## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

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.

To read the full-text of this research,

you can request a copy directly from the authors.

... The code is available as open source. 3 Rehfeldt and Koch [198] studied the MWCS and provided a combination of (NP-hard) reduction tests, primal heuristics, a problem transformation into the SAP and the integration of these techniques into a B&C framework. They further improved the computational times from Leitner et al. [153] and solved one more difficult problem instance to optimality. ...

... With data veracity in big-data applications, there is a diminishing demand in producing STP solutions with zero gap. Instead, practitioners desire stable solutions, robust with respect to minor data modifications [3], and with small duality gaps. Decomposition approaches might be promising candidates to meet these demands stemming from more computationally demanding real-world applications. ...

The Steiner tree problem (STP) in graphs is one of the most studied problems in combinatorial optimization. Since its inception in 1970, numerous articles published in the journal Networks have stimulated new theoretical and computational studies on Steiner trees: from approximation algorithms, heuristics, metaheuristics, all the way to exact algorithms based on (mixed) integer linear programming, fixed parameter tractability, or combinatorial branch‐and‐bounds. The pervasive applicability and relevance of Steiner trees have been reinforced by the recent 11th DIMACS Implementation Challenge in 2014 and the PACE 2018 Challenge. This article provides an overview of the rich developments from the last three decades for the STP in graphs and highlights the most recent computational studies for some of its closely related variants.

... Besides, Bertsimas & Sim (2003) have proposed an algorithm for robust network flows using their model. The min-max with discrete scenarios has been mostly tested on facility location problems (Averbakh & Bereg (2005)), robust prize-collecting Steiner tree problems (Álvarez-Miranda et al. (2013)), robust knapsack problem (Monaci et al. (2013)) and robust network loading problem with dynamic routing (Mattia (2013)). ...

This work extends the Vehicle Routing Problem (VRP) for addressing uncertainties via robust optimization, giving the Robust VRP (RVRP). First, uncertainties are handled on travel times/costs. Then, a bi-objective version (bi-RVRP) is introduced to handle uncertainty in both, travel times and demands. For solving the RVRP and the bi-RVRP different models and methods are proposed to determine robust solutions minimizing the worst case. A Mixed Integer Linear Program (MILP), several greedy heuristics, a Genetic Algorithm (GA), a local search procedure and four local search based algorithms are proposed: a Greedy Randomized Adaptive Search Procedure (GRASP), an Iterated Local Search (ILS), a Multi-Start ILS (MS-ILS), and a MS-ILS based on Giant Tours (MS-ILS-GT) converted into feasible routes via a lexicographic splitting procedure. Concerning the bi-RVRP, the total cost of traversed arcs and the total unmet demand are minimized over all scenarios. To solve the problem, different variations of multiobjective evolutionary metaheuristics are proposed and coupled with a local search procedure: the Multiobjective Evolutionary Algorithm (MOEA) and the Non-dominated Sorting Genetic Algorithm version 2 (NSGAII). Different metrics are used to measure the efficiency, the convergence as well as the diversity of solutions for all these algorithms

... This result is based on the work of Wald and Colbourn [44] who proved that steiner tree problem is polynomial time solvable on 2-tree graphs. An algorithm for robust prize collecting steiner tree problem was proposed by Miranda et al. [6]. There are other approaches for solving the PCST problem. ...

Constructing a steiner tree of a graph is a fundamental problem in many applications. Prize collecting steiner tree (PCST) is a special variant of the steiner tree problem and has applications in network design, content distribution etc. There are a few centralized approximation algorithms \cite{DB_MG_DS_DW_1993, GW_1995, AA_MB_MH_2011} for solving the PCST problem. However no distributed algorithm is known that solves the PCST problem with non-trivial approximation factor. In this work we present a distributed algorithm that constructs a prize collecting steiner tree for a given connected undirected graph with non-negative weight for each edge and non-negative prize value for each node. Initially each node knows its own prize value and weight of each incident edge. Our algorithm is based on primal-dual method and it achieves an approximation factor of $(2 - \frac{1}{n - 1})$ of the optimal. The total number of messages required by our distributed algorithm to construct the PCST for a graph with $|V|$ nodes and $|E|$ edges is $O(|V|^2 + |E||V|)$. The algorithm is spontaneously initiated at a special node called the root node and when the algorithm terminates each node knows whether it is in the prize part or in the steiner tree of the PCST.

... Studies [10,22,33,41] provide a collection of such problems, along with their complexity analysis. An example where constraints on the tree structure make the problem difficult is the Steiner Tree problem [3,24,34,38]. Given a subset N V of Steiner nodes (also referred as terminal nodes), the Steiner Tree problem consists in determining a particular minimum tree covering N . ...

Trees and forests have been a fascinating research topic in Operations Research (OR)/Management Science (MS) throughout the years because they are involved in numerous difficult problems, have interesting theoretical properties, and cover a large number of practical applications. A tree is a finite undirected connected simple graph with no cycles, while a set of independent trees is called a forest. A spanning tree is a tree covering all nodes of a graph. In this chapter, key components for solving difficult tree and forest problems, as well as insights to develop efficient heuristics relying on such structures, are surveyed. They are usually combined to obtain very efficient metaheuristics, hybrid methods, and matheuristics. Some emerging topics and trends in trees and forests are pointed out. This is followed by two case studies: a Lagrangian-based heuristic for the minimum degree-constrained spanning tree problem and an evolutionary algorithm for a generalization of the bounded-diameter minimum spanning tree problem. Both problems find applications in network design, telecommunication, and transportation fields, among others.

... Examples of the former are the maximal covering location (Church and ReVelle, 1974) or the competitive facility location problem (Aboolian et al., 2007). Examples of the latter are prizecollecting versions of problems that do not consider locational decisions: traveling salesman (Feillet et al., 2005), vehicle routing (Aras et al., 2011), rural postman (Aráoz et al., 2009), and prize-collecting Steiner tree problems (Álvarez-Miranda et al., 2013). ...

This paper presents a class of hub network design problems with profit-oriented objectives, which extend several families of classical hub location problems. Potential applications arise in the design of air and ground transportation networks. These problems include decisions on the origin/destination nodes that will be served as well as the activation of different types of edges, and consider the simultaneous optimization of the collected profit, setup cost of the hub network and transportation cost. Alternative models and integer programming formulations are proposed and analyzed. Results from computational experiments show the complexity of such models and highlight their superiority for decision-making.

... A survey of Steiner tree problems was given by Hwang and Richards [10], Alvarez-Miranda et al. [11] and Fu and Hao [12]. Several exact algorithms have been proposed, such as the dynamic programming technique by Dreyfuss and Wagner [13], Lagrangean relaxation approach by Beasley [14] and brand-andcut algorithm by Koch and Martin [15]. ...

A convergent product is an assembly shape concept integrating functions and sub-functions to form a final product. To conceptualize the convergent product problem, a web-based network is considered in which a collection of base functions and sub-functions configure the nodes, and each arc in the network is considered to be a link between two nodes. The aim is to find an optimal tree of functionalities in the network, adding value to the product in the web environment. First, an algorithm is proposed to assign the links among bases and sub-functions. Then, numerical values, as benefits and costs, are determined for arcs and nodes, respectively, using a mathematical approach. Also, customer value corresponding to the benefits is considered. Finally, the Steiner tree methodology is adapted to a multi-objective model optimized by an augmented epsilon-constraint method. An example is worked out to illustrate the proposed approach.

... Intell. Spring (March) given ( Konak and Smith, 2011;Khandekara, Kortsarz and Nutov, 2013;Simonetti, Protti and Frota, 2011;Alvarez-Miranda and Ljubic, 2013;Bansal, Khandekar and Konemann, 2013). In the second variant, connectivity properties of a network are a kind of surrogate for reliability. ...

In this paper we consider the NP-complete problem of finding a spanning k-tree of minimum weight in a complete weighted graph. This problem has a number of applications in designing reliable backbone telecommunication networks. We propose effective algorithms based on a greedy strategy and several variable neighborhood search metaheuristics. We also develop an integer linear programming model for calculating a lower bound. Preliminary numerical experiments using random and real-word data sets are reported to show
the effectiveness of our approach. In addition, we compare our approach with known metaheuristics.

... The first one, when a network topology is not specified and it is computed in the process of solving the problem [4,5,6,7]. The second one, when the network topology is given [8,9,10,11,12]. In the second variant, connectivity requirements of a network are a kind of surrogate for reliability [13]. Researches in this field have led to the appearance of the concept of IFI (Isolated Failure Immune) networks [14] in the 80s of the last century. ...

We consider the NP-complete problem of finding a spanning \(k\) -tree of minimum weight in a complete weighted graph. This problem has a number of applications in designing reliable backbone telecommunication networks. We propose effective algorithms based on a greedy strategy and several variable neighborhood search metaheuristics. We also develop an integer linear programming model for calculating a lower bound. Preliminary numerical experiments using random and real-word data sets are reported to show the effectiveness of our approach. In addition, we compare our approach with known metaheuristics.

... Besides, the authors have proposed an algorithm for robust network flows using their model. This criterion has been mostly used for solving facility location problems [6], robust prize-collecting Steiner tree problems [4], robust knapsack problem [42] and the robust network loading problem with dynamic routing [40]. Roy (2010) [53] argues the minmax approach fails to translate the robustness since it focuses only on minimizing the worst case. ...

Uncertain parameters appear in many optimization problems raised by real-world applications. To handle such problems, several approaches to model uncertainty are available, such as stochastic programming and robust optimization. This study is focused on robust optimization, in particular, the criteria to select and determine a robust solution. We provide an overview on robust optimization criteria and introduce two new classications criteria for measuring the robustness of both scenarios and solutions. They can be used independently or coupled with classical robust optimization criteria and could work as a complementary tool for intensification in local searches.

Let G = (V, E) be a connected graph, where V and E represent, respectively, the node-set and the edge-set. Besides, let Q ⊆ V be a set of terminal nodes, and r ∈ Q be the root node of the graph. Given a weight c ij ∈ N associated to each edge (i, j) ∈ E, the Steiner Tree Problem in graphs (STP) consists in finding a minimum-weight subgraph of G that spans all nodes in Q. In this paper, we consider the Min-max Regret Steiner Tree Problem with Interval Costs (MMR-STP), a robust variant of STP. In this variant, the weight of the edges are not known in advance, but are assumed to vary in the interval [l_{ij}, u_{ij}]. We develop an ILP formulation, an exact algorithm, and three heuristics for this problem. Computational experiments, performed on generalizations of the classical STP instances, evaluate the efficiency and the limits of the proposed methods.

We previously studied the capacitated arc routing problem over sparse underlying graphs under travel costs uncertainty. In this paper, we study the same problem by recalling the mathematical formulation of the problem given in [29]. The problem is characterized by the uncertainty of the travel costs and by the sparse network over which it is defined. In fact, a Multiple-Scenario Min-Max CARP over sparse underlying graphs is studied. More numerical instances applying the greedy heuristic algorithm developed in [29] and the adapted tabu-search algorithm are introduced in which these computational experiments show the effectiveness of these two algorithms.

Cette thèse comporte deux parties majeures : la première partie est dédiée à l'étude du problème sparse CARP déterministe où nous avons développé une transformation du sparse CARP en un sparse CVRP. La seconde est consacrée au problème sparse CARP avec coûts sous incertitude. Nous avons donné une formulation mathématique du problème en min-max. Cette modélisation a permis d'identifier le pire scénario pour le problème robuste. Deux approches algorithmiques ont été proposées pour une résolution approchée.

We introduce the bi-objective prize-collecting Steiner tree problem, whose goal is to find a subtree considering the conflicting objectives of minimizing the edge costs for building that tree, and maximizing the collected node revenues. We consider five iterative mixed-integer programming (MIP) frameworks that identify the complete Pareto front, i.e., one efficient solution for every point on the Pareto front. More precisely, the following methods are studied: an is an element of-constraint method, a two-phase method, a binary search in the objective space, a weighted Chebyshev norm method, and a method of Sylva and Crema. We also investigate how to exploit and recycle information gained during these iterative MIP procedures to accelerate the solution process. We consider (i) additional strengthening valid inequalities, (ii) procedures for initializing feasible solutions (using a solution pool), (iii) procedures for recycling violated cuts (using a cut pool), and (iv) guiding the branching process by previously detected Pareto optimal solutions. This work is a first study on exact approaches for solving the bi-objective prize-collecting Steiner tree problem. Standard benchmark instances from the literature are used to assess the efficacy of the proposed methods.

In this paper, we study the bi-objective prize-collecting Steiner tree problem, whose goal is to find a subtree that minimizes the edge costs for building that tree, and, at the same time, to maximize the collected node revenues. We propose to solve the problem using an ϵ-constraint algorithm. This is an iterative mixed-integer-programming framework that identifies one solution for every point on the Pareto front. In this framework, a branch-and-cut approach for the single-objective variant of the problem is enhanced with warm-start procedures that are used to (i) generate feasible solutions, (ii) generate violated cutting planes, and (iii) guide the branching process. Standard benchmark instances from the literature are used to assess the efficacy of our method.

This paper presents a robust optimization approach to the network design problem under traffic demand uncertainty. We consider the specific case of the network design problem in which there are several alternatives in edge capacity installations and the traffic cannot be split over several paths. A new decomposition approach is proposed that yields a strong LP relaxation and enables traffic demand uncertainty to be addressed efficiently through localization of the uncertainty to each edge of the underlying network. A branch-and-price-and-cut algorithm is subsequently developed and tested on a set of benchmark instances.

Robustness is about reducing the feasible set of a given nominal optimization problem by cutting “risky” solutions away. To this end, the most popular approach in the literature is to extend the nominal model with a polynomial number of additional variables and constraints, so as to obtain its robust counterpart. Robustness can also be enforced by adding a possibly exponential family of cutting planes, which typically leads to an exponential formulation where cuts have to be generated at run time. Both approaches have pros and cons, and it is not clear which is the best one when approaching a specific problem. In this paper we computationally compare the two options on some prototype problems with different characteristics. We first address robust optimization à la Bertsimas and Sim for linear programs, and show through computational experiments that a considerable speedup (up to 2 orders of magnitude) can be achieved by exploiting a dynamic cut generation scheme. For integer linear problems, instead, the compact formulation exhibits a typically better performance. We then move to a probabilistic setting and introduce the uncertain set covering problem where each column has a certain probability of disappearing, and each row has to be covered with high probability. A related uncertain graph connectivity problem is also investigated, where edges have a certain probability of failure. For both problems, compact ILP models and cutting plane solution schemes are presented and compared through extensive computational tests. The outcome is that a compact ILP formulation (if available) can be preferable because it allows for a better use of the rich arsenal of preprocessing/cut generation tools available in modern ILP solvers. For the cases where such a compact ILP formulation is not available, as in the uncertain connectivity problem, we propose a restart solution strategy and computationally show its practical effectiveness.

This paper introduces a robust inventory routing problem where a supplier distributes a single product to multiple customers facing dynamic uncertain demands over a finite discrete time horizon. The probability distribution of the uncertain demand at each customer is not fully specified. The only available information is that these demands are independent and symmetric random variables which can take some value from their support interval. The supplier is responsible for the inventory management of its customers, has sufficient inventory to replenish the customers, and distributes the product using a capacitated vehicle. Backlogging of the demand at customers is allowed. The problem is to determine the delivery quantities as well as the times and routes to the customers while ensuring feasibility regardless of the realized demands and minimizing the total cost composed of transportation, inventory holding and shortage costs. Using a robust optimization approach, we propose two robust mixed integer programming (MIP) formulations for the problem. We also propose a new MIP formulation for the deterministic (nominal) case of the problem. We implement these formulations within a branch-and-cut algorithm and report results on a set of instances adapted from the literature.

Given a graph G = (V,E) with a cost on each edge in E and a prize at each vertex in V, and a target set V′ ⊆ V, the Prize Collecting Steiner Tree (PCST) problem is to find a tree T interconnecting vertices in V′ that has minimum total costs on edges and maximum total prizes at vertices in T. This problem is NP-hard in general, and it is polynomial-time solvable when graphs G are restricted to 2-trees. In this paper, we study how to deal with PCST problem with uncertain costs and prizes. We assume
that edge e could be included in T by paying cost xe Î [ce-,ce+]x_e\in[c_e^-,c_e^+] while taking risk
\frac ce+-xe ce+-ce-\frac{ c_e^+-x_e}{ c_e^+-c_e^-} of losing e, and vertex v could be awarded prize pv Î [pv-,pv+]p_v\in [p_v^-,p_v^+] while taking risk
\frac yv-pv-pv+-pv-\frac{ y_v-p_v^-}{p_v^+-p_v^-} of losing the prize. We establish two risk models for the PCST problem, one minimizing the maximum risk over edges and vertices
in T and the other minimizing the sum of risks. Both models are subject to upper bounds on the budget for constructing a tree.
We propose two polynomial-time algorithms for these problems on 2-trees, respectively. Our study shows that the risk models
have advantages over the tradional robust optimization model, which yields NP-hard problems even if the original optimization
problems are polynomial-time solvable.
KeywordsPrize collecting Steiner tree-interval data-2-trees

We consider the fractional prize-collecting Steiner tree problem on trees. This problem asks for a subtree T containing the root of a given tree G=(V,E) maximizing the ratio of the vertex profits ∑
v ∈ V (T)
p(v) and the edge costs ∑
e ∈ E(T)
c(e) plus a fixed cost c
0 and arises in energy supply management. We experimentally compare three algorithms based on parametric search: the binary
search method, Newton’s method, and a new algorithm based on Megiddo’s parametric search method. We show improved bounds on
the running time for the latter two algorithms. The best theoretical worst case running time, namely O(|V|log|V|), is achieved by our new algorithm. A surprising result of our experiments is the fact that the simple Newton method is
the clear winner of the tested algorithms.

We propose a generalized version of the Prize Collecting Steiner Tree Problem (PCSTP), which offers a fundamental unifying We propose a generalized version of the Prize Collecting Steiner Tree Problem (PCSTP), which offers a fundamental unifying
model for several well-known model for several well-known
NP\mathcal{NP} NP\mathcal{NP}
-hard tree optimization problems. The PCSTP also arises naturally in a variety of network design applications including cable -hard tree optimization problems. The PCSTP also arises naturally in a variety of network design applications including cable
television and local access networks. We reformulate the PCSTP as a minimum spanning tree problem with additional packing television and local access networks. We reformulate the PCSTP as a minimum spanning tree problem with additional packing
and knapsack constraints and we explore various nondifferentiable optimization algorithms for solving its Lagrangian dual. and knapsack constraints and we explore various nondifferentiable optimization algorithms for solving its Lagrangian dual.
We report computational results for nine variants of deflected subgradient strategies, the volume algorithm (VA), and the We report computational results for nine variants of deflected subgradient strategies, the volume algorithm (VA), and the
variable target value method used in conjunction with the VA and with a generalized Polyak–Kelley cutting plane technique.nique.
The performance of these approaches is also compared with an exact stabilized constraint generation procedure. The performance of these approaches is also compared with an exact stabilized constraint generation procedure.

We consider linear programs with uncertain parameters, lying in some prescribed uncertainty set, where part of the variables must be determined before the realization of the uncertain parameters (``non-adjustable variables''), while the other part are variables that can be chosen after the realization (``adjustable variables''). We extend the Robust Optimization methodology ([1, 3-6, 9, 13, 14]) to this situation by introducing the Adjustable Robust Counterpart (ARC) associated with an LP of the above structure. Often the ARC is significantly less conservative than the usual Robust Counterpart (RC), however, in most cases the ARC is computationally intractable (NP-hard). This difficulty is addressed by restricting the adjustable variables to be affine functions of the uncertain data. The ensuing Affinely Adjustable Robust Counterpart (AARC) problem is then shown to be, in certain important cases, equivalent to a tractable optimization problem (typically an LP or a Semidefinite problem), and in other cases, having a tight approximation which is tractable. The AARC approach is illustrated by applying it to a multi-stage inventory management problem.

We propose an approach to address data uncertainty for discrete optimization and network flow problems that allows controlling the degree of conservatism of the solution, and is computationally tractable both practically and theoretically. In particular, when both the cost coefficients and the data in the constraints of an integer programming problem are subject to uncertainty, we propose a robust integer programming problem of moderately larger size that allows controlling the degree of conservatism of the solution in terms of probabilistic bounds on constraint violation. When only the cost coefficients are subject to uncertainty and the problem is a 0–1 discrete optimization problem on n variables, then we solve the robust counterpart by solving at most n+1 instances of the original problem. Thus, the robust counterpart of a polynomially solvable 0–1 discrete optimization problem remains polynomially solvable. In particular, robust matching, spanning tree, shortest path, matroid intersection, etc. are polynomially solvable. We also show that the robust counterpart of an NP-hard -approximable 0–1 discrete optimization problem, remains -approximable. Finally, we propose an algorithm for robust network flows that solves the robust counterpart by solving a polynomial number of nominal minimum cost flow problems in a modified network.

Into the cell, information from the environment is mainly propagated via signaling pathways which form a transduction network.
Here we propose a new algorithm to infer transduction networks from heterogeneous data, using both the protein interaction
network and expression datasets. We formulate the inference problem as an optimization task, and develop a message-passing,
probabilistic and distributed formalism to solve it. We apply our algorithm to the pheromone response in the baker’s yeast
S. cerevisiae. We are able to find the backbone of the known structure of the MAPK cascade of pheromone response, validating our algorithm.
More importantly, we make biological predictions about some proteins whose role could be at the interface between pheromone
response and other cellular functions.

We consider a combinatorial optimization problem that models an expansion of fiber optic telecommunication networks. Thereby, we are given a set of customers with potential gains of revenue and the set of edges with fixed installation costs. The goal is to decide which customers to connect to a given root node so that the sum of edge costs plus the node revenues for the nodes that are left out from the solution is minimized. The problem is known in the literature as the Prize-Collecting Steiner Tree Problem (PCStT). In many applications it is unrealistic to assume that the gains of revenue or installation costs are known in advance. In this paper, we extend this well-studied deterministic problem by considering the robust optimization approach in which the input parameters are subject to interval uncertainty. To control the level of conservatism of the solution, we consider Bertsimas & Sim robust optimization approach. We propose a branch-and-cut approach to solve this problem to optimality and provide an extensive computational study on a set of benchmark instances that are adapted from those previously used to solve the deterministic version of the problem. We show how the price of robustness influences the costs of the solutions and the algorithm performance.

The prize-collecting Steiner tree problem on a graph with edge costs and vertex profits asks for a subtree minimizing the sum of the total cost of all edges in the subtree plus the total profit of all vertices not contained in the subtree. For this well-known problem we develop a new algorithmic framework consisting of three main parts:
(1) An extensive preprocessing phase reduces the given graph without changing the structure of the optimal solution. (2) The central part of our approach is a memetic algorithm (MA) based on a steady-state evolutionary algorithm and an exact subroutine for the problem on trees. (3) The solution population of the memetic algorithm provides an excellent starting point for post-optimization by solving a relaxation of an integer linear programming (ILP) model constructed from a model for finding the minimum Steiner arborescence in a directed graph.
Extensive experiments on benchmark instances from the literature show that our combination of an MA with ILP-based post-optimization compares favorably with previously published results. While our solution values are almost always the same (not surprisingly, since an extension of our ILP approach shows the optimality of these values), we obtain a significant reduction of running time for medium and large instances.

We study the version of the prize collecting traveling salesman problem, where the objective is to find a tour that visits a subset of vertices such that the length of the tour plus the sum of penalties associated with vertices not in the tour is as small as possible. We present an approximation algorithm with constant bound. The algorithm is based on Christofides' algorithm for the traveling salesman problem as well as a method to round fractional solutions of a linear programming relaxation to integers, feasible for the original problem.

The Prize-Collecting Steiner Tree Problem (PCST) on a graph with edge costs and vertex profits asks for a subtree minimizing the sum of the total cost of all edges in the subtree plus the total profit of all vertices not contained in the subtree. PCST appears frequently in the design of utility networks where profit generating customers and the network connecting them have to be chosen in the most profitable way.
Our main contribution is the formulation and implementation of a branch-and-cut algorithm based on a directed graph model where we combine several state-of-the-art methods previously used for the Steiner tree problem. Our method outperforms the previously published results on the standard benchmark set of problems.
We can solve all benchmark instances from the literature to optimality, including some of them for which the optimum was not known. Compared to a recent algorithm by Lucena and Resende, our new method is faster by more than two orders of magnitude. We also introduce a new class of more challenging instances and present computational results for them. Finally, for a set of large-scale real-world instances arising in the design of fiber optic networks, we also obtain optimal solution values.

A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. A concern with such an approach is that it might be too conservative. In this paper, we propose an approach that attempts to make this trade-off more attractive; that is, we investigate ways to decrease what we call the price of robustness. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations. An attractive aspect of our method is that the new robust formulation is also a linear optimization problem. Thus we naturally extend our methods to discrete optimization problems in a tractable way. We report numerical results for a portfolio optimization problem, a knapsack problem, and a problem from the Net Lib library. Subject classifications: Programming, stochastic: robust approach for solving LP/MIP with data uncertainties. Area of review: Financial Services. History: Received September 2001; revision received August 2002; accepted December 2002.

External information propagates in the cell mainly through signaling cascades and transcriptional activation, allowing it to react to a wide spectrum of environmental changes. High-throughput experiments identify numerous molecular components of such cascades that may, however, interact through unknown partners. Some of them may be detected using data coming from the integration of a protein-protein interaction network and mRNA expression profiles. This inference problem can be mapped onto the problem of finding appropriate optimal connected subgraphs of a network defined by these datasets. The optimization procedure turns out to be computationally intractable in general. Here we present a new distributed algorithm for this task, inspired from statistical physics, and apply this scheme to alpha factor and drug perturbations data in yeast. We identify the role of the COS8 protein, a member of a gene family of previously unknown function, and validate the results by genetic experiments. The algorithm we present is specially suited for very large datasets, can run in parallel, and can be adapted to other problems in systems biology. On renowned benchmarks it outperforms other algorithms in the field.

Motivation:
In model organisms such as yeast, large databases of protein-protein and protein-DNA interactions have become an extremely important resource for the study of protein function, evolution, and gene regulatory dynamics. In this paper we demonstrate that by integrating these interactions with widely-available mRNA expression data, it is possible to generate concrete hypotheses for the underlying mechanisms governing the observed changes in gene expression. To perform this integration systematically and at large scale, we introduce an approach for screening a molecular interaction network to identify active subnetworks, i.e., connected regions of the network that show significant changes in expression over particular subsets of conditions. The method we present here combines a rigorous statistical measure for scoring subnetworks with a search algorithm for identifying subnetworks with high score.
Results:
We evaluated our procedure on a small network of 332 genes and 362 interactions and a large network of 4160 genes containing all 7462 protein-protein and protein-DNA interactions in the yeast public databases. In the case of the small network, we identified five significant subnetworks that covered 41 out of 77 (53%) of all significant changes in expression. Both network analyses returned several top-scoring subnetworks with good correspondence to known regulatory mechanisms in the literature. These results demonstrate how large-scale genomic approaches may be used to uncover signalling and regulatory pathways in a systematic, integrative fashion.

With the exponential growth of expression and protein-protein interaction (PPI) data, the frontier of research in systems biology shifts more and more to the integrated analysis of these large datasets. Of particular interest is the identification of functional modules in PPI networks, sharing common cellular function beyond the scope of classical pathways, by means of detecting differentially expressed regions in PPI networks. This requires on the one hand an adequate scoring of the nodes in the network to be identified and on the other hand the availability of an effective algorithm to find the maximally scoring network regions. Various heuristic approaches have been proposed in the literature.
Here we present the first exact solution for this problem, which is based on integer-linear programming and its connection to the well-known prize-collecting Steiner tree problem from Operations Research. Despite the NP-hardness of the underlying combinatorial problem, our method typically computes provably optimal subnetworks in large PPI networks in a few minutes. An essential ingredient of our approach is a scoring function defined on network nodes. We propose a new additive score with two desirable properties: (i) it is scalable by a statistically interpretable parameter and (ii) it allows a smooth integration of data from various sources. We apply our method to a well-established lymphoma microarray dataset in combination with associated survival data and the large interaction network of HPRD to identify functional modules by computing optimal-scoring subnetworks. In particular, we find a functional interaction module associated with proliferation over-expressed in the aggressive ABC subtype as well as modules derived from non-malignant by-stander cells.
Our software is available freely for non-commercial purposes at http://www.planet-lisa.net.

We consider the fractional prize-collecting Steiner tree problem on trees...

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.

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.

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.

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.

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.

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.

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.

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.

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.

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%.

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.

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 Â· Â¿)-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.

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.

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.

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.

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

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.

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.

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...

. 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...

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

- J E Beasly

J.E. Beasly, Collection of test data sets for a variety of operations research (OR) problems, 1990. <http://people.brunel.ac.uk/mastjjb/jeb/info. html>.

The Problem Solving Handbook for

- C Lasher
- C Poirel
- T Murali

C. Lasher, C. Poirel, T. Murali, Cellular response networks, in: L. Heath, N. Ramakrishnan (Eds.), The Problem Solving Handbook for Computational Biology and Bioinformatics, Springer-Verlag, 2010, pp. 233–250.