Polynomial Number Of Operations Research Articles

A Sequential Importance Sampling Algorithm for Counting Linear Extensions

In recent decades, a number of profound theorems concerning approximation of hard counting problems have appeared. These include estimation of the permanent, estimating the volume of a convex polyhedron, and counting (approximately) the number of linear extensions of a partially ordered set. All of these results have been achieved using probabilistic sampling methods, specifically Monte Carlo Markov Chain (MCMC) techniques. In each case, a rapidly mixing Markov chain is defined that is guaranteed to produce, with high probability, an accurate result after only a polynomial number of operations. Although of polynomial complexity, none of these results lead to a practical computational technique, nor do they claim to. The polynomials are of high degree and a non-trivial amount of computing is required to get even a single sample. Our aim in this article is to present practical Monte Carlo methods for one of these problems, counting linear extensions. Like related work on estimating the coefficients of the reliability polynomial, our technique is based on improving the so-called Knuth counting algorithm by incorporating an importance function into the node selection technique giving a sequential importance sampling (SIS) method. We define and report performance on two importance functions.

Quantum invariants and the graph isomorphism problem

Three new graph invariants are introduced which may be measured from a quantum graph state and form examples of a framework under which other graph invariants can be constructed. Each invariant is based on distinguishing a different number of qubits. This is done by applying alternate measurements to the qubits to be distinguished. The performance of these invariants is evaluated and compared to classical invariants. We verify that the invariants can distinguish all non-isomorphic graphs with 9 or fewer nodes. The invariants have also been applied to `classically hard' strongly regular graphs, successfully distinguishing all strongly regular graphs of up to 29 nodes, and preliminarily to weighted graphs. We have found that although it is possible to prepare states with a polynomial number of operations, the average number of preparations required to distinguish non-isomorphic graph states scales exponentially with the number of nodes. We have so far been unable to find operators which reliably compare graphs and reduce the required number of preparations to feasible levels.

Incremental versus non-incremental dynamic programming

Many dynamic programming algorithms for discrete optimization problems are pure in that they only use min/max and addition operations in their recursions. Some of them, in particular those for various shortest path problems, are even incremental in that one of the inputs to the addition operations is a variable. We present an explicit optimization problem such that it can be solved by a pure DP algorithm using a polynomial number of operations, but any incremental DP algorithm for this problem requires a super-polynomial number of operations.

Polynomial time algorithm for data association problem in multitarget tracking

The aim of this work is to propose a semigreedy algorithm for solving the data association problem (DAP) arising in multitarget tracking. The DAP is characterized as a maximum weighted set partitioning problem (MWSPP). A semigreedy algorithm that first generates a number of different solutions using a polynomial number of operations and then selects the best among the solutions generated is presented. Theoretically we prove that the value of the solution returned by the algorithm approximates the value of the optimal solution within a guaranteed factor. This factor only depends on the dimension of the sliding window used. Computational experiments indicate that the quality of the solutions returned is quite satisfactory.

System planning and configuration problems for optimal system design

This paper addresses the issues of server allocation and customer partition that arise from designing a service system. The service system is modelled as a GI/G/ m queueing system. In order to provide meaningful insights for system design, we consider two system design problems: the system planning problem and the system configuration problem. The former problem analyzes the tradeoff between investment cost and performance level while the latter problem examines the tradeoff between the system operating cost and the customer waiting cost. Both problems are formulated as dynamic programs that can be solved within polynomial number of operations.