A polynomial algorithm for the two-connections variant of the tree p-median problem

We consider the variant of the tree $p$-median problem where each node must be connected to the two closest centers. This problem is polynomially solved through a dynamic programming formulation that extends the solution given by A. Tamir for the classical$p$-median problem on a tree.     

A polynomial algorithm for a continuous bilevel knapsack problem

In this note, we analyze a bilevel interdiction problem, where the follower’s program is a parametrized continuous knapsack. Based on the structure of the problem and an inverse optimization strategy, we propose for its solution an algorithm with worst-case complexity $O\left(n^2\right)$.     

Finding common RNA pseudoknot structures in polynomial time

This paper presents the first polynomial time algorithm for finding common RNA substructures that include pseudoknots and similar structures. While a more general problem is known to be NP-hard, this algorithm exploits special features of RNA structures to match RNA bonds correctly in polynomial time. Although the theoretical upper bound on the algorithm?s time and space usage is high, the data-driven nature of its computation enables it to avoid computing unnecessary cases, dramatically reducing the actual running time. The algorithm works well in practice, and has been tested on sample RNA structures that include pseudoknots and pseudoknot-like tertiary structures.     

A polynomial time algorithm for a hemispherical minimax location problem

A polynomial time algorithm to obtain an exact solution for the equiweighted minimax location problem when the demand points are spread over a hemisphere is presented. It is shown that the solution of the minimax problem when the norm under consideration is geodesic is equivalent to solving a maximization problem using the Euclidean norm.     

A polynomial algorithm for an integer quadratic non-separable transportation problem

We study the problem of minimizing the total weighted tardiness when scheduling unti-length jobs on a single machine, in the presence of large sets of identical jobs. Previously known algorithms, which do not exploit the set structure, are at best pseudo-polynomial, and may be prohibitively inefficient when the set sizes are large. We give a polynomial algorithm for the problem, whose number of operations is independent of the set sizes. The problem is reformulated as an integer program with a quadratic, non-separable objective and transportation constraints. Employing methods of real analysis, we prove a tight proximity result between the integer solution to that problem and a fractional solution of a related problem. The related problem is shown to be polynomially solvable, and a rounding algorithm applied to its solution gives the optimal integer solution to the original problem.Supported in part by the National Science Foundation under grant ECS-85-01988, and by the Office of Naval Research under grant N00014-88-K-0377.Supported in part by Allon Fellowship, by Air Force grants 89-0512 and 90-0008 and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center—NSF-STC88-09648. Part of the research of this author was performed in DIMACS Center, Rutgers University.Supported in part by Air Force grant 84-0205.     

A polynomial time algorithm for a chance-constrained single machine scheduling problem

This paper gives an O(n√nlog³n) time algorithm for the chance-constrained sequencing problem on a single machine, where n is the number of jobs and the objective is to minimize the number of jobs which are early with probability not smaller than α (a given constant) against the common due time d.     

A dynamic programming algorithm for the bilevel knapsack problem

We propose an efficient dynamic programming algorithm for solving a bilevel program where the leader controls the capacity of a knapsack, and the follower solves the resulting knapsack problem. We propose new recursive rules and show how to solve the problem as a sequence of two standard knapsack problems.     

A numerical algorithm for pricing electricity derivatives for jump-diffusion processes based on continuous time lattices

We present a numerical algorithm for pricing derivatives on electricity prices. The algorithm is based on approximating the generator of the underlying price process on a lattice of prices, resulting in an approximation of the stochastic process by a continuous time Markov chain. We numerically study the rate of convergence of the algorithm for the case of the Merton jump-diffusion model and apply the algorithm to calculate prices and sensitivities of both European and Bermudan electricity derivatives when the underlying price follows a stochastic process which exhibits both fast mean-reversion and jumps of large magnitude.     

A new algorithm for multi-dimensional dynamic programming problems

The new algorithm presented here solves medium size multi-dimensional dynamic programming problems in a relatively short computational time with no fast-memory restraints. The algorithm converges to the global optimal solution under some differentiability and convexity assumptions.The procedure is to solve a succession of dynamic programming problems, the state sets of which are limited to only a very small subset of the original state space. The interrelated definition of state sets for successive subproblems facilitates an algorithmic convergence while moving the subsets to contain the optimal states at the end.     

A polynomial time algorithm for solving a quality control station configuration problem

We study unreliable serial production lines with known failure probabilities for each operation. Such a production line consists of a series of stations; existing machines and optional quality control stations (QCS). Our aim is to simultaneously decide where and if to install the QCSs along the line and to determine the production rate, so as to maximize the steady state expected net profit per time unit from the system.We use dynamic programming to solve the cost minimization auxiliary problem where the aim is to minimize the time unit production cost for a given production rate. Using the above developed O(N²) dynamic programming algorithm as a subroutine, where N stands for the number of machines in the line, we present an O(N⁴) algorithm to solve the Profit Maximization QCS Configuration Problem.     

A polynomial algorithm for minimum quadratic cost flow problems

Network flow problems with quadratic separable costs appear in a number of important applications such as; approximating input-output matrices in economy; projecting and forecasting traffic matrices in telecommunication networks; solving nondifferentiable cost flow problems by subgradient algorithms. It is shown that the scaling technique introduced by Edmonds and Karp (1972) in the case of linear cost flows for deriving a polynomial complexity bound for the out-of-kilter method, may be extended to quadratic cost flows and leads to a polynomial algorithm for this class of problems. The method may be applied to the solution of singly constrained quadratic programs and thus provides an alternative approach to the polynomial algorithm suggested by Helgason, Kennington and Lall (1980).     

Towards a strongly polynomial algorithm for strictly convex quadratic programs: An extension of Tardos' algorithm

In a recent paper Tardos described a polynomial algorithm for solving linear programming problems in which the number of arithmetic steps depends only on the size of the numbers in the constraint matrix and is independent of the size of the numbers in the right hand side and the cost coefficients. In this paper we extend Tardos' results and present a polynomial algorithm for solving strictly convex quadratic programming problems in which the number of arithmetic steps is independent of the size of the numbers in the right hand side and the linear cost coefficients.This research was partially supported by the Natural Sciences and Engineering Research Council of Canada Grant 5-83998.     

A faster strongly polynomial time algorithm for submodular function minimization

We consider the problem of minimizing a submodular function f defined on a set V with n elements. We give a combinatorial algorithm that runs in O(n ⁵EO  +  n ⁶) time, where EO is the time to evaluate f(S) for some . This improves the previous best strongly polynomial running time by more than a factor of n. We also extend our result to ring families.     

A polynomial time primal network simplex algorithm for minimum cost flows

Developing a polynomial time primal network simplex algorithm for the minimum cost flow problem has been a long standing open problem. In this paper, we develop one such algorithm that runs in O(min(n ²m lognC, n ²m² logn)) time, wheren is the number of nodes in the network,m is the number of arcs, andC denotes the maximum absolute arc costs if arc costs are integer and ∞ otherwise. We first introduce a pseudopolynomial variant of the network simplex algorithm called the “premultiplier algorithm”. We then develop a cost-scaling version of the premultiplier algorithm that solves the minimum cost flow problem in O(min(nm lognC, nm ² logn)) pivots. With certain simple data structures, the average time per pivot can be shown to be O(n). We also show that the diameter of the network polytope is O(nm logn).     

A worst case analysis of a dynamic programming-based heuristic algorithm for 2D unconstrained guillotine cutting

In this paper, a dynamic programming-based recursive method is proposed for solving an unconstrained 2D rectangular cutting problem. The algorithm is an incomplete method, in which some intricate cutting patterns may not be obtained. The worst case performance of the algorithm is evaluated and some theoretical analyses for the algorithm are performed. Compared to traditional dynamic programming, this algorithm gives a high percentage of optimal solutions (94.84%, 86.67% and 77.83% for small, medium and large sized unweighted instances, 99.67%, 99.50% and 97.00% for small, medium and large sized weighted instances) but features a far lower computational complexity. Computational results are also presented for some known benchmarks.     

A polynomial time approximation algorithm for the two-commodity splittable flow problem

We consider a generalization of the unsplittable maximum two-commodity flow problem on undirected graphs where each commodity ${i \in \{1, 2\}}$ can be split into a bounded number k _i of equally-sized chunks that can be routed on different paths. We show that in contrast to the single-commodity case this problem is NP-hard, and hard to approximate to within a factor of α > 1/2. We present a polynomial time 1/2-approximation algorithm for the case of uniform chunk size over both commodities and show that for even k _i and a mild cut condition it can be modified to yield an exact method. The uniform case can be used to derive a 1/4-approximation for the maximum concurrent (k ₁, k ₂)-splittable flow without chunk size restrictions for fixed demand ratios.     

An iterated local search algorithm for the vehicle routing problem with convex time penalty functions

We propose an iterated local search algorithm for the vehicle routing problem with time window constraints. We treat the time window constraint for each customer as a penalty function, and assume that it is convex and piecewise linear. Given an order of customers each vehicle to visit, dynamic programming (DP) is used to determine the optimal start time to serve the customers so that the total time penalty is minimized. This DP algorithm is then incorporated in the iterated local search algorithm to efficiently evaluate solutions in various neighborhoods. The amortized time complexity of evaluating a solution in the neighborhoods is a logarithmic order of the input size (i.e., the total number of linear pieces that define the penalty functions). Computational comparisons on benchmark instances with up to 1000 customers show that the proposed method is quite effective, especially for large instances.