A min-max regret approach for the Steiner Tree Problem with Interval Costs
Iago A. Carvalho, Amadeu A. Coco, Thiago F. Noronha, Christophe Duhamel
AA min-max regret approach for the Steiner Tree Problem with Interval Costs
Iago A. Carvalho
Instituto de Computac¸ ˜ao, Universidade Estadual de CampinasAv. Albert Einstein 1251, Campinas, SP, 13083-852, Brazil [email protected]
Amadeu A. Coco
Departamento de Computac¸ ˜ao, Centro Federal de Educac¸ ˜ao Tecnol´ogica de Minas GeraisAv. Amazonas 7675, Belo Horizonte, MG, 30510-000, Brazil [email protected]
Thiago F. Noronha
Departamento de Ciˆencias da Computac¸ ˜ao, Universidade Federal de Minas GeraisAv. Antˆonio Carlos 6627, Belo Horizonte, MG, 31270-901, Brazil [email protected]
Christophe Duhamel
LITIS, Universit´e Le Havre Normandie25 Rue Philippe Lebon, 76600 Le Havre, France [email protected]
ABSTRACT
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 subgraphof G that spans all nodes in Q . In this paper, we consider the Min-max Regret SteinerTree 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 a r X i v : . [ m a t h . O C ] J a n roblem. Computational experiments, performed on generalizations of the classical STPinstances, evaluate the efficiency and the limits of the proposed methods. KEYWORDS. Steiner tree problem. Min-max regret. Interval uncertainty.Combinatorial optimization, Mathematical programming
1. Introduction
Let G = ( V, E ) be a connected graph, where V is the set of nodes and E is the setof edges, where each edge ( i, j ) ∈ E is associated with a cost c ij ∈ N + . Given a set Q ⊂ V of terminal nodes, a Steiner tree is defined as a tree in G that spans all nodes in Q and maycontain additional nodes from V \ Q . The Steiner Tree Problem in graphs (STP) [Dreyfuse Wagner, 1971] consists in finding a minimum cost Steiner tree of G .STP is a well known NP-Hard problem [Karp, 1972]. This problem finds practicalapplications in areas such as telecommunication networks design, computational biology,VLSI design, among others [Pr¨omel e Steger, 2012]. In most of these applications, the costassociated with each edge is not precisely known. In this paper, we investigate how RobustOptimization (RO) [Kouvelis e Yu, 1997] can be applied to this context.RO is an approach to deal with uncertain parameters in decision making, wherethe data variability is represented by deterministic values [Kasperski e Zieli´nski, 2016;Kouvelis e Yu, 1997]. We focus on RO models where the uncertain data is modelled by aninterval of possible values. We refer to the book by Kouvelis e Yu [1997] for other robustoptimization models. In this approach, any realization of a single value for each parameteris considered as a scenario that can happen. The objective is to find a solution that isefficient for all scenarios, usually referred to as a robust solution. The RO criterion usedin this work to classify a solution as robust or not is the min-max regret . It was proposedby Wald [1939] in the context of game theory and was adapted to RO by Kouvelis e Yu[1997]. In this paper, we introduce a variant of STP where the value of c i is uncertain.However, it is assumed that this value is in the range [ l ij , u ij ] . This problem is refereed tos the Min-max Regret Steiner Tree Problem with with Interval Costs (MMR-STP) and isdefined as follows. Definition 1. A scenario S is an assignment of a single value c Sij ∈ [ l ij , u ij ] for each edge ( i, j ) ∈ E . It is worth noting that there are infinitely many scenarios, as c Sij can assume anyreal value in [ l ij , u ij ] . Let Γ be the set of all scenarios and Φ be the set of all Steiner treesin G . Definition 2.
The cost of a solution x ∈ Φ in a scenario S ∈ Γ is given by F ( x, S ) = (cid:88) ( i,j ) ∈ E c Sij x ij . Definition 3.
The cost of the optimal SPT solution z S in the scenario S is denoted by F ( z S , S ) = min z ∈ Φ F ( z, S ) = min z ∈ Φ (cid:88) ( i,j ) ∈ E c Sij z ij . Definition 4 (Kouvelis e Yu [1997]) . The regret of a solution x ∈ Φ in a scenario S ∈ Γ isthe difference between the cost of x in the scenario S and the cost of z S in S . Definition 5.
The worst-case scenario scenario of x , i.e., the one where the regret of x isthe maximum is denoted by S x = arg max S ∈ Γ (cid:8) F ( x, S ) − F ( z S , S ) (cid:9) . Lemma 1 (Averbakh [2001]) . Although | Γ | = ∞ , for any min-max regret optimizationproblem, S x is such that c S x ij = u ij if x ij = 1 , and c S x ij = l ij otherwise. That is, c S x i = l ij + ( u ij − l ij ) x ij , for all x ∈ Φ and ( i, j ) ∈ E . Definition 6.
The robust cost of a solution x ∈ Φ is defined as Z ( x ) = F ( x, S x ) − F ( z S x , S x ) , i.e. Z ( x ) is the maximum regret of x . It is worth noting that one has to solve an STP in S x in order to compute z S x . That is, it is NP-Hard to compute the robust cost of a singlesolution for MMR-STP. efinition 7. MMR-STP consists in finding the Steiner tree x ∗ ∈ Φ with the smallest robustcost Z ( x ∗ ) . MMR-STP is clearly NP-Hard, as for l ij = u ij = c ij it reduces to SPT. Therefore,in this paper we propose heuristics and exact algorithms for this problem. We evaluatehow good are the solutions provided by the state of the art exact and heuristic algorithmsdesigned for min-max regret optimization problems these problems for the case of MMR-SPT. The remainder of this work is organized as follows. Related works are presentedin Section 2. Section 3 shows the proposed ILP formulation for MMR-STP. Then, an exactalgorithm for MMR-STP are described in Section 4, while three heuristics are proposed forthis same problem in Section 5. Computation experiments, which evaluates the proposedexact and heuristic algorithms, are reported in Section 6. Finally, concluding remarks aredrawn in the last section.
2. Related work
The Steiner Tree Problem in graphs was proposed in Dreyfus e Wagner [1971],and was proven NP-Hard in Karp [1972]. Several mathematical formulations [Chopra eRao, 1994; Goemans e Myung, 1993; Polzin e Daneshmand, 2001], as well as exact algo-rithms [Lucena e Beasley, 1998], and heuristics [Duin e Voß, 1994, 1999], were proposedand evaluated for this problem. A comparison among several mathematical formulationsfor STP was shown in Polzin e Daneshmand [2001]. A compendium of STP formulationscan be found in Goemans e Myung [1993]. Furthermore, the state-of-the-art algorithmsand other recent advances regarding this problem can be found in Du et al. [2013]; Pr ¨omele Steger [2012].Many robust counterparts of classical optimization problems have been studied inthe literature, such as the Robust Shortest Path Problem [Karas¸an et al., 2001; Catanzaroet al., 2011; P´erez-Galarce et al., 2018] and the Robust Minimum Spanning Tree Pro-blem [Montemanni, 2006; Godinho e Paquete, 2019], and the Robust Shortest Path TreeProblem [Catanzaro et al., 2011; Carvalho et al., 2016b,a, 2018]. These problems are NP-hard [Aissi et al., 2009], despite the fact that their deterministic counterparts can be solvedn polynomial time. RO problems whose deterministic counterparts are already NP-hardhave also been studied, such as the Robust Restricted Shortest Path Problem [Assunc¸ ˜aoet al., 2017], the Robust Traveling Salesman Problem [Montemanni et al., 2007], and theRobust Set Covering Problem [Pereira e Averbakh, 2013; Coco et al., 2015, 2016], andthe Robust Knapsack Problem [Deineko e Woeginger, 2010; Furini et al., 2015]. As isthe case of MMR-STP, these problems are particularly harder to solve than other NP-Hardproblems, because the complexity of computing the cost of a single solution is at least thatof solving the deterministic counterpart, which is itself NP-Hard.Some works in the literature also consider robust variations of STP. A RobustPrize-Collecting Steiner Tree Problem in which both edge weights and node prizes aresubject to uncertainty was proposed in ´Alvarez-Miranda et al. [2013]. Moreover, a Two-stage Robust Steiner tree was presented in Khandekar et al. [2008]. In the initial stage, asmall subset of terminal nodes is given. In the second stage, the edge weights are increasedby a factor λ and several scenarios can occur, each one with a new set of terminal nodes.The objective is to minimize the maximum overall cost over all scenarios.Other variations of Steiner problems with data uncertainty were handled by meansof stochastic programming. The most studied of these problems is the Two-stage StochasticSteiner Tree in graphs with recourse [Bomze et al., 2010; Fleischer et al., 2006; Gupta eP´al, 2005]. In the initial stage, a known probability distribution π is set on subsets ofnodes and a cost is assigned to each edge of the graph. In the second stage, a subsetof nodes materializes, given their prior known distribution, and the cost of each edge isincreased by a factor λ . Then, an additional set of edges can be bought to build a tree thatspans all materialized nodes. The objective is to minimize the expected cost of the two-stage solution. A mathematical formulation and an exact algorithm for this problem werepresented in Bomze et al. [2010]. Approximation algorithms were presented in Fleischeret al. [2006]; Gupta e P´al [2005].To the best our knowledge, the min-max regret Robust Steiner Tree Problem ingraphs has not been studied in the literature. Therefore, we propose an Integer Linear Pro-gramming (ILP) formulation for MMR-SPT, based on STP’s bi-directed multi-commodityow formulation presented in Chopra e Rao [1994]. As this formulation has an exponen-tially large number of constraints, we extend the Benders-like Decomposition frameworkof Montemanni e Gambardella [2005] for MMR-SPT. Furthermore, we propose three heu-ristics based on the framework of Kasperski e Zieli´nski [2006]: ( i ) the Algorithm Mean(AM); ( ii ) the Algorithm Upper (AU); and ( iii ) the Algorithm Mean Upper.
3. An ILP formulation for MMR-STP
Our ILP formulation for MMR-STP is based on the multi-commodity flow for-mulation for STP proposed in Chopra e Rao [1994]. Let G (cid:48) = ( V, A ) be a directed graph,obtained by bi-directing the edges in E . Furthermore, let r ∈ Q be an arbitrary terminalnode, which is referred to as the root node. From this data, STP is formulated by means ofbinary variables x ∈ { , } | A | , such that x ij = 1 if arc ( i, j ) ∈ A belongs to the Steiner tree,and x ij = 0 otherwise. Besides, we make use of auxiliary binary variables y ∈ { , } | A × Q | ,such that y kij = 1 if arc ( i, j ) ∈ A is used to send an unit of flow from the root r to theterminal k ∈ Q , and y kij = 0 otherwise. The resulting formulation consists of the objectivefunction (1) and the constraints in (2)–(5). min (cid:88) ( i,j ) ∈ A c ij x ij (1) s.t. (cid:88) ( j,i ) ∈ A y kji − (cid:88) ( i,j ) ∈ A y kij =
1, if j = r -1, if j = k
0, otherwise , ∀ j ∈ N, k ∈ Q (2) y kij + y kij ≤ x ij , ∀ ( i, j ) ∈ A, k ∈ Q (3) x ij ∈ { , } , ∀ ( i, j ) ∈ A (4) y kij ∈ { , } , ∀ ( i, j ) ∈ A, k ∈ Q (5)The objective function (1) minimizes the cost of the arcs in the Steiner tree. Theconstraints in (2) are the classic flow conservation constraints that enforce a path from theroot r to every other terminal k ∈ Q . The inequalities in (3) project the variables y into theariables x . Besides, together with (2), they enforce that x induce a spanning tree of theterminals in Q . The domain of the variables x and y are defined by (4) and (5), respectively.From the STP formulation described above, we have that the polytope that descri-bes the set Φ of Steiner trees of G can be formulated by (2)–(5). We have that MMR-STPcan be written as min x ∈ Φ Z ( x ) = min x ∈ Φ F ( x, S x ) − F ( z S x , S x ) . Besides, from definition 1.2, we have that F ( x, S x ) = (cid:88) ( i,j ) ∈ e u ij x ij , and from definition 1.3 and lemma 1.6, we have that F ( z S x , S x ) = min z ∈ Φ (cid:88) ( i,j ) ∈ A c S x ij z ij = min z ∈ Φ (cid:88) ( i,j ) ∈ A (cid:0) l ij + ( u ij − l ij ) x ij (cid:1) z ij . Therefore, MMR-SPT can be formulated by the 0-1 Bilevel Integer Linear Program definedby the objective function (6) and the constraints (2)–(5). We note that (6) is indeed linearas x is constant in the inner optimization problem. min x ∈ Φ (cid:88) ( i,j ) ∈ A u ij x ij − min z ∈ Φ (cid:88) ( i,j ) ∈ A (cid:0) l ij + ( u ij − l ij (cid:1) x ij ) z ij (6)We can then obtain a MIP formulation by linearizing F ( y S x , S x ) , as explainedin Aissi et al. [2009]. Let y kij and x ij be the binary variables defined in the STP formula-tion (1)–(4). Besides, let variable θ ∈ R be the cost of the Steiner tree in the worst-casescenario defined by variables z ij . The resulting formulation is defined by the objectivefunction (7) and constraints (8)–(13). in (cid:88) ( i,j ) ∈ A u ij x ij − θ (7) s.t.θ (cid:54) (cid:88) ( i,j ) ∈ A (cid:0) l ij + ( u ij − l ij (cid:1) x ij ) z ij , ∀ z ∈ Γ (8) (cid:88) ( j,i ) ∈ A y kji − (cid:88) ( i,j ) ∈ A y kij =
1, if j = r -1, if j = k
0, otherwise , ∀ j ∈ N, k ∈ Q (9) y kij + y kij (cid:54) x ij , ∀ ( i, j ) ∈ A, k ∈ Q (10) x ij ∈ { , } , ∀ ( i, j ) ∈ A (11) y kij ∈ { , } , ∀ ( i, j ) ∈ A, k ∈ Q (12) θ ∈ R (13)The objective function (7) aims at minimizing the maximum regret. The cons-traints in (9)–(12) are as previously defined for the STP. The inequalities in (8) computesthe cost of each solution z ∈ Γ in the worst-case scenario. One can see that we have aninequality (8) for each possible solution. Therefore, the number of these inequalities isexponential. In order to satisfy these inequalities, the value of θ should not be greater thanthe cost of any solution z ∈ Γ . Finally, the constraint in (13) defines the domain of variable θ .
4. A Benders-like Decomposition for MMR-STP
The Benders-like Decomposition (Benders) for MMR-STP is inspired by the Ben-ders Decomposition and based on the approaches used to solve other interval data min-maxrobust optimization problems, as the Robust Set Covering Problem Pereira e Averbakh[2013] and the Robust Minimal Spanning Tree Problem Montemanni [2006]. Benders isbased on formulation (7)–(13). As the number of constraints (8) grows exponentially withthe number of nodes, they are relaxed in the master problem. At each iteration, one of theseconstraints is separated and added to the master problem. Benders stops when the loweround obtained by solving the master problem is equal to the cost of the best (in this caseoptimal) solution.Let Γ h ⊆ Γ be the subset of constraints (8) that are known in the master problem atiteration h of Benders, and X h be the optimal solution of this problem. The value of θ maynot be equal to the cost of the optimal solution Y h ∈ Γ of scenario s ( X h ) , as constraints(8) are relaxed. Therefore, a new constraint (8), generated from Y h , must be added to Γ h +1 in order to update the value of θ for X h . Y h can be obtained by solving a STP subproblemin s ( X h ) .In the first iteration, in order to avoid an unbounded master problem, Γ is initi-alized with two solutions obtained by the Algorithm Mean and Algorithm Upper heuris-tics proposed in Section 5, as suggested in Pereira e Averbakh [2013]. At each iteration,the master problem and the corresponding STP subproblem (Equations (1)–(5)) are sol-ved. Given the lower bound z h obtained by solving the master problem at iteration h , if z h < min l ∈{ ,...,h } ρ s ( X h ) ( X h ) , Γ h +1 = Γ h ∪ { Y h } and a new iteration starts. Otherwise,Benders stops since an optimal solution was found.
5. Heuristics for MMR-STP
A framework for building heuristics that can be applied to any interval data min-max robust optimization problem was introduced in Kasperski e Zieli´nski [2006]. Thecomplexity of the algorithms developed through this framework are the same of solvingthe classical counterpart of the robust optimization problem studied. In this work, weapplied this framework to develop three heuristics for MMR-STP.The first heuristic, called Algorithm Mean (AM), uses a branch-and-bound algo-rithm based on the flow formulation presented in Polzin e Daneshmand [2001] to solve aSTP at the midpoint scenario s = . In s = , the weight of each edge is set to its mean value, i.e. c s = ij = ( u ij + l ij ) / , for all edges ( i, j ) ∈ E . Next, the maximum regret of the computed so-lution is evaluated and returned. The cost of the solution obtained through AM is boundedby a factor of from the optimal solution, as proved in Kasperski e Zieli´nski [2006].The second heuristic, called Algorithm Upper (AU), is similar to AM. However,instead of solving a STP for scenario s = , AU solves the STP for the upper scenario s + ,here the weight of each edge is set to its upper value, i.e. c s + ij = u ij , for all edges ( i, j ) ∈ E . Unlike AM, the cost of the solution obtained by AU is not bounded.The last heuristic, called Algorithm Mean Upper (AMU), combines AM and AU.Next, it returns the smallest computed maximum regret . As AMU runs AM, it is also a -approximation algorithm for any interval data min-max robust optimization problem.
6. Computational experiments
Computational experiments have been performed on an Intel Xeon CPU E5645with . GHz clock and GB of RAM memory, running under Linux operating system.The branch-and-bound implementation of the ILOG CPLEX version . with defaultparameter settings was used to solve the mixed integer linear programs. The algorithmswere implemented in C++ using the ILOG Concert Technology and compiled with GNUg++ 5.4.0. The running time of all algorithms has been limited to 10800 seconds (3 hours).The instances used in the experiments are generalizations of classical STP instan-ces. The 5 first instances (WRP3-11 to WRP3-15) from the SteinLib WRP3 set were used.Their sizes range from 128 nodes, 227 edges, and 11 terminal nodes (WRP3-11) to 138 no-des, 257 edges, and 15 terminal nodes (WRP3-15). Next, three different methods, namelyBeasley (BE), Montemanni (MO) and Kasperski-Zielinski (KZ), are used to generate theedge weights interval as in Pereira e Averbakh [2013]. A parameter β = { . , . , . } is used in BE, while M = { , , } is used as parameter in MO and KZ. Foreach method, the higher the parameter value, the larger the edge weight interval. Thesemethods are applied to the selected WRP3 instances. Therefore, nine sets of 5 instanceswere generated by using different interval sizes. They are used in the experiment describedbelow. The performance of Benders, AM, AU, and AMU for these sets is displayed inTable 1. The name of each instance set is shown in Column 1. The average relative opti-mality gap of Benders, as well as the average computation time of these runs are reported incolumns 2 and 3, respectively. Then, columns 4 and 5 present respectively ( i ) the averagepercent relative deviation to the Bender’s upper bound and ( ii ) the average computational http://elib.zib.de/steinlib ime of AM for each set. The same information is given for AU and for AMU.Benders AM AU AMUInstance set gap% t(s) %dev t(s) %dev t(s) %dev t(s)WRP3-BE-0.1 15.48 5948.72 3.70 1.91 1.83 1.94 1.46 3.85WRP3-BE-0.3 14.80 4763.78 3.12 2.03 1.03 2.74 0.91 3.77WRP3-BE-0.5 6.25 4825.14 0.85 1.76 1.65 0.97 0.14 3.26WRP3-MO-750 6.32 6059.24 4.33 238.80 2.29 240.00 1.21 478.80WRP3-MO-1000 12.60 7434.67 4.22 467.28 2.80 637.26 1.96 1104.54WRP3-MO-1250 23.64 8583.08 5.93 1231.83 2.79 1658.32 2.79 2890.15WRP3-KZ-750 28.37 10800.00 3.94 2916.45 3.78 2508.25 2.02 5424.70WRP3-KZ-1000 27.88 10800.00 1.25 2247.66 4.10 2894.98 0.91 5142.64WRP3-KZ-1250 27.92 8926.96 1.83 1510.39 3.12 1486.42 0.93 2996.81Tabela 1: Evaluation of Benders, AM, AU, and AMUOne can see from Table 1 that Benders achieves a smaller relative optimality gapand running times in BE and MO instances than in KZ. It indicates that the KZ instancesare the most difficult ones among the three proposed sets. Regarding the interval sizes,we obtained different results for each instance set. For the BE instances, the smaller isthe interval size, the greater is the average optimality gap. On the other hand, for theMO instances, the greater is the interval size, the smaller is the average optimality gap.However, the Benders’s algorithm average optimality gap was almost the same for all ofthe KZ instances.Regarding the heuristics, one can see from this same table that, for BE instances,AM, AU and AMU average running times never exceeds 4 seconds, but grow quickly forMO and KZ instances. The maximum average relative deviations for AM, AU, and AMUare respectively 5.93%, 4.10%, and 2.79%. These results indicate that the Bender’s algo-rithm did not greatly improved its initial solution (which is given by AMU, as explained inection 4).
7. Conclusions
This paper considers a new combinatorial optimization problem that arises fromthe uncertain nature of Steiner tree problem applications. We propose a mathematical for-mulation based on the robust optimization framework presented in Kouvelis e Yu [1997]and an exact and three heuristic algorithms to solve it. Computational experiments showthat Benders-like decomposition did not solve all proposed instances to optimality. Howe-ver, the heuristics achieve good results in a small running time. Future works should focuson the development of new exact and heuristic methods for the studied problem. Moreover,other mathematical formulations for the Steiner tree problem in graphs presented in Polzine Daneshmand [2001] can be extended to handle the uncertain data for this problem.
Aknowledgments
This study was financed in part by the
Coordenac¸ ˜ao de Aperfeic¸oamento de Pes-soal de N´ıvel Superior - Brasil (CAPES) - Finance Code 001, the
Conselho Nacional deDesenvolvimento Cient´ıfico e Tecnol´ogico - Brasil (CNPq), the
Fundac¸ ˜ao de Amparo `aPesquisa do Estado de Minas Gerais - Brasil (FAPEMIG), and the
Fundac¸ ˜ao de Amparo `aPesquisa do Estado de S˜ao Paulo - Brasil (FAPESP).
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