arXiv subjects

David Gamarnik

Publications and source records attributed to David Gamarnik.

102 records · Page 6Linked to original sources

arXiv2007

A Deterministic Approximation Algorithm for Computing a Permanent of a 0,1 matrix

We construct a deterministic approximation algorithm for computing a permanent of a $0,1$ $n$ by $n$ matrix to within a multiplicative factor $(1+ε)^n$, for arbitrary $ε>0$. When the graph underlying the matrix is a constant degree expander our algorithm runs in polynomial time (PTAS). In the general case the running time of the algorithm is $\exp(O(n^{2\over 3}\log^3n))$. For the class of graphs which are constant degree expanders the first result is an improvement over the best known approximation factor $e^n$ obtained in \cite{LinialSamorodnitskyWigderson}. Our results use a recently developed deterministic approximation algorithm for counting partial matchings of a graph Bayati et al., and Jerrum-Vazirani decomposition method.

math.CO↗

arXiv2006

On Exponential Ergodicity of Multiclass Queueing Networks

One of the key performance measures in queueing systems is the exponential 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 \pr(\cdot >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}.

math.PR↗

arXiv2006

Validity of heavy traffic steady-state approximations in generalized Jackson Networks

We consider a single class open queueing network, also known as a generalized Jackson network (GJN). A classical result in heavy-traffic theory asserts that the sequence of normalized queue length processes of the GJN converge weakly to a reflected Brownian motion (RBM) in the orthant, as the traffic intensity approaches unity. However, barring simple instances, it is still not known whether the stationary distribution of RBM provides a valid approximation for the steady-state of the original network. In this paper we resolve this open problem by proving that the re-scaled stationary distribution of the GJN converges to the stationary distribution of the RBM, thus validating a so-called ``interchange-of-limits'' for this class of networks. Our method of proof involves a combination of Lyapunov function techniques, strong approximations and tail probability bounds that yield tightness of the sequence of stationary distributions of the GJN.

math.PR↗

arXiv2005

Counting without sampling. New algorithms for enumeration problems using statistical physics

We propose a new type of approximate counting algorithms for the problems of enumerating the number of independent sets and proper colorings in low degree graphs with large girth. Our algorithms are not based on a commonly used Markov chain technique, but rather are inspired by developments in statistical physics in connection with correlation decay properties of Gibbs measures and its implications to uniqueness of Gibbs measures on infinite trees, reconstruction problems and local weak convergence methods. On a negative side, our algorithms provide $ε$-approximations only to the logarithms of the size of a feasible set (also known as free energy in statistical physics). But on the positive side, our approach provides deterministic as opposed to probabilistic guarantee on approximations. Moreover, for some regular graphs we obtain explicit values for the counting problem. For example, we show that every 4-regular $n$-node graph with large girth has approximately $(1.494...)^n$ independent sets, and in every $r$-regular graph with $n$ nodes and large girth the number of $q\geq r+1$-proper colorings is approximately $[q(1-{1\over q})^{r\over 2}]^n$, for large $n$. In statistical physics terminology, we compute explicitly the limit of the log-partition function. We extend our results to random regular graphs. Our explicit results would be hard to derive via the Markov chain method.

math.PR↗

arXiv2005

A Transposition Rule Analysis Based on a Particle Process

A linear list is a collection of items that can be accessed sequentially. The cost of a request is the number of items that need to be examined before the desired item is located, i.e., the distance of the requested item from the beginning of the list. The transposition rule is one of the algorithms designed to reduce the search cost by organizing the list. In particular, upon a request for a given item, the item is transposed with the preceding one. We develop a new approach for analyzing the algorithm. The approach is based on a coupling with a certain constrained asymmetric exclusion process. This allows us to establish an asymptotic optimality of the rule for two families of request distributions.

math.PR↗

arXiv2004

Instability in Stochastic and Fluid Queueing Networks

The fluid model has proven to be one of the most effective tools for the analysis of stochastic queueing networks, specifically for the analysis of stability. It is known that stability of a fluid model implies positive (Harris) recurrence (stability) of a corresponding stochastic queueing network, and weak stability implies rate stability of a corresponding stochastic network. These results have been established both for cases of specific scheduling policies and for the class of all work conserving policies. However, only partial converse results have been established and in certain cases converse statements do not hold. In this paper we close one of the existing gaps. For the case of networks with two stations we prove that if the fluid model is not weakly stable under the class of all work conserving policies, then a corresponding queueing network is not rate stable under the class of all work conserving policies. We establish the result by building a particular work conserving scheduling policy which makes the associated stochastic process transient. An important corollary of our result is that the condition $ρ^*\leq 1$, which was proven in \cite{daivan97} to be the exact condition for global weak stability of the fluid model, is also the exact global rate stability condition for an associated queueing network. Here $ρ^*$ is a certain computable parameter of the network involving virtual station and push start conditions.

math.PR↗

arXiv2004

Dynamics of exponential linear map in functional space

We consider the question of existence of a unique invariant probability distribution which satisfies some evolutionary property. The problem arises from the random graph theory but to answer it we treat it as a dynamical system in the functional space, where we look for a global attractor. We consider the following bifurcation problem: Given a probability measure $μ$, which corresponds to the weight distribution of a link of a random graph we form a positive linear operator $Φ$ (convolution) on distribution functions and then we analyze a family of its exponents with a parameter $λ$ which corresponds to connectivity of a sparse random graph. We prove that for every measure $μ$ (\emph{i.e.}, convolution $Φ$) and every $λ< e$ there exists a unique globally attracting fixed point of the operator, which yields the existence and uniqueness of the limit probability distribution on the random graph. This estimate was established earlier \cite{KarpSipser} for deterministic weight distributions (Dirac measures $μ$) and is known as $e$-cutoff phenomena, as for such distributions and $λ>e$ there is no fixed point attractor. We thus establish this phenomenon in a much more general sense.

math.DS↗

arXiv2003

Maximum Weight Independent Sets and Matchings in Sparse Random Graphs. Exact Results using the Local Weak Convergence Method

Let $G(n,c/n)$ and $G_r(n)$ be an $n$-node sparse random graph and a sparse random $r$-regular graph, respectively, and let ${\cal I}(n,r)$ and ${\cal I}(n,c)$ be the sizes of the largest independent set in $G(n,c/n)$ and $G_r(n)$. The asymptotic value of ${\cal I}(n,c)/n$ as $n\to\infty$, can be computed using the Karp-Sipser algorithm when $c\leq e$. For random cubic graphs, $r=3$, it is only known that $.432\leq\liminf_n {\cal I}(n,3)/n \leq \limsup_n {\cal I}(n,3)\leq .4591$ with high probability (w.h.p.) as $n\to\infty$, as shown by Frieze and Suen and by Bollobas, respectively. In this paper we assume in addition that the nodes of the graph are equipped with non-negative weights, independently generated according to some common distribution, and we consider instead the maximum weight of an independent set. Surprisingly, we discover that for certain weight distributions, the limit $\lim_n {\cal I}(n,c)/n$ can be computed exactly even when $c>e$, and $\lim_n {\cal I}(n,r)/n$ can be computed exactly for some $r\geq 2$. For example, when the weights are exponentially distributed with parameter 1, $\lim_n {\cal I}(n,2e)/n\approx .5517$, and $\lim_n {\cal I}(n,3)/n\approx .6077$. Our results are established using the recently developed local weak convergence method further reduced to a certain local optimality property exhibited by the models we consider.

math.PR↗

arXiv2003

Random MAX SAT, Random MAX CUT, and Their Phase Transitions

Given a 2-SAT formula $F$ consisting of $n$ variables and $\cn$ random clauses, what is the largest number of clauses $\max F$ satisfiable by a single assignment of the variables? We bound the answer away from the trivial bounds of $(3/4)cn$ and $cn$. We prove that for $c<1$, the expected number of clauses satisfiable is $\cn-Θ(1/n)$; for large $c$, it is $((3/4)c + Θ(\sqrt{c}))n$; for $c = 1+\eps$, it is at least $(1+\eps-O(\eps^3))n$ and at most $(1+\eps-Ω(\eps^3/\ln \eps))n$; and in the ``scaling window'' $c= 1+Θ(n^{-1/3})$, it is $cn-Θ(1)$. In particular, just as the decision problem undergoes a phase transition, our optimization problem also undergoes a phase transition at the same critical value $c=1$. Nearly all of our results are established without reference to the analogous propositions for decision 2-SAT, and as a byproduct we reproduce many of those results, including much of what is known about the 2-SAT scaling window. We consider ``online'' versions of MAX-2-SAT, and show that for one version, the obvious greedy algorithm is optimal. We can extend only our simplest MAX-2-SAT results to MAX-k-SAT, but we conjecture a ``MAX-k-SAT limiting function conjecture'' analogous to the folklore satisfiability threshold conjecture, but open even for $k=2$. Neither conjecture immediately implies the other, but it is natural to further conjecture a connection between them. Finally, for random MAXCUT (the size of a maximum cut in a sparse random graph) we prove analogous results.

math.CO↗

arXiv2003

Linear Phase Transition in Random Linear Constraint Satisfaction Problem

Our model is a generalized linear programming relaxation of a much studied random K-SAT problem. Specifically, a set of linear constraints C on K variables is fixed. From a pool of n variables, K variables are chosen uniformly at random and a constraint is chosen from C also uniformly at random. This procedure is repeated m times independently. We ask the following question: is the resulting linear programming problem feasible? We show that the feasibility property experiences a linear phase transition, when n diverges to infinity and m=cn for some constant c. Namely, there exists a critical value c* such that, when c c*, the "distance" from feasibility is at least a positive constant independent of n. Our results are obtained using powerful local weak convergence methods developed by Aldous and Steele. By exploiting a linear programming duality, our theorem implies the following result in the context of sparse random graphs G(n, cn) on n nodes with cn edges, where edges are equipped with randomly generated weights. Let M(n,c) denote maximum weight matching in G(n, cn). We prove that when c is a constant and n\to\infty, the limit \lim_n M(n,c)/n exists, with high probability. We further extend this result to maximum weight b-matchings also in G(n,cn).

math.PR↗

arXiv2002

Computing stationary probability distributions and large deviation rates for constrained random walks. The undecidability results

Our model is a constrained homogeneous random walk in a nonnegative orthant Z_+^d. The convergence to stationarity for such a random walk can often be checked by constructing a Lyapunov function. The same Lyapunov function can also be used for computing approximately the stationary distribution of this random walk, using methods developed by Meyn and Tweedie. In this paper we show that, for this type of random walks, computing the stationary probability exactly is an undecidable problem: no algorithm can exist to achieve this task. We then prove that computing large deviation rates for this model is also an undecidable problem. We extend these results to a certain type of queueing systems. The implication of these results is that no useful formulas for computing stationary probabilities and large deviations rates can exist in these systems.

math.PR↗

arXiv2001

The diameter of a long range percolation graph

We consider the following long range percolation model: an undirected graph with the node set $\{0,1,...,N\}^d$, has edges $(\x,\y)$ selected with probability $\approx β/||\x-\y||^s$ if $||\x-\y||>1$, and with probability 1 if $||\x-\y||=1$, for some parameters $β,s>0$. This model was introduced by Benjamini and Berger, who obtained bounds on the diameter of this graph for the one-dimensional case $d=1$ and for various values of $s$, but left cases $s=1,2$ open. We show that, with high probability, the diameter of this graph is $Θ(\log N/\log\log N)$ when $s=d$, and, for some constants $0<η_1<η_2<1$, it is at most $N^{η_2}$, when $s=2d$ and is at least $N^{η_1}$ when $d=1,s=2,β<1$ or $s>2d$. We also provide a simple proof that the diameter is at most $\log^{O(1)}N$ with high probability, when $d<s<2d$, established previously by Berger and Benjamini.

math.PR↗