Radislav Vaisman

The Splitting Method for Decision Making

We show how a simple modification of the splitting method based on Gibbs sampler can be efficiently used for decision making in the sense that one can efficiently decide whether or not a given set of integer program constraints has at least one feasible solution. We also show how to incorporate the classic capture-recapture method into the splitting algorithm in order to obtain a low variance estimator for the counting quantity representing, say the number of feasible solutions on the set of the constraints of an integer program. We finally present numerical with with both, the decision making and the capture-recapture estimators and show their superiority as compared to the conventional one, while solving quite general decision making and counting ones, like the satisfiability problems.

Permutational methods for performance analysis of stochastic flow networks

In this paper we show how the permutation Monte Carlo method, originally developed for reliability networks, can be successfully adapted for stochastic flow networks, and in particular for estimation of the probability that the maximal flow in such a network is above some fixed level, called the threshold. A stochastic flow network is defined as one, where the edges are subject to random failures. A failed edge is assumed to be erased (broken) and, thus, not able to deliver any flow. We consider two models; one where the edges fail with the same failure probability and another where they fail with different failure probabilities. For each model we construct a different algorithm for estimation of the desired probability; in the former case it is based on the well known notion of the D-spectrum and in the later one—on the permutational Monte Carlo. We discuss the convergence properties of our estimators and present supportive numerical results.

Monte Carlo for Estimating Exponential Convolution

We study the numerical stability problem that may take place when calculating the cumulative distribution function (CDF) of the Hypoexponential random variable. This computation is extensively used during the execution of Monte Carlo network reliability estimation algorithms. In spite of the fact that analytical formulas are available, they can be unstable in practice. This instability occurs frequently when estimating very small failure probabilities that can happen for example while estimating the unreliability of telecommunication systems. In order to address this problem, we propose a simple unbiased estimation algorithm that is capable of handling a large number of variables.

On the Use of Smoothing to Improve the Performance of the Splitting Method

We present an enhanced version of the splitting method, called the smoothed splitting method (SSM), for counting associated with complex sets, such as the set defined by the constraints of an integer program and in particular for counting the number of satisfiability assignments. Like the conventional splitting algorithms, ours uses a sequential sampling plan to decompose a “difficult” problem into a sequence of “easy” ones. The main difference between SSM and splitting is that it works with an auxiliary sequence of continuous sets instead of the original discrete ones. The rationale of doing so is that continuous sets are easier to handle. We show that while the proposed method and its standard splitting counterpart are similar in their CPU time and variability, the former is more robust and more flexible than the latter.

Improved Sampling Plans for Combinatorial Invariants of Coherent Systems

Terminal network reliability problems appear in many real-life applications, such as transportation grids, social and computer networks, communication systems, etc. In this paper, we focus on monotone binary systems with identical component reliabilities. The reliability of such systems depends only on the number of failure sets of all possible sizes, which is an essential system invariant. For large problems, no analytical solution for calculating this invariant in a reasonable time is known to exist, and one has to rely on different approximation techniques. An example of such a method is Permutation Monte Carlo. It is known that this simple plan is not sufficient for adequate estimation of network reliability due to the rare-event problem. As an alternative, we propose a different sampling strategy that is based on the recently pioneered Stochastic Enumeration algorithm for tree cost estimation. We show that, thanks to its built-in splitting mechanism, this method is able to deliver accurate results while employing a relatively modest sample size. Moreover, our numerical results indicate that the proposed sampling scheme is capable of solving problems that are far beyond the reach of the simple Permutation Monte Carlo approach.

Decision-making with cross-entropy for self-adaptation

Approaches to decision-making in self-adaptive systems are increasingly becoming more effective at managing the target system by taking into account more elements of the decision problem that were previously ignored. These approaches have to solve complex optimization problems at run time, and even though they have been shown to be suitable for different kinds of systems, their time complexity can make them excessively slow for systems that have a large adaptation-relevant state space, or that require a tight control loop driven by fast decisions. In this paper we present an approach to speed up complex proactive latency-aware self-adaptation decisions, using the cross-entropy (CE) method for combinatorial optimization. The CE method is an any-time algorithm based on random sampling from the solution space, and is not guaranteed to find an optimal solution. Nevertheless, our experiments using two very different systems show that in practice it finds solutions that are close to optimum even when its running time is restricted to a fraction of a second, attaining speedups of up to 40 times over the previous fastest solution approach.

Sequential Monte Carlo for counting vertex covers in general graphs

In this paper we describe a sequential importance sampling (SIS) procedure for counting the number of vertex covers in general graphs. The optimal SIS proposal distribution is the uniform over a suitably restricted set, but is not implementable. We will consider two proposal distributions as approximations to the optimal. Both proposals are based on randomization techniques. The first randomization is the classic probability model of random graphs, and in fact, the resulting SIS algorithm shows polynomial complexity for random graphs. The second randomization introduces a probabilistic relaxation technique that uses Dynamic Programming. The numerical experiments show that the resulting SIS algorithm enjoys excellent practical performance in comparison with existing methods. In particular the method is compared with cachet—an exact model counter, and the state of the art SampleSearch, which is based on Belief Networks and importance sampling.

How to generate uniform samples on discrete sets using the splitting method

The goal of this work is twofold. We show the following: 1. In spite of the common consensus on the classic Markov chain Monte Carlo (MCMC) as a universal tool for generating samples on complex sets, it fails to generate points uniformly distributed on discrete ones, such as that defined by the constraints of integer programming. In fact, we will demonstrate empirically that not only does it fail to generate uniform points on the desired set, but typically it misses some of the points of the set. 2. The splitting, also called the cloning method – originally designed for combinatorial optimization and for counting on discrete sets and presenting a combination of MCMC, like the Gibbs sampler, with a specially designed splitting mechanism—can also be efficiently used for generating uniform samples on these sets. Without introducing the appropriate splitting mechanism, MCMC fails. Although we do not have a formal proof, we guess (conjecture) that the main reason that the classic MCMC is not working is that its resulting chain is not irreducible. We provide valid statistical tests supporting the uniformity of generated samples by the splitting method and present supportive numerical results.

The Multilevel Splitting algorithm for graph colouring with application to the Potts model

Calculating the partition function of the zero-temperature antiferromagnetic model is an important problem in statistical physics. However, an exact calculation is hard since it is strongly connected to a fundamental combinatorial problem of counting proper vertex colourings in undirected graphs, for which an efficient algorithm is not known to exist. Thus, one has to rely on approximation techniques. In this paper, we formulate the problem of the partition function approximation in terms of rare-event probability estimation and investigate the performance of a particle-based algorithm, called Multilevel Splitting, for handling this setting. The proposed method enjoys a provable probabilistic performance guarantee and our numerical study indicates that this algorithm is capable of delivering accurate results using a relatively modest amount of computational resources.

Model Counting of Monotone Conjunctive Normal Form Formulas with Spectra

Model counting is the #P problem of counting the number of satisfying solutions of a given propositional formula. Here we focus on a restricted variant of this problem, where the input formula is monotone i.e., there are no negations. A monotone conjunctive normal formCNF formula is sufficient for modeling various graph problems, e.g., the vertex covers of a graph. Even for this restricted case, there is no known efficient approximation scheme. We show that the classical Spectra technique that is widely used in network reliability can be adapted for counting monotone CNF formulas. We prove that the proposed algorithm is logarithmically efficient for random monotone 2-CNF instances. Although we do not prove the efficiency of Spectra for k-CNF where k > 2, our experiments show that it is effective in practice for such formulas.