deserves attention from the theoretical standpoint, and here not much is known at the present stage. In order to fill this gap we consider the performance of the BP algorithm in the context of the capacitated minimum-cost network flow problem - the classical problem in the operations research field. We prove that BP converges to the optimal solution in the pseudo-polynomial time, provided that the optimal solution of the underlying problem is unique and the problem input is integral. Moreover, we present a simple modification of the BP algorithm which gives a fully polynomial-time randomized approximation scheme (FPRAS) for the same problem, which no longer requires the uniqueness of the optimal solution. This is the first instance where BP is proved to have fully-polynomial running time. Our results thus provide a theoretical justification for the viability of BP as an attractive method to solve an important class of optimization problems.

cs.DM↗

arXiv2012

Strong spatial mixing for list coloring of graphs

The property of spatial mixing and strong spatial mixing in spin systems has been of interest because of its implications on uniqueness of Gibbs measures on infinite graphs and efficient approximation of counting problems that are otherwise known to be #P hard. In the context of coloring, strong spatial mixing has been established for regular trees when $q \geq α^{*} Δ+ 1$ where $q$ the number of colors, $Δ$ is the degree and $α^* = 1.763..$ is the unique solution to $xe^{-1/x} = 1$. It has also been established for bounded degree lattice graphs whenever $q \geq α^* Δ- β$ for some constant $β$, where $Δ$ is the maximum vertex degree of the graph. The latter uses a technique based on recursively constructed coupling of Markov chains whereas the former is based on establishing decay of correlations on the tree. We establish strong spatial mixing of list colorings on arbitrary bounded degree triangle-free graphs whenever the size of the list of each vertex $v$ is at least $αΔ(v) + β$ where $Δ(v)$ is the degree of vertex $v$ and $α> α^*$ and $β$ is a constant that only depends on $α$. We do this by proving the decay of correlations via recursive contraction of the distance between the marginals measured with respect to a suitably chosen error function.

math.PR↗

arXiv2012

Right-convergence of sparse random graphs

The paper is devoted to the problem of establishing right-convergence of sparse random graphs. This concerns the convergence of the logarithm of number of homomorphisms from graphs or hyper-graphs $\G_N, N\ge 1$ to some target graph $W$. The theory of dense graph convergence, including random dense graphs, is now well understood, but its counterpart for sparse random graphs presents some fundamental difficulties. Phrased in the statistical physics terminology, the issue is the existence of the log-partition function limits, also known as free energy limits, appropriately normalized for the Gibbs distribution associated with $W$. In this paper we prove that the sequence of sparse \ER graphs is right-converging when the tensor product associated with the target graph $W$ satisfies certain convexity property. We treat the case of discrete and continuous target graphs $W$. The latter case allows us to prove a special case of Talagrand's recent conjecture (more accurately stated as level III Research Problem 6.7.2 in his recent book), concerning the existence of the limit of the measure of a set obtained from $\R^N$ by intersecting it with linearly in $N$ many subsets, generated according to some common probability law. Our proof is based on the interpolation technique, introduced first by Guerra and Toninelli and developed further in a series of papers. Specifically, Bayati et al establish the right-convergence property for Erdos-Renyi graphs for some special cases of $W$. In this paper most of the results in this paper follow as a special case of our main theorem.

math.PR↗

arXiv2011

Multiclass multiserver queueing system in the Halfin-Whitt heavy traffic regime. Asymptotics of the stationary distribution

We consider a heterogeneous queueing system consisting of one large pool of $O(r)$ identical servers, where $r\to\infty$ is the scaling parameter. The arriving customers belong to one of several classes which determines the service times in the distributional sense. The system is heavily loaded in the Halfin-Whitt sense, namely the nominal utilization is $1-a/\sqrt{r}$ where $a>0$ is the spare capacity parameter. Our goal is to obtain bounds on the steady state performance metrics such as the number of customers waiting in the queue $Q^r(\infty)$. While there is a rich literature on deriving process level (transient) scaling limits for such systems, the results for steady state are primarily limited to the single class case. This paper is the first one to address the case of heterogeneity in the steady state regime. Moreover, our results hold for any service policy which does not admit server idling when there are customers waiting in the queue. We assume that the interarrival and service times have exponential distribution, and that customers of each class may abandon while waiting in the queue at a certain rate (which may be zero). We obtain upper bounds of the form $O(\sqrt{r})$ on both $Q^r(\infty)$ and the number of idle servers. The bounds are uniform w.r.t. parameter $r$ and the service policy. In particular, we show that $\limsup_r E \exp(θr^{-1/2}Q^r(\infty))<\infty$. Therefore, the sequence $r^{-1/2}Q^r(\infty)$ is tight and has a uniform exponential tail bound. We further consider the system with strictly positive abandonment rates, and show that in this case every weak limit $\hat{Q}(\infty)$ of $r^{-1/2}Q^r(\infty)$ has a sub-Gaussian tail. Namely $E[\exp(θ(\hat{Q}(\infty))^2)]<\infty$, for some $θ>0$.

math.PR↗

arXiv2010

Performance Analysis of Queueing Networks via Robust Optimization

Performance analysis of queueing networks is one of the most challenging areas of queueing theory. Barring very specialized models such as product-form type queueing networks, there exist very few results which provide provable non-asymptotic upper and lower bounds on key performance measures. In this paper we propose a new performance analysis method, which is based on the robust optimization. The basic premise of our approach is as follows: rather than assuming that the stochastic primitives of a queueing model satisfy certain probability laws, such as, for example, i.i.d. interarrival and service times distributions, we assume that the underlying primitives are deterministic and satisfy the implications of such probability laws. These implications take the form of simple linear constraints, namely, those motivated by the Law of the Iterated Logarithm (LIL). Using this approach we are able to obtain performance bounds on some key performance measures. Furthermore, these performance bounds imply similar bounds in the underlying stochastic queueing models. We demonstrate our approach on two types of queueing networks: a) Tandem Single Class queue- ing network and b) Multiclass Single Server queueing network. In both cases, using the proposed robust optimization approach, we are able to obtain explicit upper bounds on some steady-state performance measures. For example, for the case of TSC system we obtain a bound of the form $\frac{C}{1-ρ} \ln \ln(1/(1-ρ))$ on the expected steady-state sojourn time, where C is an explicit constant and $ρ$ is the bottleneck traffic intensity. This qualitatively agrees with the correct heavy traffic scaling of this performance measure up to the $ln ln(1/(1-ρ))$ correction factor.

math.OC↗

arXiv2010

Stability of Skorokhod problem is undecidable

Skorokhod problem arises in studying Reflected Brownian Motion (RBM) on an non-negative orthant, specifically in the context of queueing networks in the heavy traffic regime. One of the key problems is identifying conditions for stability of a Skorokhod problem, defined as the property that trajectories are attracted to the origin. The stability conditions are known in dimension up to three, but not for general dimensions. In this paper we explain the fundamental difficulties encountered in trying to establish stability conditions for general dimensions. We prove that stability of Skorokhod problem is an undecidable property when the starting state is a part of the input. Namely, there does not exist an algorithm (a constructive procedure) for identifying stable Skorokhod problem in general dimensions.

math.PR↗

arXiv2009

Correlation Decay in Random Decision Networks

We consider a decision network on an undirected graph in which each node corresponds to a decision variable, and each node and edge of the graph is associated with a reward function whose value depends only on the variables of the corresponding nodes. The goal is to construct a decision vector which maximizes the total reward. This decision problem encompasses a variety of models, including maximum-likelihood inference in graphical models (Markov Random Fields), combinatorial optimization on graphs, economic team theory and statistical physics. The network is endowed with a probabilistic structure in which costs are sampled from a distribution. Our aim is to identify sufficient conditions to guarantee average-case polynomiality of the underlying optimization problem. We construct a new decentralized algorithm called Cavity Expansion and establish its theoretical performance for a variety of models. Specifically, for certain classes of models we prove that our algorithm is able to find near optimal solutions with high probability in a decentralized way. The success of the algorithm is based on the network exhibiting a correlation decay (long-range independence) property. Our results have the following surprising implications in the area of average case complexity of algorithms. Finding the largest independent (stable) set of a graph is a well known NP-hard optimization problem for which no polynomial time approximation scheme is possible even for graphs with largest connectivity equal to three, unless P=NP. We show that the closely related maximum weighted independent set problem for the same class of graphs admits a PTAS when the weights are i.i.d. with the exponential distribution. Namely, randomization of the reward function turns an NP-hard problem into a tractable one.

math.PR↗

arXiv2009

On deciding stability of multiclass queueing networks under buffer priority scheduling policies

One of the basic properties of a queueing network is stability. Roughly speaking, it is the property that the total number of jobs in the network remains bounded as a function of time. One of the key questions related to the stability issue is how to determine the exact conditions under which a given queueing network operating under a given scheduling policy remains stable. While there was much initial progress in addressing this question, most of the results obtained were partial at best and so the complete characterization of stable queueing networks is still lacking. In this paper, we resolve this open problem, albeit in a somewhat unexpected way. We show that characterizing stable queueing networks is an algorithmically undecidable problem for the case of nonpreemptive static buffer priority scheduling policies and deterministic interarrival and service times. Thus, no constructive characterization of stable queueing networks operating under this class of policies is possible. The result is established for queueing networks with finite and infinite buffer sizes and possibly zero service times, although we conjecture that it also holds in the case of models with only infinite buffers and nonzero service times. Our approach extends an earlier related work [Math. Oper. Res. 27 (2002) 272--293] and uses the so-called counter machine device as a reduction tool.

math.PR↗

arXiv2008

Sequential cavity method for computing free energy and surface pressure

We propose a new method for the problems of computing free energy and surface pressure for various statistical mechanics models on a lattice $\Z^d$. Our method is based on representing the free energy and surface pressure in terms of certain marginal probabilities in a suitably modified sublattice of $\Z^d$. Then recent deterministic algorithms for computing marginal probabilities are used to obtain numerical estimates of the quantities of interest. The method works under the assumption of Strong Spatial Mixing (SSP), which is a form of a correlation decay. We illustrate our method for the hard-core and monomer-dimer models, and improve several earlier estimates. For example we show that the exponent of the monomer-dimer coverings of $\Z^3$ belongs to the interval $[0.78595,0.78599]$, improving best previously known estimate of (approximately) $[0.7850,0.7862]$ obtained in \cite{FriedlandPeled},\cite{FriedlandKropLundowMarkstrom}. Moreover, we show that given a target additive error $ε>0$, the computational effort of our method for these two models is $(1/ε)^{O(1)}$ \emph{both} for free energy and surface pressure. In contrast, prior methods, such as transfer matrix method, require $\exp\big((1/ε)^{O(1)}\big)$ computation effort.

math.PR↗

arXiv2008

Randomized greedy algorithms for independent sets and matchings in regular graphs: Exact results and finite girth corrections

We derive new results for the performance of a simple greedy algorithm for finding large independent sets and matchings in constant degree regular graphs. We show that for $r$-regular graphs with $n$ nodes and girth at least $g$, the algorithm finds an independent set of expected cardinality $f(r)n - O\big(\frac{(r-1)^{\frac{g}{2}}}{\frac{g}{2}!} n\big)$, where $f(r)$ is a function which we explicitly compute. A similar result is established for matchings. Our results imply improved bounds for the size of the largest independent set in these graphs, and provide the first results of this type for matchings. As an implication we show that the greedy algorithm returns a nearly perfect matching when both the degree $r$ and girth $g$ are large. Furthermore, we show that the cardinality of independent sets and matchings produced by the greedy algorithm in \emph{arbitrary} bounded degree graphs is concentrated around the mean. Finally, we analyze the performance of the greedy algorithm for the case of random i.i.d. weighted independent sets and matchings, and obtain a remarkably simple expression for the limiting expected values produced by the algorithm. In fact, all the other results are obtained as straightforward corollaries from the results for the weighted case.

cs.DM↗

arXiv2008

Steady-state analysis of a multi-server queue in the Halfin-Whitt regime

We consider a multi-server queue in the Halfin-Whitt regime: as the number of servers $n$ grows without a bound, the utilization approaches 1 from below at the rate $Θ(1/\sqrt{n})$. Assuming that the service time distribution is lattice-valued with a finite support, we characterize the limiting stationary queue length distribution in terms of the stationary distribution of an explicitly constructed Markov chain. Furthermore, we obtain an explicit expression for the critical exponent for the moment generating function of a limiting (scaled) steady-state queue length. This exponent has a compact representation in terms of three parameters: the amount of spare capacity and the coefficients of variation of interarrival and service times. Interestingly, it matches an analogous exponent corresponding to a single-server queue in the conventional heavy-traffic regime.

math.PR↗

arXiv2007

Correlation decay and deterministic FPTAS for counting list-colorings of a graph

We propose a deterministic algorithm for approximately counting the number of list colorings of a graph. Under the assumption that the graph is triangle free, the size of every list is at least $αΔ$, where $α$ is an arbitrary constant bigger than $α^{**}=2.8432...$, and $Δ$ is the maximum degree of the graph, we obtain the following results. For the case when the size of the each list is a large constant, we show the existence of a \emph{deterministic} FPTAS for computing the total number of list colorings. The same deterministic algorithm has complexity $2^{O(\log^2 n)}$, without any assumptions on the sizes of the lists, where $n$ is the instance size. We further extend our method to a discrete Markov random field (MRF) model. Under certain assumptions relating the size of the alphabet, the degree of the graph and the interacting potentials we again construct a deterministic FPTAS for computing the partition function of a MRF. Our results are not based on the most powerful existing counting technique -- rapidly mixing Markov chain method. Rather we build upon concepts from statistical physics, in particular, the decay of correlation phenomena and its implication for the uniqueness of Gibbs measures in infinite graphs. This approach was proposed in two recent papers \cite{BandyopadhyayGamarnikCounting} and \cite{weitzCounting}. The principle insight of this approach is that the correlation decay property can be established with respect to certain \emph{computation tree}, as opposed to the conventional correlation decay property with respect to graph theoretic neighborhoods of a given node. This allows truncation of computation at a logarithmic depth in order to obtain polynomial accuracy in polynomial time.

math.CO↗