Random Walks on Combinatorial Objects
نویسندگان
چکیده
Approximate sampling from combinatorially-defined sets, using the Markov chain Monte Carlo method, is discussed from the perspective of combinatorial algorithms. We also examine the associated problem of discrete integration over such sets. Recent work is reviewed, and we re-examine the underlying formal foundational framework in the light of this. We give a detailed treatment of the coupling technique, a classical method for analysing the convergence rates of Markov chains. The related topic of perfect sampling is examined. In perfect sampling, the goal is to sample exactly from the target set. We conclude with a discussion of negative results in this area. These are results which imply that there are no polynomial time algorithms of a particular type for a particular problem.
منابع مشابه
Random Walks in Random Environment
My main research interest is in theoretical and applied probability mainly focusing on discrete problems arising out of combinatorics, statistical physics and computer science. In particular I am interested in random graphs, probability on tress, combinatorial optimization and statistical physics problems, recursive distributional equations, branching random walks, percolation theory, interacti...
متن کاملFirst Hitting times of Simple Random Walks on Graphs with Congestion Points
We derive the explicit formulas of the probability generating functions of the first hitting times of simple random walks on graphs with congestion points using group representations. 1. Introduction. Random walk on a graph is a Markov chain whose state space is the vertex set of the graph and whose transition from a given vertex to an adjacent vertex along an edge is defined according to some ...
متن کاملRandom Walks and Evolving Sets: Faster Convergences and Limitations
Analyzing the mixing time of random walks is a well-studied problem with applications in random sampling and more recently in graph partitioning. In this work, we present new analysis of random walks and evolving sets using more combinatorial graph structures, and show some implications in approximating small-set expansion. On the other hand, we provide examples showing the limitations of using...
متن کامل1 M ay 2 00 8 RANDOM WALKS , ARRANGEMENTS , CELL COMPLEXES , GREEDOIDS , AND SELF - ORGANIZING LIBRARIES
The starting point is the known fact that some much-studied random walks on permutations, such as the Tsetlin library, arise from walks on real hyperplane arrangements. This paper explores similar walks on complex hyperplane arrangements. This is achieved by involving certain cell complexes naturally associated with the arrangement. In a particular case this leads to walks on libraries with sev...
متن کاملTwo non-holonomic lattice walks in the quarter plane
We present two classes of random walks restricted to the quarter plane whose generating function is not holonomic. The non-holonomy is established using the iterated kernel method, a recent variant of the kernel method. This adds evidence to a recent conjecture on combinatorial properties of walks with holonomic generating functions. The method also yields an asymptotic expression for the numbe...
متن کاملTableau sequences, open diagrams, and Baxter families
Walks on Young’s lattice of integer partitions encode many objects of algebraic and combinatorial interest. Chen et al. established connections between such walks and arc diagrams. We show that walks that start at ∅, end at a row shape, and only visit partitions of bounded height are in bijection with a new type of arc diagram – open diagrams. Remarkably two subclasses of open diagrams are equi...
متن کامل