Parametric Maximum Flow Research Articles

The Application of Stochastic Mine Production Scheduling in the Presence of Geological Uncertainty

The scheduling of open-pit mine production is a large-scale, mixed-integer linear programming problem that is computationally expensive. The purpose of this study is to create a computationally efficient algorithm for solving open-pit production scheduling problems with uncertain geological parameters. To demonstrate the effectiveness of the proposed research, a case study of an Indian iron ore mine is presented. Multiple realizations of the resource models were developed and integrated within the stochastic production scheduling framework to capture uncertainty and incorporate it into the mine plan. In this case study, two hybrid methods were developed to evaluate their performance. Model 1 is a combined branch and cut with the longest path, whereas Model 2 is a sequential parametric maximum flow and branch and cut. The results show that both methods produce similar materials, ore, metal, and risk profiles; however, Model 2 generates slightly more (4 percent) discounted cash flow from this study mine than Model 1. The results also show that Model 2’s computational time is 46.64 percent less than that of Model 1.

A Parametric Network Approach for Concepts Hierarchy Generation in Text Corpus

Abstract The article presents a preflow approach for the parametric maximum flow problem, derived from the rules of constructing concepts hierarchy in text corpus. Just as generating a taxonomy can be equivalently reduced to ranking concepts within a text corpus according to a defined criterion, the proposed preflow bipush-relabel algorithm computes the maximum flow - the optimum ow that respects certain ranking constraints. The parametric preflow algorithm for generating two level concepts hierarchy in text corpus works in a parametric bipartite association network and, on each step, the maximum possible amount of ow is pushed along conditional augmenting two-arcs directed paths in the parametric residual network, for the maximum interval of the parameter values. The obtained parametric maximum ow generates concepts hierarchies (taxonomies) in text corpus for different degrees of association values described by the parameter values.

Solving the parametric bipartite maximum flow problem in unbalanced and closure bipartite graphs

Given a directed bipartite graph $$G=(V,E)$$ with a node set $$V=\{s\} \cup V_1 \cup V_2 \cup \{t\}$$ , and an arc set $$E=E_1\cup E_2\cup E_3$$ , where $$E_1=\{s\} \times V_1$$ , $$E_2= V_1 \times V_2$$ , and $$E_3=V_2 \times \{t\}$$ . Chen (1995) presented an $$O(|V| |E| \log (\frac{|V|^2}{|E|}))$$ time algorithm to solve the parametric bipartite maximum flow problem. In this paper, we assume all arcs in $$E_2$$ have infinite capacity [such a graph is called closure graph Hochbaum (1998)], and present a new approach to solve the problem, which runs in $$O(|V_1| |E| \log (\frac{|V_1|^2}{|E|}+2))$$ time using Gallo et al’s parametric maximum flow algorithm, see Gallo et al. (1989). In unbalanced bipartite graphs, we have $$|V_1| << |V_2|$$ , so, our algorithm improves Chens’s algorithm in unbalanced and closure bipartite graphs.

Implementing a parametric maximum flow algorithm for optimal open pit mine design under uncertain supply and demand

Conventional open pit mine optimization models for designing mining phases and ultimate pit limit do not consider expected variations and uncertainty in metal content available in a mineral deposit (supply) and commodity prices (market demand). Unlike the conventional approach, a stochastic framework relies on multiple realizations of the input data so as to account for uncertainty in metal content and financial parameters, reflecting potential supply and demand. This paper presents a new method that jointly considers uncertainty in metal content and commodity prices, and incorporates time-dependent discounted values of mining blocks when designing optimal production phases and ultimate pit limit, while honouring production capacity constraints. The structure of a graph representing the stochastic framework is proposed, and it is solved with a parametric maximum flow algorithm. Lagragnian relaxation and the subgradient method are integrated in the proposed approach to facilitate producing practical designs. An application at a copper deposit in Canada demonstrates the practical aspects of the approach and quality of solutions over conventional methods, as well as the effectiveness of the proposed stochastic approach in solving mine planning and design problems.

Parametric maximum flow methods for minimax approximation of target quotas in biproportional apportionment

AbstractIn this article, we study the biproportional apportionment problem, which deals with the assignment of seats to parties within regions. We consider the minimization of both the maximum absolute error and the maximum relative error of the apportioned seats with respect to target quotas. We show that this can be done polynomially through a reduction to a parametric maximum flow problem. Moreover, the maximum absolute error can be minimized in strongly polynomial time. More generally, our method can be used for computing $\ell_{\infty}$ projections onto a flow polytope. We also address the issue of uniqueness of the solution, proposing a method based on finding unordered lexicographic minima. Our procedure is compared to other well‐known ones available in the literature. Finally we apply our procedures to the data of the 2008 Italian political elections, for which the procedure stated by the law produced an inconsistent assignment of seats. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012

On Total Variation Minimization and Surface Evolution Using Parametric Maximum Flows

In a recent paper Boykov et al. (LNCS, Vol. 3953, pp. 409–422, 2006) propose an approach for computing curve and surface evolution using a variational approach and the geo-cuts method of Boykov and Kolmogorov (International conference on computer vision, pp. 26–33, 2003). We recall in this paper how this is related to well-known approaches for mean curvature motion, introduced by Almgren et al. (SIAM Journal on Control and Optimization 31(2):387–438, 1993) and Luckhaus and Sturzenhecker (Calculus of Variations and Partial Differential Equations 3(2):253–271, 1995), and show how the corresponding problems can be solved with sub-pixel accuracy using Parametric Maximum Flow techniques. This provides interesting algorithms for computing crystalline curvature motion, possibly with a forcing term.

Parametric Maximum Flow Algorithms for Fast Total Variation Minimization

This report studies the global minimization of anisotropically discretized total variation (TV) energies with an $L^p$ (in particular, $L^1$ and $L^2$) fidelity term using parametric maximum flow algorithms to minimize $s$-$t$ cut representations of these energies. The TV/$L^2$ model, also known as the Rudin-Osher-Fatemi (ROF) model, is suitable for restoring images contaminated by Gaussian noise, while the TV/$L^1$ model is able to remove impulsive noise from grayscale images and perform multiscale decompositions of them. Preliminary numerical results on large-scale two-dimensional CT and three-dimensional brain MR images are presented to illustrate the effectiveness of these approaches.

Flow Faster: Efficient Decision Algorithms for Probabilistic Simulations

Strong and weak simulation relations have been proposed for Markov chains, while strong simulation and strong probabilistic simulation relations have been proposed for probabilistic automata. However, decision algorithms for strong and weak simulation over Markov chains, and for strong simulation over probabilistic automata are not efficient, which makes it as yet unclear whether they can be used as effectively as their non-probabilistic counterparts. This paper presents drastically improved algorithms to decide whether some (discrete- or continuous-time) Markov chain strongly or weakly simulates another, or whether a probabilistic automaton strongly simulates another. The key innovation is the use of parametric maximum flow techniques to amortize computations. We also present a novel algorithm for deciding strong probabilistic simulation preorders on probabilistic automata, which has polynomial complexity via a reduction to an LP problem. When extending the algorithms for probabilistic automata to their continuous-time counterpart, we retain the same complexity for both strong and strong probabilistic simulations.

A Faster Algorithm for Computing the Principal Sequence of Partitions of a Graph

We consider the following problem: given an undirected weighted graph G=(V,E,c) with nonnegative weights, minimize function c(δ(Π))−λ|Π| for all values of parameter λ. Here Π is a partition of the set of nodes, the first term is the cost of edges whose endpoints belong to different components of the partition, and |Π| is the number of components. The current best known algorithm for this problem has complexity O(|V|2) maximum flow computations. We improve it to |V| parametric maximum flow computations. We observe that the complexity can be improved further for families of graphs which admit a good separator, e.g. for planar graphs.

A note on the parametric maximum flow problem and some related reoptimization issues

In this paper, we will extend the results about the parametric maximum flow problem to networks in which the parametrization of the arc capacities can involve both the source and the sink, as in Gallo, Grigoriadis, and Tarjan (1989), and also an additional node. We will show that the minimum cuts of the investigated networks satisfy a relaxed form of the generalized nesting property (Arai, Ueno, and Kajitani, 1993). A consequence is that the corresponding parametric maximum flow value function has at most n −1 breakpoints. All the minimum cut capacities can therefore be computed by O(1) maximum flow computations.

Ordered optimal solutions and parametric minimum cut problems

In this paper, we present an algebraic sufficient condition for the existence of a selection of optimal solutions in a parametric optimization problem that are totally ordered, but not necessarily monotone. Based on this result, we present necessary and sufficient conditions that ensure the existence of totally ordered selections of minimum cuts for some classes of parametric maximum flow problems. These classes subsume the class studied by Arai et al. [Discrete Appl. Math. 41 (1993) 69–74] as a special case.

Parametric Maximum Flows And The Calculation Of Optimal Intermediate Contours In Open Pit Mine Design

The open pit mine design problem is that of deciding which blocks of an ore deposit should be mined so as to maximize total profit while respecting certain digging constraints. In this article we show, through a parametric maximum flow analysis, how intermediale contours should be mined so as to maximize the benefit-cost ratio while proceeding toward the ultimate mine contour.

A Fast Parametric Submodular Intersection Algorithm for Strong Map Sequences

This paper presents a fast algorithm to solve the intersection problem for a pair of nondecreasing and nonincreasing strong map sequences of submodular systems. The worst-case time bound is the same as that of the push/relabel algorithm for a single intersection problem. This extends the Gallo-Grigoriadis-Tarjan (GGT) method for parametric maximum flow problems and reveals an algorithmic significance of the concept of strong maps for submodular systems.

On strongly polynomial dual simplex algorithms for the maximum flow problem

Several pivot rules for the dual network simplex algorithm that enable it to solve a maximum flow problem on an n-node, m-arc network in at most 2nm pivots and O(n 2 m) time are presented. These rules are based on the concept of a preflow and depend upon the use of node labels which are either the lengths of a shortest pseudoaugmenting path from those nodes to the sink node or valid underestimates of those lengths. Extended versions of our algorithms are shown to solve an important class of parametric maximum flow problems with no increase in the worst-case pivot and time bounds of these algorithms.

A parametric maximum flow algorithm for bipartite graphs with applications

Suppose we are given a capacitated bipartite network G with node sets S and T. In network G, the arc capacities are not fixed values but are functions of a single parameter λ, where λ is a continuous real variable. Our problem is to determine the minimum value of λ such that the maximum flow value in the corresponding network equals a given threshold. For this problem, an algorithm of time complexity O( nm 2 log( m/ n)) is presented, where n is the number of nodes in S, m is the number of nodes in T and n ≤ m. Examples are then given to show how to use this parametric algorithm to solve practical problems.

Generalization of a theorem on the parametric maximum flow problem

A general theorem on the nesting property of minimum cuts in a parametric network and its consequence to the parametric maximum flow problem are presented.

Distance-directed augmenting path algorithms for maximum flow and parametric maximum flow problems

Until recently, fast algorithms for the maximum flow problem have typically proceeded by constructing layered networks and establishing blocking flows in these networks. However, in recent years, new distance-directed algorithms have been suggested that do not construct layered networks but instead maintain a distance label with each node. The distance label of a node is a lower bound on the length of the shortest augmenting path from the node to the sink. In this article we develop two distance-directed augmenting path algorithms for the maximum flow problem. Both the algorithms run in O(n2m) time on networks with n nodes and m arcs. We also point out the relationship between the distance labels and layered networks. Using a scaling technique, we improve the complexity of our distance-directed algorithms to O(nm log U), where U denotes the largest arc capacity. We also consider applications of these algorithms to unit capacity maximum flow problems and a class of parametric maximum flow problems.

A Fast Parametric Maximum Flow Algorithm and Applications

The classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter, and the goal is to find the value of the parameter such that the corresponding maximum flow or minimum cut satisfies some side condition. Finding the desired parameter value requires solving a sequence of related maximum flow problems. In this paper it is shown that the recent maximum flow algorithm of Goldberg and Tarjan can be extended to solve an important class of such parametric maximum flow problems, at the cost of only a constant factor in its worst-case time bound. Faster algorithms for a variety of combinatorial optimization problems follow from the result.