Decomposition Methods for Stochastic Steiner Trees M. Leitner 2 c 1 M. Luipersbeck 2 M. Sinnl 2 I. Ljubi´ 1 ESSEC Business School of Paris, France 2 ISOR, University of Vienna, Austria 2nd European Conference on Stochastic Optimization (ECSO 2017) September 20-22, 2017 Roma Tre University, Rome, Italy
Deterministic Steiner Tree Problem (STP) Deterministic STP • Given: undirected graph G = ( V , E ), positive edge costs c e , set of terminals T ⊂ V , T � = ∅ . • Objective: min { c ( E 0 ) : E 0 ⊂ E , E 0 spans R } . Decision problem NP-complete. Well studied, many applications, recent DIMACS Challenge (non-trivial graphs with 100 000’s of nodes solved to optimality). I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 2
WHY DO WE STUDY STEINER TREES UNDER UNCERTAINTY? I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 3
Steiner Tree Problem (STP) Under Uncertainty In practice, two sources of uncertainty: • Who are the terminals? No precise knowledge of future customer demands. • What are the edge installation costs? Future edge costs may be more expensive and prices are highly volatile (“wait and see” can be costly). One possible approach: Stochastic Optimization Estimate possible outcomes and derive scenarios: • Each scenario k assumes terminals T k ⊂ V are given and edge costs c k are specified. Decision Process: Two Stages • First Stage : (“now”, Monday): buy cheap/profitable edges now. Difficulty: we only know possible outcomes and their probabilities. • Second Stage : (“future”, Tuesday, one scenario is realized): additional edges are purchased to make the solution feasible ( recourse action ). I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 4
SSTP: Formal Problem Definition SSTP • Given: Undirected graph G = ( V , E ), root r ∈ V , positive edge costs c 0 e , e ∈ E . Set of scenarios K , s.t. k ∈ K : ◮ probability p k > 0, ◮ edge costs c k e , e ∈ E , ◮ set of terminals T k ⊂ V , r ∈ T k . • Objective: Find E 0 ⊂ E (purchased in the first-stage) and E k ⊂ E (purchased in the second-stage, if scenario k is realized ), for all k ∈ K such that expected solution cost is minimized, i.e.: � c 0 � p k � c k min e + e e ∈ E 0 k ∈ K e ∈ E k s.t. E 0 ∪ E k spans T k , ∀ k ∈ K I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 5
WHAT IS KNOWN ABOUT SSTP SO FAR? I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 6
Previous Work • introduced by Gupta et al. [2007a] (approximation and complexity results) • approximation algorithms [Gupta and P´ al, 2005, Gupta et al., 2004, 2007b, Swamy and Shmoys, 2006] ◮ In general, SSTP is NP-hard to approximate within a constant factor. Constant approximation possible only for special cases. • fixed-parameter tractability [Kurz et al., 2013] • heuristics [Hokama et al., 2014] ( genetic algorithm, DIMACS Challenge 2014 ) • exact two-stage branch-and-cut based on Benders decomposition : ◮ stochastic STP [Bomze et al., 2010], ◮ stochastic survivable network design [Ljubi´ c et al., 2017], ◮ PhD thesis Bernd Zey (upcoming 2017). I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 7
Our Contribution • we introduce a new ILP formulation for the SSTP ◮ strongest among existing formulations • we design a solution framework based on this formulation ◮ exploits the decomposability of the formulation in various ways Figure: Algorithmic framework. • we present a computational study comparing our approach with ◮ state-of-the-art exact approach from [Bomze et al., 2010, Ljubi´ c et al., 2017] (Benders decomposition based on two-stage branch-and-cut) ◮ genetic algorithm from [Hokama et al., 2014] • presented method significantly outperforms these approaches I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 8
STEP 1: A STRONGER FORMULATION I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 9
Two Semi-Directed Models for SSTP [Bomze et al., 2010, Zey, 2016, Ljubi´ c et al., 2017] It is impossible to orient the first- stage solution, so we derive semi- directed formulations. I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 10
Hierarchy of Formulations (SDC FB 3 ) (SDC FB 2 ) (SDC ∗ (SDF) (SDC 3 ) (SDC 2 ) 2 ) (SDC 1 ) (UF) (UC) Figure: Directed arcs indicate that the target formulation is stronger than the source formulation. Blue boxes: the formulation has been introduced by us, all the others are from Bomze et al. [2010], Zey [2016] . Flow-Balance constraints (FB): • strengthening: ensure, that only terminals can be leaf-nodes • added to (SDC 2 ) from Bomze et al. [2010], Zey [2016] → (SDC FB 2 ) • added to our (SDC 3 ) → (SDC FB 3 ) I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 11
(SDC 3 ): A Strong Formulation for SSTP • idea : Steiner arborscence rooted at r for each k ∈ K , using arcs bought in first and second stage ◮ binary w k ij = 1, iff arc ( i , j ) is selected in the first stage for scenario k ◮ binary z k ij = 1, iff arc ( i , j ) is selected in the second stage for scenario k ◮ binary x e = 1, iff edge e is selected in the first stage • W k : set of directed Steiner cuts for scenario k � c 0 � p k � c k e ( z k ij + z k min e x e + ji ) e ∈ E k ∈ K e = { i , j }∈ E w k ( δ − ( W )) + z k ( δ − ( W )) ≥ 1 ∀ W ∈ W k , ∀ k ∈ K s.t. (SDC 3 :1) w k ij + w k ji ≤ x e ∀ e = { i , j } ∈ E , ∀ k ∈ K (SDC 3 :2) ( x , z , w ) ∈ { 0 , 1 } | E | +2 | A || K | (SDC 3 :3) I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 12
The Framework Advantages of ( SDC 3 ) : It decomposes nicely, and gives the strongest bounds with (SDC FB 3 ). How does it work? Dual ascent: greedy heuristic that changes dual multipliers λ while 1 monotonically increasing LB. Gives also an UB. Lagrangian: takes UB and final λ from DA to initialize the subgradient 2 method. Improves UB and LB. Applies reduction techniques. Generates a collection of useful dual multipliers λ . Benders: takes UB and optimality cuts associated to Langrangian λ found 3 during the subgradient procedure. OBSERVE: Steps 1 and 2 give valid LB and UB and are purely combinatorial (no MIP solver needed!) Step 3 is a branch-and-cut (CPLEX). I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 13
STEP 2: DUAL ASCENT I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 14
Dual Ascent • let β and λ be the dual multipliers of (SDC 3 :1) (connectivity) and (SDC 3 :2) (linking) (SDC D � � β k 3 ) max W k ∈ K W ∈W k � λ k e ≤ c 0 ∀ e ∈ E e k ∈ K (SDC D 3 :1) β ( W k ij ) ≤ p k c k ∀ ( i , j ) ∈ A , ∀ k ∈ K , e = { i , j } e (SDC D 3 :2) β ( W k ij ) − λ k e ≤ 0 ∀ ( i , j ) ∈ A , ∀ k ∈ K , e = { i , j } (SDC D 3 :3) ( β k , λ k ) ∈ R |W k | + | E | ∀ k ∈ K ≥ 0 • dual ascent works similar to dual ascent for STP Wong [1984] ◮ start from initial solution ¯ β = 0 ◮ each iteration: increase one dual variable β k W = 0 while preserving feasibility ◮ The worst-case time complexity: | A | min {| A | , | T k || V |} ). O ( � k ∈ K I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 15
STEP 3: LAGRANGIAN HEURISTIC I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 16
Lagrangian Relaxation • relax constraints (SDC 3 :2) using Lagrangian dual multipliers λ ≥ 0 • we obtain the relaxation � � c 0 � � p k c k e ( z k ij + z k L ( λ ) := min e x e + ji )+ e ∈ E k ∈ K e = { i , j }∈ E � � � λ k e ( w k ij + w k ji − x e ) : (SDC 3 :1) , (SDC 3 :3) k ∈ K e = { i , j }∈ E c e := c 0 • define Lagrangian cost as ˜ e − � k ∈ K λ k e , e ∈ E • problem decomposes into | K | + 1 independent subproblems ◮ one in x � � c e x e : x ∈ { 0 , 1 } | E | � L 0 ( λ ) := min ˜ e ∈ E ◮ and one in z k , w k for k ∈ K � � � � L k ( λ ) := min p k c k e ( z k ij + z k ji ) + λ k e ( w k ij + w k ji ) : e = { i , j }∈ E (SDC 3 :1) , ( z k , w k ) ∈ { 0 , 1 } 2 | A | � I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 17
Lagrangian Relaxation • the Lagrangian dual problem is � � (SDC LD L 0 ( λ ) + � L k ( λ ) 3 ) max λ ≥ 0 k ∈ K • L 0 ( λ ) can be computed by inspection • L k ( λ ): solving an instance of the Steiner arborescence problem (SAP) Theorem v ( LP-SDC FB 3 ) ≤ v ( SDC LD 3 ) = v ( SDC 3 ) • we solve (SDC LD 3 ) using a subgradient scheme • dual variables at the end of the dual ascent are used to initialize λ • subproblems L k ( λ ) are solved heuristically ◮ using a dual ascent for SAP together with a primal heuristic • two different heuristics to calculate high-quality feasible solutions • we designed reduction tests to fix nodes and edges I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 18
STEP 4: BENDERS DECOMPOSITION I. Ljubi´ c (ESSEC Business School) Decomposition for SSTP ECSO2017 19
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