Models and Algorithms for the Balance-Constrained Stochastic Bottleneck Spanning Tree Problem Jue Wang 1 Siqian Shen 1 Murat Kurt 2 1 Department of Industrial and Operations Engineering University of Michigan 2 Department of Industrial and Systems Engineering University at Buffalo (State University of New York) The 13th INFORMS Computing Society Conference January 08, 2013 Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 1 / 29
Outline ◮ Introduction ◮ Basic and MINLP formulations for the BCSBSTP ◮ SOS1- and SOS2-based formulations and algorithm ◮ SAA-based MILP formulation ◮ Computational results Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 2 / 29
What is the BCSBSTP? ◮ BCSBSTP: Balance-Constrained Stochastic Bottleneck Spanning Tree Problem (a stochastic MST problem) ◮ Each edge weight is characterized by a probability distribution; all weights are independently distributed. ◮ Goal: minimize an upper bound imposed on the maximum edge weight in a spanning tree with certain probability. ◮ “Balanced-Constrained” implies an additional chance constraint on the minimum edge weight in a spanning tree. ◮ SBSTP: A special case of the BCSBSTP without bounding the minimum edge weight. Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 3 / 29
Applications ◮ Telecommunication, e.g., wireless sensor networks ◮ Post-disaster relief ◮ Epidemic spread ◮ Network reliability Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 4 / 29
Previous work ◮ Ishii and Nishida (1983) studied the SBSTP with normally and independently distributed edge weights. ◮ Ishii and Shiode (1995) continued to discuss variants and extensions of the SBSTP. ◮ Kurt (2012) proposed a polynomial-time approximation for solving the generalized SBSTP and showed that 1. the exact optimal solution can be obtained when edge weights have the same distribution type, 2. BCSBSTP is in general NP-Complete. Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 5 / 29
Notation • Graph Configuration G = ( V , E ) An undirected connected graph. T ( G ) Set of all spanning trees of graph G . T = ( V , E T ) A spanning tree of G . Random edge weight for every edge e j ∈ E . w j Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 6 / 29
Notation • Graph Configuration G = ( V , E ) An undirected connected graph. T ( G ) Set of all spanning trees of graph G . T = ( V , E T ) A spanning tree of G . Random edge weight for every edge e j ∈ E . w j • Decision Variable ℓ an upper bound variable on the maximum edge weight. Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 6 / 29
Notation • Graph Configuration G = ( V , E ) An undirected connected graph. T ( G ) Set of all spanning trees of graph G . T = ( V , E T ) A spanning tree of G . Random edge weight for every edge e j ∈ E . w j • Decision Variable ℓ an upper bound variable on the maximum edge weight. • Parameters a given lower bound on the minimum edge weight. κ α, β probability levels associated with the upper and lower bound chance constraints, respectively. Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 6 / 29
Basic formulation for the BCSBSTP � � � � � � w j ≤ ℓ ≥ α, Pr w j ≥ κ ≥ β Q := min ℓ : Pr max min , (1) T ∈ T ( G ) j : e j ∈ E T j : e j ∈ E T Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 7 / 29
Basic formulation for the BCSBSTP � � � � � � w j ≤ ℓ ≥ α, Pr w j ≥ κ ≥ β Q := min ℓ : Pr max min , (1) T ∈ T ( G ) j : e j ∈ E T j : e j ∈ E T Because all distributions are independent, we have � � � � Pr max w j ≤ ℓ ≥ α ⇔ F j ( ℓ ) ≥ α ⇔ log F j ( ℓ ) ≥ log α, and j : e j ∈ E T j : e j ∈ E T j : e j ∈ E T � � � � min w j ≥ κ ≥ β ⇔ � 1 − F j ( κ ) � ≥ β ⇔ log � 1 − F j ( κ ) � ≥ log β, Pr j : e j ∈ E T j : e j ∈ E T j : e j ∈ E T Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 7 / 29
Basic formulation for the BCSBSTP � � � � � � w j ≤ ℓ ≥ α, Pr w j ≥ κ ≥ β Q := min ℓ : Pr max min , (1) T ∈ T ( G ) j : e j ∈ E T j : e j ∈ E T Because all distributions are independent, we have � � � � Pr max w j ≤ ℓ ≥ α ⇔ F j ( ℓ ) ≥ α ⇔ log F j ( ℓ ) ≥ log α, and j : e j ∈ E T j : e j ∈ E T j : e j ∈ E T � � � � min w j ≥ κ ≥ β ⇔ � 1 − F j ( κ ) � ≥ β ⇔ log � 1 − F j ( κ ) � ≥ log β, Pr j : e j ∈ E T j : e j ∈ E T j : e j ∈ E T which transform Problem Q into an equivalent nonlinear problem: Q ′ := � � log F j ( ℓ ) ≥ log α, � 1 − F j ( κ ) � ≥ log β min ℓ : log . (2) T ∈T ( G ) j : e j ∈ E T j : e j ∈ E T Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 7 / 29
MINLP formulation for the BCSBSTP � 1 if edge e j ∈ E T , Introduce new decision variables : x j = 0 otherwise. min: ℓ � s.t. x j log F j ( ℓ ) ≥ log α (3a) j : e j ∈ E � � � x j log 1 − F j ( κ ) ≥ log β (3b) j : e j ∈ E � x j = n − 1 (3c) j : e j ∈ E � x j ≤ | V s | − 1 ∀ V s ⊂ V , V s � = ∅ (3d) j : e j ∈ E V s x j ∈ { 0 , 1 } ∀ e j ∈ E (3e) Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 8 / 29
SOS1-based formulation Special Ordered Sets of type 1 (SOS1): a set of variables, at most one of which can take a strictly positive value with all others being at 0.
SOS1-based formulation Special Ordered Sets of type 1 (SOS1): a set of variables, at most one of which can take a strictly positive value with all others being at 0. Define binary variables � 1 n z k = � z k = 1 (4c) 0 k =1 z k ∈ { 0 , 1 } ∀ k = 1 , . . . , n . (4d) Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 9 / 29
SOS1-based formulation n � min: z k ℓ k Special Ordered Sets of type 1 k =1 (SOS1): a set of variables, at most one of which can take a strictly positive value with all others being at 0. Define binary variables � 1 if ℓ = ℓ k , n z k = � z k = 1 (4c) 0 otherwise. k =1 z k ∈ { 0 , 1 } ∀ k = 1 , . . . , n . (4d) Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 9 / 29
SOS1-based formulation n � min: z k ℓ k Special Ordered Sets of type 1 k =1 (SOS1): a set of variables, at s.t. (3c)–(3e) most one of which can take a � strictly positive value with all x j log F j ( ℓ ) ≥ log α (4a) others being at 0. j : e j ∈ E � � � x j log 1 − F j ( κ ) ≥ log β (4b) Define binary variables j : e j ∈ E � 1 n if ℓ = ℓ k , � z k = z k = 1 (4c) 0 otherwise. k =1 z k ∈ { 0 , 1 } ∀ k = 1 , . . . , n . (4d) Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 9 / 29
SOS1-based formulation n � min: z k ℓ k Special Ordered Sets of type 1 k =1 (SOS1): a set of variables, at s.t. (3c)–(3e) most one of which can take a n strictly positive value with all � � z k x j log F j ( ℓ k ) ≥ log α (4a) others being at 0. j : e j ∈ E k =1 � � � x j log 1 − F j ( κ ) ≥ log β (4b) Define binary variables j : e j ∈ E � 1 if ℓ = ℓ k , n z k = � z k = 1 (4c) 0 otherwise. k =1 z k ∈ { 0 , 1 } ∀ k = 1 , . . . , n . (4d) Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 9 / 29
SOS1-based formulation Compute the upper bound ℓ and lower bound ℓ of ℓ a priori , dissect the whole interval equally, treat each sample point as parameter. n � min: z k ℓ k k =1 s.t. (3c)–(3e) n � � z k x j log F j ( ℓ k ) ≥ log α j : e j ∈ E k =1 � x j log � 1 − F j ( κ ) � ≥ log β j : e j ∈ E n � z k = 1 k =1 z k ∈ { 0 , 1 } ∀ k = 1 , . . . , n . Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 10 / 29
SOS1-based formulation Compute the upper bound ℓ and lower bound ℓ of ℓ a priori , dissect the whole interval equally, treat each sample point as parameter. n � min: z k ℓ k k =1 s.t. (3c)–(3e) n � � z k x j log F j ( ℓ k ) ≥ log α j : e j ∈ E k =1 � x j log � 1 − F j ( κ ) � ≥ log β j : e j ∈ E n � z k = 1 k =1 z k ∈ { 0 , 1 } ∀ k = 1 , . . . , n . Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 10 / 29
SOS1-based formulation Compute the upper bound ℓ and lower bound ℓ of ℓ a priori , dissect the whole interval equally, treat each sample point as parameter. n � min: z k ℓ k k =1 s.t. (3c)–(3e) n � � z k x j log F j ( ℓ k ) ≥ log α j : e j ∈ E k =1 � x j log � 1 − F j ( κ ) � ≥ log β j : e j ∈ E n � z k = 1 k =1 z k ∈ { 0 , 1 } ∀ k = 1 , . . . , n . Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 10 / 29
SOS1-based formulation Compute the upper bound ℓ and lower bound ℓ of ℓ a priori , dissect the whole interval equally, treat each sample point as parameter. n � min: z k ℓ k k =1 s.t. (3c)–(3e) n � � z k x j log F j ( ℓ k ) ≥ log α j : e j ∈ E k =1 � x j log � 1 − F j ( κ ) � ≥ log β j : e j ∈ E n � z k = 1 k =1 z k ∈ { 0 , 1 } ∀ k = 1 , . . . , n . Wang, Shen, Kurt Models and Algorithms for the BCSBSTP 10 / 29
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