Perturbation series expansions for nearly completely-decomposable Markov chains
Proc. of the international seminar on Teletraffic analysis and computer performance evaluation
An algorithm for finding Hamilton paths and cycles in random graphs
Combinatorica - Theory of Computing
Hamiltonian cycles, quadratic programming, and ranking of extreme points
Recent advances in global optimization
Hamiltonian cycles and Markov chains
Mathematics of Operations Research
Multiple centrality corrections in a primal-dual method for linear programming
Computational Optimization and Applications
ACM Computing Surveys (CSUR)
An efficient algorithm for the knight's tour problem
Discrete Applied Mathematics
Competitive Markov decision processes
Competitive Markov decision processes
Primal-dual interior-point methods
Primal-dual interior-point methods
Constrained Discounted Markov Decision Processes and Hamiltonian Cycles
Mathematics of Operations Research
Finite State Markovian Decision Processes
Finite State Markovian Decision Processes
Inversion of Analytic Matrix Functions That are Singular at the Origin
SIAM Journal on Matrix Analysis and Applications
Fast Algorithm for the Cutting Angle Method of Global Optimization
Journal of Global Optimization
Hamiltonian Cycles and Singularly Perturbed Markov Chains
Mathematics of Operations Research
The Cross Entropy Method: A Unified Approach To Combinatorial Optimization, Monte-carlo Simulation (Information Science and Statistics)
An Interior Point Heuristic for the Hamiltonian Cycle Problem via Markov Decision Processes
Journal of Global Optimization
Directed graphs, Hamiltonicity and doubly stochastic matrices
Random Structures & Algorithms
Local Optimization Method with Global Multidimensional Search
Journal of Global Optimization
Graph theory: An algorithmic approach (Computer science and applied mathematics)
Graph theory: An algorithmic approach (Computer science and applied mathematics)
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
Fast generation of regular graphs and construction of cages
Journal of Graph Theory
Determinants and Longest Cycles of Graphs
SIAM Journal on Discrete Mathematics
Computational Commutative Algebra 1
Computational Commutative Algebra 1
Refined MDP-Based Branch-and-Fix Algorithm for the Hamiltonian Cycle Problem
Mathematics of Operations Research
Markov Chains and Optimality of the Hamiltonian Cycle
Mathematics of Operations Research
Refined MDP-Based Branch-and-Fix Algorithm for the Hamiltonian Cycle Problem
Mathematics of Operations Research
Hamiltonian Cycles, Random Walks, and Discounted Occupational Measures
Mathematics of Operations Research
Hamiltonian transition matrices
Proceedings of the 5th International ICST Conference on Performance Evaluation Methodologies and Tools
Splitting Randomized Stationary Policies in Total-Reward Markov Decision Processes
Mathematics of Operations Research
Markov chains, Hamiltonian cycles and volumes of convex bodies
Journal of Global Optimization
Hi-index | 0.00 |
This manuscript summarizes a line of research that maps certain classical problems of discrete mathematics -- such as the Hamiltonian Cycle and the Traveling Salesman Problems -- into convex domains where continuum analysis can be carried out. Arguably, the inherent difficulty of these, now classical, problems stems precisely from the discrete nature of domains in which these problems are posed. The convexification of domains underpinning the reported results is achieved by assigning probabilistic interpretation to key elements of the original deterministic problems. In particular, approaches summarized here build on a technique that embeds Hamiltonian Cycle and Traveling Salesman Problems in a structured singularly perturbed Markov Decision Process. The unifying idea is to interpret subgraphs traced out by deterministic policies (including Hamiltonian Cycles, if any) as extreme points of a convex polyhedron in a space filled with randomized policies. The topic has now evolved to the point where there are many, both theoretical and algorithmic, results that exploit the nexus between graph theoretic structures and both probabilistic and algebraic entities of related Markov chains. The latter include moments of first return times, limiting frequencies of visits to nodes, or the spectra of certain matrices traditionally associated with the analysis of Markov chains. Numerous open questions and problems are described in the presentation.