Combinatorial Minimization Problem Research Articles

A combinatorial algorithm and its application in computing all minimum toll sets of graphs

Abstract This article formalizes an algorithm that computes the minimum toll sets in an undirected graph. A core process in our algorithm is to check vertex subsets in order of size. We add a new flavor to the implementation of this process; when the k − 1 k-1 -vertex subsets are already constructed, our algorithm produces the k k -vertex subsets building on the k − 1 k-1 -vertex subsets rather than reconstructing the k k -vertex subsets from the ground up as the existing algorithms would do. Our implementation is usable in combinatorial minimization problems that require checking variable-size combinations in order of size.

Extending single tolerances to set tolerances

The theory of single upper and lower tolerances for combinatorial minimization problems was formalized in 2005 for the three types of cost functions sum, product, and maximum, and since then it has shown to be rather useful in creating heuristics and exact algorithms. However, such single tolerances are often used because the assessment of multiple cost changes is considered too complicated. This paper addresses that issue. In this paper we extend this theory from single to set tolerances for these three types of cost functions. In particular, we characterize specific values of set upper and lower tolerances as positive and infinite, and we show a criterion for the uniqueness of an optimal solution to a combinatorial minimization problem. Furthermore, we present one exact formula and several bounds for computing set upper and lower tolerances using the relation to their corresponding single tolerance counterparts.

Solving Linear Conditional Completely Combinatorial Optimization Problems on Permutations by the Branch and Bound Method

A conditional linear fully combinatorial minimization problem on permutations is analyzed. The methods of branching, pruning, and estimating in the branch and bound method are proposed for this problem. An illustrative example of applying the method to the problem is presented. The property of the proposed estimation of the admissible subset, which increases the efficiency of branching and pruning, is proved.

Stackelberg Network Pricing Games

We study a multi-player one-round game termed Stackelberg Network Pricing Game, in which a leader can set prices for a subset of m priceable edges in a graph. The other edges have a fixed cost. Based on the leader’s decision one or more followers optimize a polynomial-time solvable combinatorial minimization problem and choose a minimum cost solution satisfying their requirements based on the fixed costs and the leader’s prices. The leader receives as revenue the total amount of prices paid by the followers for priceable edges in their solutions. Our model extends several known pricing problems, including single-minded and unit-demand pricing, as well as Stackelberg pricing for certain follower problems like shortest path or minimum spanning tree. Our first main result is a tight analysis of a single-price algorithm for the single follower game, which provides a (1+e)log m-approximation. This can be extended to provide a (1+e)(log k+log m)-approximation for the general problem and k followers. The problem is also shown to be hard to approximate within $\mathcal{O}(\log^{\varepsilon}k + \log^{\varepsilon}m)$ for some e>0. If followers have demands, the single-price algorithm provides an $\mathcal{O}(m^{2})$ -approximation, and the problem is hard to approximate within $\mathcal{O}(m^{\epsilon})$ for some e>0. Our second main result is a polynomial time algorithm for revenue maximization in the special case of Stackelberg bipartite vertex-cover, which is based on non-trivial max-flow and LP-duality techniques. This approach can be extended to provide constant-factor approximations for any constant number of followers.

Buyer–supplier games: Optimization over the core

In a buyer–supplier game, a special type of assignment game, a distinguished player, called the buyer, wishes to purchase some combinatorial structure. A set of players, called suppliers, offer various components of the structure for sale. Any combinatorial minimization problem can be transformed into a buyer–supplier game. While most previous work has been concerned with characterizing the core of buyer–supplier games, in this paper we study optimization over the set of core vectors. We give a polynomial time algorithm for optimizing over the core of any buyer–supplier game for which the underlying minimization problem is solvable in polynomial time. In addition, we show that it is hard to determine whether a given vector belongs to the core if the base minimization problem is not solvable in polynomial time. Finally, we introduce and study the concept of focus point price, which answers the question: If we are constrained to play in equilibrium, how much can we lose by playing the wrong equilibrium?

Case study on statistically estimating minimum makespan for flow line scheduling problems

Lower bounds are typically used to evaluate the performance of heuristics for solving combinatorial minimization problems. In the absence of tight analytical lower bounds, optimal objective-function values may be estimated statistically. In this paper, extreme value theory is used to construct confidence-interval estimates of the minimum makespan achievable when scheduling nonsimilar groups of jobs on a two-stage flow line. Experimental results based on randomly sampled solutions to each of 180 randomly generated test problems revealed that (i) least-squares parameter estimators outperformed standard analytical estimators for the Weibull approximation to the distribution of the sample minimum makespan; (ii) to evaluate each Weibull fit reliably, both the Anderson–Darling and Kolmogorov–Smirnov goodness-of-fit tests should be used; and (iii) applying a local improvement procedure to a large sample of randomly generated initial solutions improved the probability that the resulting Weibull fit yielded a confidence interval covering the minimum makespan.

Hereditary systems and greedy-type algorithms

We define a hereditary system on a finite set U as a partition of the family 2 U of all subsets of U into disjoint families A and D satisfying (A∈ A,A′⊆A)⇒A′∈ A and (D∈ D,D⊆D′)⇒D′∈ D , respectively. The members of A are called independent sets, the sets D∈ D are called dependent. We consider two important special cases of hereditary systems, matroids and comatroids, and study the structure of these objects. Two general combinatorial optimization problems on a hereditary system, the maximum independent set problem max{f(X) : X∈ A} and the minimum dependent set problem min{f(X) : X∈ D} , are considered. Jenkyns, Korte and Hausmann obtained a performance guarantee of the greedy heuristic (‘best in’) for the maximum independent set problem. We present a greedy-type approximation algorithm for solving the minimum dependent set problem, the steepest descent heuristic (‘worst out’), study interconnections between the above-mentioned problems and derive performance guarantees of the steepest descent algorithm. Finally, we apply our results to obtain performance guarantees of the steepest descent algorithm for some known special combinatorial minimization problems.

A method for classifying metabolites in topological pathway analyses based on minimization of pathway number

Metabolic pathway analysis based on the concept of elementary flux mode is a valuable tool for reconstruction of bacterial metabolisms and in predicting optimal conversion yields in biotechnology. However, pathway analysis of large and highly entangled metabolic networks meets the problem of combinatorial explosion of possible routes across the networks. Here we propose a method for coping with this problem by suitably classifying metabolites as external or internal. External metabolites are considered to have buffered concentrations while internal metabolites have to fulfil a balance condition at steady state. For many substances such as nutrients and excreted products, there are biochemical reasons to classify them as external. In addition, other substances (especially at central branching points) can operationally be considered external in order to avoid combinatorial explosion. We suggest to find such a classification of metabolites that minimizes the number of elementary flux modes (pathways). This is motivated by the objectives of finding such a description of the system that reduces as much as possible the amount of necessary data and of removing the ambiguity and arbitrariness in the classification of metabolites in an automated, systematic way. For networks of moderate size, the solution to this combinatorial minimization problem can be found by exhaustive search. To tackle also larger systems, a stochastic optimization program based on the Metropolis algorithm was developed. Both methods are applied, for illustration, to several reaction schemes including a larger network representing glutathione metabolism.

Using simulated annealing to obtain optimal linear end-member mixtures of hyperspectral data

Current thinking in hyperspectral remote sensing suggests the existence of “pure” end-member components. Once identified these components become apices for an n-dimensional convex hull, or minimum encompassing hypervolume, containing all remaining data points. Data in the hypervolume are considered linear mixtures of the apex components. As such, each spectra can be constructed of linear combinations of end-member components. Deriving the relative proportions of end-member components is non-trivial and is often computed using variations of least-squares techniques. These solutions have not been shown to be optimal for combinatorial minimization problems. An exhaustive search of the domain space is not practical and the discrete nature of the data precludes the notion of search based on direction, i.e., hill climbing or steepest descent approaches. Simulated annealing (SA) is an optimization strategy easily applied to this domain. Loosely based on thermodynamic cooling models for crystalline materials, SA combines a “random walk” model with a Boltzmann's probability distribution based upon the temperature ( t) of a system. SA operates much like a phase cooling within a magma body, allowing the user to control the “cooling” schedule and preventing “quenching” to a non-optimal state or local minimum. For this study SA is applied to linear combinations of mineral spectra and reproduces the component mixtures within acceptable tolerances.

An analog hopfield network to overcome excessive smoothing in shape from shading

The regularization term used in the cost function of the shape from shading (SFS) problem is necessary to make the problem well-posed. However, it has the disadvantage of producing excessive smoothing of the surface even at places where the gradient of the surface slope is very high. To overcome this we propose the use of a line function depending on the gradient of the surface slope in the regularization term. For an analog line function, the SFS energy function is then minimized using an analog Hopfield network. For a binary line function, the SFS energy minimization problem is formulated as a combinatorial minimization problem; the minimization is implemented using a Boltzmann machine (simulated annealing). Simulation results are presented to substantiate our approach.

Nonequilibrium Simulated Annealing: A Faster Approach to Combinatorial Minimization

A nonequilibrium simulated annealing (NESA) algorithm is presented for solving combinatorial minimization problems. The original Metropolis algorithm and its variant due to Glauber were modified by enforcing the cooling schedule as soon as an improved solution is obtained, without the need to reach near-equilibrium conditions at each temperature level. The original and modified algorithms were applied to the classical traveling salesman problem and to the optimization of a pressure relief header network. Statistical evaluation of the performance of the algorithms revealed that the proposed modification resulted in a faster approach to the global optimum. A simple stopping criterion, based on an averaged gradient of the objective function with respect to the number of function evaluations, provides a convenient method to control the convergence of the modified algorithms.

Magnet sorting algorithms for insertion devices for the Advanced Light Source

Insertion devices for the Advanced Light Source (ALS) incorporate up to 3000 magnet blocks each for pole energization. In order to minimize field errors, these magnets must be measured, sorted and assigned appropriate locations and orientations in the magnetic structures. Sorting must address multiple objectives, including pole excitation and minimization of integrated multipole fields from minor field components in the magnets. This is equivalent to a combinatorial minimization problem with a large configuration space. Multi-stage sorting algorithms use ordering and pairing schemes in conjunction with other combinatorial methods to solve the minimization problem. This paper discusses objective functions, solution algorithms and results of application to magnet block measurement data.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A Combinatorial Problem Related to Multimodule Memory Organizations

This paper deals with a combinatorial minimization problem arising from studies on multimodule memory organizations. Instead of searching for an optimum solution, a particular solution is proposed and it is demonstrated that it is close to optimum. Lower bounds for the objective functions are obtained and compared with the corresponding values of the particular solution. The maximum percentage deviation of this solution from optimum is also established.