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We study support recovery for a $k \times k$ principal submatrix with elevated mean $\lambda/N$, hidden in an $N\times N$ symmetric mean zero Gaussian matrix. Here $\lambda>0$ is a universal constant, and we assume $k = N \rho$ for some constant $\rho \in (0,1)$. We establish that {there exists a constant $C>0$ such that} the MLE recovers a constant proportion of the hidden submatrix if $\lambda {\geq C} \sqrt{\frac{1}{\rho} \log \frac{1}{\rho}}$, {while such recovery is information theoretically impossible if $\lambda = o( \sqrt{\frac{1}{\rho} \log \frac{1}{\rho}} )$}. The MLE is computationally intractable in general, and in fact, for $\rho>0$ sufficiently small, this problem is conjectured to exhibit a \emph{statistical-computational gap}. To provide rigorous evidence for this, we study the likelihood landscape for this problem, and establish that for some $\varepsilon>0$ and $\sqrt{\frac{1}{\rho} \log \frac{1}{\rho} } \ll \lambda \ll \frac{1}{\rho^{1/2 + \varepsilon}}$, the problem exhibits a variant of the \emph{Overlap-Gap-Property (OGP)}. As a direct consequence, we establish that a family of local MCMC based algorithms do not achieve optimal recovery. Finally, we establish that for $\lambda > 1/\rho$, a simple spectral method recovers a constant proportion of the hidden submatrix., Comment: 42 pages, 1 figure

We consider a heterogeneous distributed service system, consisting of $n$ servers with unknown and possibly different processing rates. Jobs with unit mean and independent processing times arrive as a renewal process of rate $\lambda n$, with $0, Comment: arXiv admin note: text overlap with arXiv:1807.02882

We consider the following distributed service model: jobs with unit mean, general distribution, and independent processing times arrive as a renewal process of rate $\lambda n$, with $0, Comment: 44 pages

We consider the problem of finding a two-layer neural network with sigmoid, rectified linear unit, or binary step activation functions that “fits” a training data set as accurately as possible as quantified by the training error; and study the following question: does a low training error guarantee that the norm of the output layer (outer norm) itself is small? We address this question for the case of non-negative output weights. Using a simple covering number argument, we establish that under quite mild distributional assumptions on the input/label pairs; any such network achieving a small training error on polynomially many data necessarily has a well-controlled outer norm. Notably, our results (a) have a good sample complexity, (b) are independent of the number of hidden units, (c) are oblivious to the training algorithm; and (d) require quite mild assumptions on the data (in particular the input vector $X$ ∊ Rd need not have independent coordinates). We then show how our bounds can be leveraged to yield generalization guarantees for such networks.

We establish the average-case hardness of the algorithmic problem of exact computation of the partition function associated with the Sherrington–Kirkpatrick model of spin glasses with Gaussian couplings and random external field. In particular, we establish that unless P=#P, there does not exist a polynomial-time algorithm to exactly compute the partition function on average. This is done by showing that if there exists a polynomial time algorithm, which exactly computes the partition function for inverse polynomial fraction (1/nO(1)) of all inputs, then there is a polynomial time algorithm, which exactly computes the partition function for all inputs, with high probability, yielding P=#P. The computational model that we adopt is finite-precision arithmetic, where the algorithmic inputs are truncated first to a certain level N of digital precision. The ingredients of our proof include the random and downward self-reducibility of the partition function with random external field; an argument of Cai et al. (In STACS 99 (Trier) (1999) 90–99 Springer) for establishing the average-case hardness of computing the permanent of a matrix; a list-decoding algorithm of Sudan (In 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996) (1996) 164–172 IEEE Comput. Soc. Press), for reconstructing polynomials intersecting a given list of numbers at sufficiently many points; and near-uniformity of the log-normal distribution, modulo a large prime p. To the best of our knowledge, our result is the first one establishing a provable hardness of a model arising in the field of spin glasses. Furthermore, we extend our result to the same problem under a different real-valued computational model, for example, using a Blum–Shub–Smale machine (In [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science (1988) 387–397 IEEE) operating over real-valued inputs. We establish that, if there exists a polynomial time algorithm which exactly computes the partition function for 34+1nO(1) fraction of all inputs, then there exists a polynomial time algorithm, which exactly computes the partition function for all inputs, with high probability, yielding P=#P. Our proof uses the random self-reducibility of the partition function, together with a control over the total variation distance for log-normal random variables in presence of a convex perturbation, and the Berlekamp–Welch algorithm.

We consider the algorithmic problem of finding a near ground state (near optimal solution) of a $p$-spin model. We show that for a class of algorithms broadly defined as Approximate Message Passing (AMP), the presence of the Overlap Gap Property (OGP), appropriately defined, is a barrier. We conjecture that when $p\ge 4$ the model does indeed exhibits OGP (and prove it for the space of binary solutions). Assuming the validity of this conjecture, as an implication, the AMP fails to find near ground states in these models, per our result. We extend our result to the problem of finding pure states by means of Thouless, Anderson and Palmer (TAP) based iterations, which is yet another example of AMP type algorithms. We show that such iterations fail to find pure states approximately, subject to the conjecture that the space of pure states exhibits the OGP, appropriately stated, when $p\ge 4$., 27 pages

We consider the problem of finding nearly optimal solutions of optimization problems with random objective functions. Such problems arise widely in the theory of random graphs, theoretical computer science, and statistical physics. Two concrete problems we consider are (a) optimizing the Hamiltonian of a spherical or Ising p-spin glass model, and (b) finding a large independent set in a sparse Erdos-Renyi graph. Two families of algorithms are considered: (a) low-degree polynomials of the input—a general framework that captures methods such as approximate message passing and local algorithms on sparse graphs, among others; and (b) the Langevin dynamics algorithm, a canonical Monte Carlo analogue of the gradient descent algorithm (applicable only for the spherical p-spin glass Hamiltonian). We show that neither family of algorithms can produce nearly optimal solutions with high probability. Our proof uses the fact that both models are known to exhibit a variant of the overlap gap property (OGP) of near-optimal solutions. Specifically, for both models, every two solutions whose objective values are above a certain threshold are either close or far from each other. The crux of our proof is the stability of both algorithms: a small perturbation of the input induces a small perturbation of the output. By an interpolation argument, such a stable algorithm cannot overcome the OGP barrier. The stability of the Langevin dynamics is an immediate consequence of the well-posedness of stochastic differential equations. The stability of low-degree polynomials is established using concepts from Gaussian and Boolean Fourier analysis, including noise sensitivity, hypercontractivity, and total influence.

We establish the average-case hardness of the algorithmic problem of exactly computing the partition function of the Sherrington-Kirkpatrick model of spin glasses with Gaussian couplings. In particular, we establish that unless P=#P, there does not exist a polynomial-time algorithm to exactly compute this object on average. This is done by showing that if there exists a polynomial-time algorithm exactly computing the partition function for a certain fraction of all inputs, then there is a polynomial-time algorithm exactly computing this object for all inputs, with high probability, yielding P =#P. Our results cover both finite-precision arithmetic as well as the real-valued computational models. The ingredients of our proofs include Berlekamp-Welch algorithm, a list-decoding algorithm by Sudan for reconstructing a polynomial from its noisy samples, near-uniformity of log-normal distribution modulo a large prime; and a control over total variation distance for log-normal distribution under convex perturbation. To the best of our knowledge, this is the first average-case hardness result pertaining a statistical physics model with random parameters.

Matrices $\Phi\in\R^{n\times p}$ satisfying the Restricted Isometry Property (RIP) are an important ingredient of the compressive sensing methods. While it is known that random matrices satisfy the RIP with high probability even for $n=\log^{O(1)}p$, the explicit construction of such matrices defied the repeated efforts, and the most known approaches hit the so-called $\sqrt{n}$ sparsity bottleneck. The notable exception is the work by Bourgain et al \cite{bourgain2011explicit} constructing an $n\times p$ RIP matrix with sparsity $s=\Theta(n^{{1\over 2}+\epsilon})$, but in the regime $n=\Omega(p^{1-\delta})$. In this short note we resolve this open question in a sense by showing that an explicit construction of a matrix satisfying the RIP in the regime $n=O(\log^2 p)$ and $s=\Theta(n^{1\over 2})$ implies an explicit construction of a three-colored Ramsey graph on $p$ nodes with clique sizes bounded by $O(\log^2 p)$ -- a question in the extremal combinatorics which has been open for decades.

We focus on the high-dimensional linear regression problem, where the algorithmic goal is to efficiently infer an unknown feature vector $\beta^*\in\mathbb{R}^p$ from its linear measurements, using a small number $n$ of samples. Unlike most of the literature, we make no sparsity assumption on $\beta^*$, but instead adopt a different regularization: In the noiseless setting, we assume $\beta^*$ consists of entries, which are either rational numbers with a common denominator $Q\in\mathbb{Z}^+$ (referred to as $Q$-rationality); or irrational numbers supported on a rationally independent set of bounded cardinality, known to learner; collectively called as the mixed-support assumption. Using a novel combination of the PSLQ integer relation detection, and LLL lattice basis reduction algorithms, we propose a polynomial-time algorithm which provably recovers a $\beta^*\in\mathbb{R}^p$ enjoying the mixed-support assumption, from its linear measurements $Y=X\beta^*\in\mathbb{R}^n$ for a large class of distributions for the random entries of $X$, even with one measurement $(n=1)$. In the noisy setting, we propose a polynomial-time, lattice-based algorithm, which recovers a $\beta^*\in\mathbb{R}^p$ enjoying $Q$-rationality, from its noisy measurements $Y=X\beta^*+W\in\mathbb{R}^n$, even with a single sample $(n=1)$. We further establish for large $Q$, and normal noise, this algorithm tolerates information-theoretically optimal level of noise. We then apply these ideas to develop a polynomial-time, single-sample algorithm for the phase retrieval problem. Our methods address the single-sample $(n=1)$ regime, where the sparsity-based methods such as LASSO and Basis Pursuit are known to fail. Furthermore, our results also reveal an algorithmic connection between the high-dimensional linear regression problem, and the integer relation detection, randomized subset-sum, and shortest vector problems., Comment: 56 pages. Parts of the material of this manuscript were presented at NeurIPS 2018, and ISIT 2019. This submission subsumes the content of arXiv:1803.06716

We study high-dimensional linear regression problem without sparsity, and address the question of efficient recovery with small number of measurements. We propose an algorithm which efficiently recovers an unknown feature vector β∗ ∈ ℝp from its linear measurements Y = Xβ∗ in polynomially many steps, with high probability (as p → ∞), even with a single measurement, provided elements of β∗ are supported on a rationally independent set of at most polynomial in p size known to learner. We use a combination of PSLQ integer relation and LLL lattice basis reduction algorithms to achieve our goal. We then apply our ideas to develop an efficient, single-sample algorithm for the phase retrieval problem, where ${\beta ^ * } \in {\mathbb{C}^p}$ is to be recovered from magnitude-only observations Y = |〈X, β∗〉|.

We formulate a continuous version of the well known discrete hardcore (or independent set) model on a locally finite graph, parameterized by the so-called activity parameter $\lambda > 0$. In this version, the state or "spin value" $x_u$ of any node $u$ of the graph lies in the interval $[0,1]$, the hardcore constraint $x_u + x_v \leq 1$ is satisfied for every edge $(u,v)$ of the graph, and the space of feasible configurations is given by a convex polytope. When the graph is a regular tree, we show that there is a unique Gibbs measure associated to each activity parameter $\lambda>0$. Our result shows that, in contrast to the standard discrete hardcore model, the continuous hardcore model does not exhibit a phase transition on the infinite regular tree. We also consider a family of continuous models that interpolate between the discrete and continuous hardcore models on a regular tree when $\lambda = 1$ and show that each member of the family has a unique Gibbs measure, even when the discrete model does not. In each case, the proof entails the analysis of an associated Hamiltonian dynamical system that describes a certain limit of the marginal distribution at a node. Furthermore, given any sequence of regular graphs with fixed degree and girth diverging to infinity, we apply our results to compute the asymptotic limit of suitably normalized volumes of the corresponding sequence of convex polytopes of feasible configurations. In particular, this yields an approximation for the partition function of the continuous hard core model on a regular graph with large girth in the case $\lambda = 1$., Comment: 34 pages, 1 figure

We show that in random $K$-uniform hypergraphs of constant average degree, for even $K \geq 4$, local algorithms defined as factors of i.i.d. can not find nearly maximal cuts, when the average degree is sufficiently large. These algorithms have been used frequently to obtain lower bounds for the max-cut problem on random graphs, but it was not known whether they could be successful in finding nearly maximal cuts. This result follows from the fact that the overlap of any two nearly maximal cuts in such hypergraphs does not take values in a certain non-trivial interval - a phenomenon referred to as the overlap gap property - which is proved by comparing diluted models with large average degree with appropriate fully connected spin glass models and showing the overlap gap property in the latter setting., Final version; to appear in Ann. Probab

We formalize the class of “sequential local algorithms" and show that these algorithms fail to find satisfying assignments on random instances of the “Not-All-Equal-$K$-SAT” (NAE-$K$-SAT) problem if the number of message passing iterations is bounded by a function moderately growing in the number of variables and if the clause-to-variable ratio is above $(1+o_K(1)){2^{K-1}\over K}\ln^2 K$ for sufficiently large $K$. Sequential local algorithms are those that iteratively set variables based on some local information and/or local randomness and then recurse on the reduced instance. Our model captures some weak abstractions of natural algorithms such as Survey Propagation (SP)-guided as well as Belief Propagation (BP)-guided decimation algorithms---two widely studied message-passing--based algorithms---when the number of message-passing rounds in these algorithms is restricted to be growing only moderately with the number of variables. The approach underlying our paper is based on an intricate geometry of th...

We consider the problem of estimating the size of a maximum cut (Max-Cut problem) in a random Erd\H{o}s-R\'{e}nyi graph on $n$ nodes and $\lfloor cn \rfloor$ edges. It is shown in Coppersmith et al. ~\cite{Coppersmith2004} that the size of the maximum cut in this graph normalized by the number of nodes belongs to the asymptotic region $[c/2+0.37613\sqrt{c},c/2+0.58870\sqrt{c}]$ with high probability (w.h.p.) as $n$ increases, for all sufficiently large $c$. In this paper we improve both upper and lower bounds by introducing a novel bounding technique. Specifically, we establish that the size of the maximum cut normalized by the number of nodes belongs to the interval $[c/2+0.47523\sqrt{c},c/2+0.55909\sqrt{c}]$ w.h.p. as $n$ increases, for all sufficiently large $c$. Instead of considering the expected number of cuts achieving a particular value as is done in the application of the first moment method, we observe that every maximum size cut satisfies a certain local optimality property, and we compute the expected number of cuts with a given value satisfying this local optimality property. Estimating this expectation amounts to solving a rather involved two dimensional large deviations problem. We solve this underlying large deviation problem asymptotically as $c$ increases and use it to obtain an improved upper bound on the Max-Cut value. The lower bound is obtained by application of the second moment method, coupled with the same local optimality constraint, and is shown to work up to the stated lower bound value $c/2+0.47523\sqrt{c}$. It is worth noting that both bounds are stronger than the ones obtained by standard first and second moment methods. Finally, we also obtain an improved lower bound of $1.36000n$ on the Max-Cut for the random cubic graph or any cubic graph with large girth, improving the previous best bound of $1.33773n$., Comment: To appear in Random Structures & Algorithms

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. Research over the years has shown that such algorithms can be surprisingly powerful in terms of computing structures like large independent sets in graphs locally. These algorithms have also been implicitly considered in the work on graph limits, where a conjecture due to Hatami, Lovász and Szegedy [17] implied that local algorithms may be able to compute near-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 smaller that the largest one by a multiplicative factor of at least 1/2+1/(2√2) ≈ .853, asymptotically as d → ∞. 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., National Science Foundation (U.S.) (NSF grants CMMI-1031332)

We consider the problem of finding a $k\times k$ submatrix of an $n\times n$ matrix with i.i.d. standard Gaussian entries, which has a large average entry. It was shown earlier by Bhamidi et al. that the largest average value of such a matrix is $2\sqrt{\log n/k}$ with high probability. In the same paper an evidence was provided that a natural greedy algorithm called Largest Average Submatrix ($\LAS$) should produce a matrix with average entry approximately $\sqrt{2}$ smaller. In this paper we show that the matrix produced by the $\LAS$ algorithm is indeed $\sqrt{2\log n/k}$ w.h.p. Then by drawing an analogy with the problem of finding cliques in random graphs, we propose a simple greedy algorithm which produces a $k\times k$ matrix with asymptotically the same average value. Since the greedy algorithm is the best known algorithm for finding cliques in random graphs, it is tempting to believe that beating the factor $\sqrt{2}$ performance gap suffered by both algorithms might be very challenging. Surprisingly, we show the existence of a very simple algorithm which produces a matrix with average value $(4/3)\sqrt{2\log n/k}$. To get an insight into the algorithmic hardness of this problem, and motivated by methods originating in the theory of spin glasses, we conduct the so-called expected overlap analysis of matrices with average value asymptotically $\alpha\sqrt{2\log n/k}$. The overlap corresponds to the number of common rows and common columns for pairs of matrices achieving this value. We discover numerically an intriguing phase transition at $\alpha^*\approx 1.3608..$: when $\alpha\alpha^*$. We conjecture that $\alpha>\alpha^*$ marks the onset of the algorithmic hardness., Comment: 38 pages 6 figures

The Skorokhod problem arises in studying reflected Brownian motion (RBM) and the associated fluid model on the non-negative orthant. This problem specifically arises in the context of queueing networks in the heavy traffic regime. One of the key problems is that of determining, for a given deterministic Skorokhod problem, whether for every initial condition all solutions of the problem staring from the initial condition are attracted to the origin. The conditions for this attraction property, called stability, 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 the existence of dimension d[subscript 0] such that stability of the Skorokhod problem associated with a fluid model of an RBM in dimension d ≥ d[subscript 0] 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 dimensions d ≥ d[subscript 0]., National Science Foundation (U.S.) (Grant CMMI-0726733)

We consider a heterogeneous queueing system consisting of one large pool of O(r) identical servers, where r→∞ 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/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 [superscript r] (∞). 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(r√) on both Q [superscript r] (∞) and the number of idle servers. The bounds are uniform w.r.t. parameter r and the service policy. In particular, we show that lim sup[subscript r]Eexp(θr[superscript −1/2)Q[superscript r](∞))0., National Science Foundation (U.S.) (Grant CMMI-0726733)

Consider algorithms with unbounded computation time that probe the entries of the adjacency matrix of an $n$ vertex graph, and need to output a clique. We show that if the input graph is drawn at random from $G_{n,\frac{1}{2}}$ (and hence is likely to have a clique of size roughly $2\log n$), then for every $\delta < 2$ and constant $\ell$, there is an $\alpha < 2$ (that may depend on $\delta$ and $\ell$) such that no algorithm that makes $n^{\delta}$ probes in $\ell$ rounds is likely (over the choice of the random graph) to output a clique of size larger than $\alpha \log n$., Comment: 15 pages

We consider the problem of computing the permanent of a 0,1n by n matrix. For a class of matrices corresponding to constant degree expanders we construct a deterministic polynomial time approximation algorithm to within a multiplicative factor ([email protected])^n, for arbitrary @e>0. This is an improvement over the best known approximation factor e^n obtained in Linial, Samorodnitsky and Wigderson (2000) [9], though the latter result was established for arbitrary non-negative matrices. Our results use a recently developed deterministic approximation algorithm for counting partial matchings of a graph (Bayati, Gamarnik, Katz, Nair and Tetali (2007) [2]) and Jerrum-Vazirani method (Jerrum and Vazirani (1996) [8]) of approximating permanent by near perfect matchings.

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 forr-regular graphs withnnodes and girth at leastg, the algorithm finds an independent set of expected cardinalitywheref(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 degreerand girthgare large. Furthermore, we show that the cardinality of independent sets and matchings produced by the greedy algorithm inarbitrarybounded-degree graphs is concentrated around the mean. Finally, we analyse 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.

This paper studies the time constant for first-passage percolation, and the Vickrey-Clarke-Groves (VCG) payment, for the shortest path on a ladder graph (a width-2 strip) with random edge costs, treating both in a unified way based on recursive distributional equations. For first-passage percolation where the edge costs are independent Bernoulli random variables we find the time constant exactly; it is a rational function of the Bernoulli parameter. For first-passage percolation where the edge costs are uniform random variables we present a reasonably efficient means for obtaining arbitrarily close upper and lower bounds. Using properties of Harris chains we also show that the incremental cost to advance through the medium has a unique stationary distribution, and we compute stochastic lower and upper bounds. We rely on no special properties of the uniform distribution: the same methods could be applied to any well-behaved, bounded cost distribution. For the VCG payment, with Bernoulli-distributed costs the payment for an n-long ladder, divided by n, tends to an explicit rational function of the Bernoulli parameter. Again, our methods apply more generally. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 350-364, 2011 © 2011 Wiley Periodicals, Inc.

One of the key performance measures in queueing systems is the decay rate of the steady-state tail probabilities of the queue lengths. It is known that if a corresponding fluid model is stable and the stochastic primitives have finite moments, then the queue lengths also have finite moments, so that the tail probability ℙ(⋅>s) decays faster than s −n for any n. It is natural to conjecture that the decay rate is in fact exponential. In this paper an example is constructed to demonstrate that this conjecture is false. For a specific stationary policy applied to a network with exponentially distributed interarrival and service times, it is shown that the corresponding fluid limit model is stable, but the tail probability for the buffer length decays slower than s −log s .

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 $\epsilon>0$, the computational effort of our method for these two models is $(1/\epsilon)^{O(1)}$ \emph{both} for free energy and surface pressure. In contrast, prior methods, such as transfer matrix method, require $\exp\big((1/\epsilon)^{O(1)}\big)$ computation effort., Comment: 33 pages, 4 figures

We consider the problem of the asymptotic size of the random maximum-weight matching of a sparse random graph, which we translate into dynamics of the operator in the space of distribution functions. A tight condition for the uniqueness of the globally attracting fixed point is provided, which extends the result of Karp and Sipser [Maximum matchings in sparse random graphs. 22nd Ann. Symp. on Foundations of Computer Science (Nashville, TN, 28–30 October, 1981). IEEE, New York, 1981, pp. 364–375] from deterministic weight distributions (Dirac measures μ) to general ones. Given a probability measure μ which corresponds to the weight distribution of a link of a random graph, we form a positive linear operator Φμ (convolution) on distribution functions and then analyze a family of its exponents, with parameter λ, which corresponds to the connectivity of a sparse random graph. The operator 𝕋 relates the distribution F on the subtrees to the distribution 𝕋F on the node of the tree by 𝕋F=exp (−λΦμF). We prove that for every probability measure μ and every λe, there is no fixed point attractor. We further establish that the uniqueness of the invariant measure of the underlying operator is not a monotone property of the average connectivity; this parallels similar non-monotonicity results in the statistical physics context.

We consider a multiserver 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 Assuming that the service time distribution is lattice valued with a finite support, we characterize the limiting scaled 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 stationary 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.

We consider a constrained homogeneous random walk in ℤ+d. Such random walks are used to model various stochastic processes, most importantly multiclass Markovian queueing networks operating under state-dependent scheduling policies. These applications motivate the need to compute various performance measures, including stationary probability distributions and large deviations rates. In this paper, we show that computing the stationary probability distributions exactly is an algorithmically undecidable problem. That is no algorithm can possibly exist to solve this task. The problem remains undecidable even if a linear Lyapunov function that verifies positive recurrence of the underlying random walk is available as a part of the input. We then prove that computing large deviation rates for this model is also an undecidable problem. Specifically, we show that it is an undecidable problem to determine finiteness of large deviations rates.

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 GN,N≥1 to some target graph W . The theory of dense graph convergence, including random dense graphs, is now well understood (Borgs et al. in Ann Math 176:151–219, 2012; Borgs et al. in Adv Math 219:1801–1851, 2008; Chatterjee and Varadhan in Eur J Comb 32:1000–1017, 2011; Lovász and Szegedy in J Comb Theory Ser B 96:933–957, 2006), but its counterpart for sparse random graphs presents some fundamental difficulties. Phrased in the statistical physics terminology, the issue is the existence of the limits of appropriately normalized log-partition functions, also known as free energy limits, for the Gibbs distribution associated with W . In this paper we prove that the sequence of sparse Erdös-Rényi graphs is right-converging when the tensor product associated with the target graph W satisfies a 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 (Talagrand in Mean Field Models for Spin Glasses: Volume I: Basic examples. Springer, Berlin, 2010)], concerning the existence of the limit of the measure of a set obtained from RN 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 (Commun Math Phys 230:71–79, 2002) and developed further in (Abbe and Montanari in On the concentration of the number of solutions of random satisfiability formulas, 2013; Bayati et al. in Ann Probab Conference version in Proceedings of 42nd Ann. Symposium on the Theory of Computing (STOC), 2010; Contucci et al. in Antiferromagnetic Potts model on the Erdös-Rényi random graph, 2011; Franz and Leone in J Stat Phys 111(3/4):535–564, 2003; Franz et al. in J Phys A Math Gen 36:10967–10985, 2003; Montanari in IEEE Trans Inf Theory 51(9):3221–3246, 2005; Panchenko and Talagrand in Probab Theory Relat Fields 130:312–336, 2004). Specifically, Bayati et al. (Ann Probab Conference version in Proceedings of 42nd Ann. Symposium on the Theory of Computing (STOC), 2010) establishes the right-convergence property for Erdös-Rényi graphs for some special cases of W . In this paper most of the results in Bayati et al. (Ann Probab Conference version in Proceedings of 42nd Ann. Symposium on the Theory of Computing (STOC), 2010) follow as a special case of our main theorem., National Science Foundation (U.S.) (NSF grant CMMI-1031332)

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 $q \geq \alpha^{*} \Delta + 1$ where $q$ the number of colors, $\Delta$ is the degree and $\alpha^* = 1.763..$ is the unique solution to $xe^{-1/x} = 1$. It has also been established for bounded degree lattice graphs whenever $q \geq \alpha^* \Delta - \beta$ for some constant $\beta$, where $\Delta$ 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 $\alpha \Delta(v) + \beta$ where $\Delta(v)$ is the degree of vertex $v$ and $\alpha > \alpha ^*$ and $\beta$ is a constant that only depends on $\alpha$. 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.

Given a (directed or undirected) graph G, finding the smallest number of additional edges which make the graph Hamiltonian is called the Hamiltonian Completion Problem (HCP). We consider this problem in the context of sparse random graphs G(n,c/n) on n nodes, where each edge is selected independently with probability c/n. We give a complete asymptotic answer to this problem when c

We consider the travelling salesman problem (TSP) problem on (the metric completion of) 3-edge-connected cubic graphs. These graphs are interesting because of the connection between their optimal solutions and the subtour elimination LP relaxation. Our main result is an approximation algorithm better than the 3/2-approximation algorithm for TSP in general.

Consider a game in which edges of a graph are provided a pair at a time, and the player selects one edge from each pair, attempting to construct a graph with a component as large as possible. This game is in the spirit of recent papers on avoiding a giant component, but here we embrace it. We analyze this game in the offline and online setting, for arbitrary and random instances, which provides for interesting comparisons. For arbitrary instances, we find that the competitive ratio (the best possible solution value divided by best possible online solution value) is large. For "sparse" random instances the competitive ratio is also large, with high probability (whp): If the instance has 1/4(1 + e)n random edge pairs, with 0 < e ≤ 0.003, then any online algorithm generates a component of size O((log n)3/2) whp, while the optimal offline solution contains a component of size Ω(n) whp. For "dense" random instances, the average-case competitive ratio is much smaller. If the instance has ½(1 - e)n random edge pairs, with 0 < e ≤ 0.015, we give an online algorithm which finds a component of size Ω(n) whp.

A stochastic online version of the classical bin packing problem, where a bin corresponds to the capacity of a resource allocated among streams of requests at discrete time units, is a fundamental problem that arises in a wide variety of application areas including bandwidth allocation in networks, memory management in computers, and message transmission in slotted network channels. We derive a mathematical analysis of the corresponding multi-dimensional stochastic process, potentially infinite in each dimension, under a general class of scheduling policies based on a combination of a Lyapunov function technique and matrix-analytic methods. Our analysis yields stability conditions and stationary distributions for this stochastic bin packing process under general probability distributions. We further provide some algorithmic techniques for the numerical computation of these measures.

We consider the following stochastic bandwidth packing process: the requests for communication bandwidth of different sizes arrive at times te0,1,2,… and are allocated to a communication link using “largest first” rule. Each request takes a unit time to complete. The unallocated requests form queues. Coffman and Stolyar [6] introduced this system and posed the following question: under which conditions do the expected queue lengths remain bounded over time (queueing system is stable)? We derive exact constructive conditions for the stability of this system using the Lyapunov function technique. The result holds under fairly general assumptions on the distribution of the arrival processes.