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Abstract

In the Prize-Collecting Steiner Tree Problem (PCStT) we are given a set of customers with potential revenues and a set of possible links connecting these customers with fixed installation costs. The goal is to decide which customers to connect into a tree structure so that the sum of the link costs plus the revenues of the customers that are left out is minimized. The problem, as well as some of its variants, is used to model a wide range of applications in telecommunications, gas distribution networks, protein–protein interaction networks, or image segmentation.In many applications it is unrealistic to assume that the revenues or the installation costs are known in advance. In this paper we consider the well-known Bertsimas and Sim (B&S) robust optimization approach, in which the input parameters are subject to interval uncertainty, and the level of robustness is controlled by introducing a control parameter, which represents the perception of the decision maker regarding the number of uncertain elements that will present an adverse behavior.We propose branch-and-cut approaches to solve the robust counterparts of the PCStT and the Budget Constraint variant and provide an extensive computational study on a set of benchmark instances that are adapted from the deterministic PCStT inputs. We show how the Price of Robustness influences the cost of the solutions and the algorithmic performance.Finally, we adapt our recent theoretical results regarding algorithms for a general class of B&S robust optimization problems for the robust PCStT and its budget and quota constrained variants.

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Thesis
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Article
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Chapter
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Thesis
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We consider the fractional prize-collecting Steiner tree problem on trees...
Article
Cellular signaling and regulatory networks underlie fundamental biological processes such as growth, differentiation, and response to the environment. Although there are now various high-throughput methods for studying these processes, knowledge of them remains fragmentary. Typically, the majority of hits identified by transcriptional, proteomic, and genetic assays lie outside of the expected pathways. In addition, not all components in the regulatory networks can be exposed in one experiment because of systematic biases in the assays. These unexpected and hidden components of the cellular response are often the most interesting, because they can provide new insights into biological processes and potentially reveal new therapeutic approaches. However, they are also the most difficult to interpret. We present a technique, based on the Steiner tree problem, that uses a probabilistic protein-protein interaction network and high confidence measurement and prediction of protein-DNA interactions, to determine how these hits are organized into functionally coherent pathways, revealing many components of the cellular response that are not readily apparent in the original data. We report the results of applying this method to (1) phosphoproteomic and transcriptional data from the pheromone response in yeast, and (2) phosphoproteomic, DNaseI hypersensitivity sequencing and mRNA profiling data from the U87MG glioblastoma cell lines over-expressing the variant III mutant of the epidermal growth factor receptor (EGFRvIII). In both cases the method identifies changes in diverse cellular processes that extend far beyond the expected pathways. Analysis of the EGFRVIII network connectivity property and transcriptional regulators that link observed changes in protein phosphorylation and differential expression suggest a few intriguing hypotheses that may lead to improved therapeutic strategy for glioblastoma.
Article
We improve the well-known result presented in Bertsimas and Sim (Math Program B98:49–71, 2003) regarding the computation of optimal solutions of Robust Combinatorial Optimization problems with interval uncertainty in the objective function coefficients. We also extend this improvement to a more general class of Combinatorial Optimization problems with interval uncertainty.
Article
The following is a valid model for an important class of scheduling and routing problems. A salesman who travels between pairs of cities at a cost depending only on the pair, gets a prize in every city that he vitis and pays a penalty to every city that he fails to visit, wishes to minimize his travel costs and net penalties, while visiting enough cities to collect a prescribed amount of prize money. We call this problem the Prize Collecting Traveling Salesman Problem (PCTSP). This paper discusses structural properties of the PCTS polytope, the convex hull of solutions to the PCTSP. In particular, it identifies several families of facet defining inequalities for this polytope. Some of these inequalities are related to facets of the ordinary TS polytope, others to facets of the knapsack polytope. They can be used in algorithms for the PCTSP either as cutting planes or as ingredients of a Lagrangean optimand.
Article
The general Node-Weighted Steiner Tree problem is an extension of the standard Steiner Tree problem by the addition of node-associated weights. This article analyzes a special case of that problem, where the set of nodes, which must be included in the solution tree, consists of a single node, and all node weights are negative. The special case is shown to be NP-Complete, its integer programming formulation is presented, and heuristic procedures are proposed. Using Lagrangian relaxation and subgradient optimization, tight lower bounds were derived and utilized by a branch and bound algorithm. The effectiveness of the developed procedures is demonstrated by a set of computational experiments.
Chapter
Dedicated to the memory of Albert W. Tucker The primal-dual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primal-dual method can be modified to provide good approximation algorithms for a wide variety of NP-hard problems. We concentrate on results from recent research applying the primal-dual method to problems in network design.
Article
Given an undirected graph with prizes associated with its nodes and weights associated with its edges, the prize-collecting Steiner tree problem consists of finding a subtree of this graph which minimizes the sum of the weights of its edges plus the prizes of the nodes not spanned. In this paper, we describe a multistart local search algorithm for the prize-collecting Steiner tree problem, based on the generation of initial solutions by a primal-dual algorithm using perturbed node prizes. Path-relinking is used to improve the solutions found by local search and variable neighborhood search is used as a post-optimization procedure. Computational experiments involving different algorithm variants are reported. Our results show that the local search with perturbations approach found optimal solutions on nearly all of the instances tested. © 2001 John Wiley & Sons, Inc.
Article
We study efficient implementations of the push—relabel method for the maximum flow problem. The resulting codes are faster than the previous codes, and much faster on some problem families. The speedup is due to the combination of heuristics used in our implementations: we show that the highest-level selection strategy gives better results when combined with both global and gap relabeling heuristics. We also exhibit a family of problems for which the running time of all implementations we consider is quadratic.
Chapter
Complex networks of interactions between genes, proteins, and other molecules choreograph cellular processes. The interactions that are active in the cell change over time, both as a natural outcome of the cell‘s natural life cycle and in response to external signals. The set of active interactions, called the response network, are likely to be significantly different between a normally-functioning cell and a diseased cell. The wide availability of DNA microarray data and experimentallydetermined interaction networks has made it possible to automatically compute response networks. This chapter surveys algorithms that have been developed to compute response networks.
Article
This paper provides a survey of the research in and an annotated bibliography of multiple objective combinatorial optimization, MOCO. We present a general formulation of MOCO problems, describe the main characteristics of MOCO problems, and review the main properties and theoretical results for these problems. The main parts of the paper are a section on the review of the available solution methodology, both exact and heuristic, and a section on the annotation of the existing literature in the field organized problem by problem. We conclude the paper by stating open questions and areas of future research. Der Artikel bietet einen Überblick und eine kommentierte Bibliographie über die Forschung in multikriterieller kombinatorischer Optimierung (MOCO, multiple objective combinatorial optimization). Wir stellen eine allgemeine Formulierung von MOCO Problemen vor, beschreiben die wichtigsten Charakteristika und Eigenschaften solcher Probleme und fassen die wesentlichen theoretischen Ergebnisse in diesem Forschungsgebiet zusammen. Die Hauptteile des Artikels sind die Abschnitte 4 über exakte und heuristsiche Lösungsverfahren und 6, der – problemweise untergliedert – die vorhandene Literatur kommentiert. Am Ende des Artikels steht ein Abschnitt zu offenen Fragen und Richtungen für zukünftige Forschung.
We propose a branch-and-cut strategy for efficient region-based object detection. Given an oversegmented image, our method determines the subset of spatially contiguous regions whose collective features will maximize a classifier's score. We formulate the objective as an instance of the prize-collecting Steiner tree problem, and show that for a family of additive classifiers this enables fast search for the optimal object region via a branch-and-cut algorithm. Unlike existing branch-and-bounddetection methods designed for bounding boxes, our approach allows scoring of irregular shapes - which is especially critical for objects that do not conform to a rectangular window. We provide results on three challenging object detection datasets, and demonstrate the advantage of rapidly seeking best-scoring regions rather than subwindow rectangles.
Article
We consider the version of prize collecting Steiner tree problem (PCSTP) where each node of a given weighted graph is associated with a prize and where the objective is to find a minimum weight tree spanning a subset of nodes and collecting a total prize not less that a given quota Q. We present a lower bound and a genetic algorithm for the PCSTP. The lower bound is based on a Lagrangian decomposition of a minimum spanning tree formulation of the problem. The volume algorithm is used to solve the Lagrangian dual. The genetic algorithm incorporates several enhancements. In particular, it fully exploits both primal and dual information produced by Lagrangian decomposition. The proposed lower and upper bounds are assessed through computational experiments on randomly generated instances with up to 500 nodes and 5000 edges. For these instances, the proposed lower and upper bounds exhibit consistently a tight gap: in 76% of the cases the gap is strictly less than 2%.
Article
This article introduces a proper redefinition of the concept of bottleneck Steiner distance for the prize-collecting Steiner problem. This allows the application of reduction tests known to be effective on Steiner problem in graphs in their full power. Computational experiments attest the effectiveness of the proposed tests.
Article
In this paper, we present an integer programming formulation of the prize collecting Steiner problem in graphs (PCSPG) and describe an algorithm to obtain lower bounds for the problem. The algorithm is based on polyhedral cutting planes and is initiated with tests that attempt to reduce the size of the input graph. Computational experiments were carried out to evaluate the strength of the formulation through its linear programming relaxation. On 96 out of the 114 instances tested, integer solutions were found (thus generating optimal PCSPG solutions).
We study the prize-collecting versions of the Steiner tree, traveling salesman, and stroll (a.k.a. Path-TSP) problems (PCST, PCTSP, and PCS, respectively): given a graph (V, E) with costs on each edge and a penalty (a.k.a. prize) on each node, the goal is to find a tree (for PCST), cycle (for PCTSP), or stroll (for PCS) that minimizes the sum of the edge costs in the tree/cycle/stroll and the penalties of the nodes not spanned by it. In addition to being a useful theoretical tool for helping to solve other optimization problems, PCST has been applied fruitfully by AT&T to the optimization of real-world telecommunications networks. The most recent improvements for the first two problems, giving a 2-approximation algorithm for each, appeared first in 1992. (A 2-approximation for PCS appeared in 2003.) The natural linear programming (LP) relaxation of PCST has an integrality gap of 2, which has been a barrier to further improvements for this problem. We present (2 · ¿)-approximation algorithms for all three problems, connected by a unified technique for improving prize-collecting algorithms that allows us to circumvent the integrality gap barrier.
Article
We consider the problem of optimizing a novel acoustic leakage detection system for urban water distribution networks. The system is composed of a number of detectors and transponders to be placed in a choice of hydrants such as to provide a desired coverage under given budget restrictions. The problem is modeled as a particular Prize-Collecting Steiner Arborescence Problem. We present a branch-and-cut-and-bound approach taking advantage of the special structure at hand which performs well when compared to other approaches. Furthermore, using a suitable stopping criterion, we obtain approximations of provably excellent quality (in most cases actually optimal solutions). The test bed includes the real water distribution network from the Lausanne region, as well as carefully randomly generated realistic instances.
Article
Given an undirected graph G with penalties associated with its vertices and costs associated with its edges, a Prize Collecting Steiner (PCS) tree is either an isolated vertex of G or else any tree of G, be it spanning or not. The weight of a PCS tree equals the sum of the costs for its edges plus the sum of the penalties for the vertices of G not spanned by the PCS tree. Accordingly, the Prize Collecting Steiner Problem in Graphs (PCSPG) is to find a PCS tree with the lowest weight. In this paper, after reformulating and re-interpreting a given PCSPG formulation, we use a Lagrangian Non Delayed Relax and Cut (NDRC) algorithm to generate primal and dual bounds to the problem. The algorithm was capable of adequately dealing with the expo-nentially many candidate inequalities to dualize. It incorporates ingredients such as a new PCSPG reduction test, an effective Lagrangian heuristic and a modification in the NDRC framework that allowed duality gaps to be further reduced. The Lagrangian heuristic suggested here dominates their PCSPG counterparts in the literature. The NDRC PCSPG lower bounds, most of the time, nearly matched corresponding Linear Programming relaxation bounds.
Article
In this paper, we establish a new model for path planning with interval data which arises in a variety of applications. It is formulated as minimum risk-sum path problem: given a source-destination pair in a network G=(V,E), traveling on each link e in G may take time xe in a prespecified interval [le,ue] and take risk (ue-xe)/(ue-le), the goal is to find a path in G from the source to the destination, together with an allocation of travel times along each link on the path, so that the total travel time of links on the path is no more than a given time bound and the risk-sum over the links on the path is minimized. Our study shows that this new model has two features that make it different from the existing models. First, the minimum risk-sum path problem is polynomial-time solvable, and second, it provides many solutions that vary with time bounds and risk sums and leaves the choice for decision makers. Therefore, the new model is more flexible and easier to use for the path planning with interval data.
Article
The robust optimization framework proposed by Bertsimas and Sim accounts for data uncertainty in integer linear programs. This article investigates the polyhedral impacts of this robust model for the 0-1 knapsack problem. In particular, classical cover cuts are adapted to provide valid inequalities for the robust knapsack problem. The strength of the proposed inequalities is studied theoretically. Then, experiments on the robust bandwidth packing problem illustrate the practical interest of these inequalities for solving hard robust combinatorial problems. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012
Article
This paper investigates the Prize Collecting Steiner Tree Problem (PCSTP) on a graph, which is a generalization of the well-known Steiner tree problem. Given a root node, edge costs, node prizes and penalties, as well as a preset quota, the PCSTP seeks to find a subtree that includes the root node and collects a total prize not smaller than the specified quota, while minimizing the sum of the total edge costs of the tree plus the penalties associated with the nodes that are not included in the subtree. For this challenging network design problem that arises in telecommunication settings, we present two valid 0-1 programming formulations and use them to develop preprocessing procedures for reducing the graph size. Also, we design an optimization-based heuristic that requires solving a PCSTP on a specific tree-subgraph. Although, this latter special case is shown to be NP-hard, it is effectively solvable in pseudo-polynomial time. The worst-case performance of the proposed heuristic is also investigated. In addition, we describe new valid inequalities for the PCSTP and embed all the aforementioned constructs in an exact row-generation approach. Our computational study reveals that the proposed approach can solve relatively large-scale PCSTP instances having up to 1000 nodes to optimality.
Article
Cellular signaling and regulatory networks underlie fundamental biological processes such as growth, differentiation, and response to the environment. Although there are now various high-throughput methods for studying these processes, knowledge of them remains fragmentary. Typically, the majority of hits identified by transcriptional, proteomic, and genetic assays lie outside of the expected pathways. These unexpected components of the cellular response are often the most interesting, because they can provide new insights into biological processes and potentially reveal new therapeutic approaches. However, they are also the most difficult to interpret. We present a technique, based on the Steiner tree problem, that uses previously reported protein-protein and protein-DNA interactions to determine how these hits are organized into functionally coherent pathways, revealing many components of the cellular response that are not readily apparent in the original data. Applied simultaneously to phosphoproteomic and transcriptional data for the yeast pheromone response, it identifies changes in diverse cellular processes that extend far beyond the expected pathways.
Article
We consider variants on the Prize Collecting Steiner Tree problem and on the primal-dual 2-approximation algorithm devised for it by Goemans and Williamson. We introduce an improved pruning rule for the algorithm that is slightly faster and provides solutions that are at least as good and typically significantly better. On a selection of real-world instances whose underlying graphs are county street maps, the improvement in the standard objective function ranges from 1.7% to 9.2%. Substantially better improvements are obtained for the complementary "net worth" objective function and for randomly generated instances. We also show that modifying the growth phase of the GoemansWilliamson algorithm to make it independent of the choice of root vertex does not significantly affect the algorithm's worst-case guarantee or behavior in practice. The resulting algorithm can be further modified so that, without an increase in running time, it becomes a 2-approximation algorithm for finding the bes...
Conference Paper
. We study efficient implementations of the push-relabel method for the maximum flow problem. The resulting codes are faster than the previous codes, and much faster on some problem families. The speedup is due to the combination of heuristics used in our implementation. We also exhibit a family of problems for which all known methods seem to have almost quadratic time growth rate. Andrew V. Goldberg was supported in part by NSF Grant CCR-9307045 and a grant from Powell Foundation. This work was done while Boris V. Cherkassky was visiting Stanford University Computer Science Department and supported by the above-mentioned NSF and Powell Foundation grants. 1 1. Introduction The maximum flow problem is a classical combinatorial problem that comes up in a wide variety of applications. In this paper we study implementations of the push-relabel [13, 17] method for the problem. The basic methods for the maximum flow problem include the network simplex method of Dantzig [6, 7], the augmen...
Article
In this paper we present the implementation of a branch-and-cut algorithm for solving Steiner tree problems in graphs. Our algorithm is based on an integer programming formulation for directed graphs and comprises preprocessing, separation algorithms and primal heuristics. We are able to solve all problem instances discussed in literature to optimality, including one to our knowledge not yet solved problem. We also report on our computational experiences with some very large Steiner tree problems arising from the design of electronic circuits. All test problems are gathered in a newly introduced library called SteinLib that is accessible via World Wide Web. Keywords. Branch-and-Cut, Cutting Planes, Reduction Methods, Steiner tree, Steiner tree library. Mathematical Subject Classification (1995): 05C40, 90C06, 90C10, 90C35. 1 Introduction Given an undirected graph G = (V; E) and a node set T ` V , a Steiner tree for T in G is a subset S ` E of the edges such that (V (S); S) contains...
Collection of test data sets for a variety of operations research (OR) problems
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