Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion Hop Constrained Connected Facility Location Stefan Gollowitzer and Ivana Ljubi´ c Department of Statistics and Decision Support Systems, Faculty of Business, Economics, and Statistics, University of Vienna, Austria Aussois, January 2010
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion Outline Introduction ConFL Connectivity Concepts Quality of LP bounds Hop Constrained ConFL Cut Based Formulations Layered Graphs Polyhedral Results Computational Results Conclusion
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion What is Connected Facility Location?
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion What is Connected Facility Location? ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion What is Connected Facility Location? ⋆ ⋆ ⋆ � � ⋆ ⋆ � � � ⋆ ⋆ � ⋆ ⋆ ⋆ ⋆ �
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion What is Connected Facility Location? ⋆ ⋆ ⋆ � � ⋆ ⋆ � � � ⋆ ◦ ◦ ⋆ � ⋆ ⋆ ⋆ ⋆ �
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion What is Connected Facility Location? ⋆ ⋆ ⋆ � � ⋆ ⋆ � � � ⋆ ◦ ◦ ⋆ � ⋆ ⋆ ⋆ ⋆ �
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion What is Connected Facility Location? ⋆ ⋆ ⋆ � � � � � � � � � � � � � � � � � � � � � � � � � ⋆ ⋆ � � � � � � � � � � � � � � � � � � ⋆ ◦ ◦ ⋆ � � � � � � � � � � � � � � � � � ⋆ ⋆ ⋆ ⋆ �
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion What is Connected Facility Location? ⋆ ⋆ ⋆ � � � � � � � � � � � � � � � � � � � � � � � � � ⋆ ⋆ � � � � � � � � � � � � � � � � � � � � � � � � � � ⋆ ◦ ◦ ⋆ � � � � � � � � � � � � � � � � � ⋆ ⋆ ⋆ ⋆ �
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion ConFL Given • a graph G = ( V , E ), a set of customers ( R ⊆ V ), a set of facilities ( F ⊆ V ) and a set of Steiner nodes (˜ S ⊆ V ) such that S := ˜ S ∪ F and S ∩ R = ∅ , • edge cost c e ≥ 0 for all e ∈ E and • facility opening costs f i for all i ∈ F we try to find • a subset of open facilities such that • each customer is assigned to exactly one facility, • a Steiner tree connects the open facilities and • facility opening and edge costs are minimized.
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion Applications and Previous Work Applications • Telecommunications • Fiber-to-the-Curb strategy for fiber-optic broadband networks • Multicasting Previous Work • Approximation Algorithms: 4.23 (4.00) (Eisenbrand et al., 2008), ..., (Karger & Minkoff, 2000) • Heuristics: Dual-Ascent (Bardossy & Raghavan, 2010), GRASP (Tomazic & Ljubi´ c, 2008), VNS (Ljubi´ c, 2007) • Exact: Branch-and-Cut (Gollowitzer & Ljubi´ c, 2010)
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion Notation Sets • A S : core graph (bidirected), connecting Steiner nodes and facilities • A R : assignment graph (directed), connecting facilities to customers • A := A R ∪ A S Variables and Parameters • x ij indicates whether arc ij is in the solution (1) or not (0) • z j indicates whether facility j is in the solution (1) or not (0) • c ij costs for opening arc ij • f j costs for opening facility j
� � � Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion Cuts based on customers ⋆ � � ���������������������� � �������������������������� � � � � � � � � � � � � � � � � � � � � � � � � 3 � 4 � ������������������� � ��������������� � � � � � � � � � � � � � � � � � � � � � � � � ≥ 1 � r � 1 � 2
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion Model CUT R � � min x ij c ij + z i f i ij ∈ A i ∈ F � x uv ≥ 1 ∀ W ∩ R � = ∅ , r �∈ W (1) s.t. uv ∈ δ − ( W ) x jk ≤ z j ∀ jk ∈ A R z r = 1 x ij ∈ { 0 , 1 } ∀ ij ∈ A z i ∈ { 0 , 1 } ∀ i ∈ F
� � � � � � � � � � � � � Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion Cuts based on facilities ⋆ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � = 1 � � � � � � � � � � � � � � � 3 � � � � � � � � � � � � � � � � � � � � 4 � r � 1 � 2
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion Cuts based on facilities ⋆ � 3 ���������������� ���������� � � � � � � � � � � � � � � � � � � � � � � � � � ≥ z 3 � � � � � � 4 � r � 1 � 2
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion Model CUT F � � min x ij c ij + z i f i ij ∈ A i ∈ F � x uv ≥ z i ∀ W ⊆ S \ { r } , ∀ i ∈ W ∩ F � = ∅ (2) s.t. uv ∈ δ − ( W ) � ∀ k ∈ R x jk = 1 jk ∈ A R x jk ≤ z j ∀ jk ∈ A R z r = 1 x ij ∈ { 0 , 1 } ∀ ij ∈ A z i ∈ { 0 , 1 } ∀ i ∈ F
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion Quality of LP bounds?
� � � � Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion Quality of LP bounds? Example � 1 Facility opening and assignment � 2 costs are 1. c rs = L and c si = 1, for all i ∈ { 1 , . . . , n } . . � r ◦ s ⋆ k . . L � n − 1 � n Solutions Lemma υ LP = L / n + 4 There are instances for which υ IP = L + 4 υ LP ( CUT F ) ≈ 1 | F | OPT .
� � � � Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion Quality of LP bounds? Example � 1 Facility opening and assignment 1 / n 1 / n � 2 costs are 1. c rs = L and c si = 1, for all i ∈ { 1 , . . . , n } . 1 / n . � r ◦ s ⋆ k . . L � n − 1 � n Solutions Lemma υ LP = L / n + 4 There are instances for which υ IP = L + 4 υ LP ( CUT F ) ≈ 1 | F | OPT .
� � Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion Quality of LP bounds? Example � 1 Facility opening and assignment � � � � � 1 � � 1 � 2 costs are 1. c rs = L and c si = 1, for � � � � � � � � all i ∈ { 1 , . . . , n } . � 1 � ◦ s . � r ⋆ k . . L � n − 1 � n Solutions Lemma υ LP = L / n + 4 There are instances for which υ IP = L + 4 υ LP ( CUT F ) ≈ 1 | F | OPT .
� � Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion Quality of LP bounds? Example � 1 Facility opening and assignment � � � � � 1 � � 1 � 2 costs are 1. c rs = L and c si = 1, for � � � � � � � � all i ∈ { 1 , . . . , n } . � 1 � ◦ s . � r ⋆ k . . L � n − 1 � n Solutions Lemma υ LP = L / n + 4 There are instances for which υ IP = L + 4 υ LP ( CUT F ) ≈ 1 | F | OPT .
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion Example: Neusiedl am See – Last Mile Network customers Steiner nodes open facilities
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion Example: Neusiedl am See – Last Mile Network customers Steiner nodes open facilities
Introduction ConFL Hop Constrained ConFL Polyhedral Results Computational Results Conclusion Hop Constrained ConFL HC ConFL: Connected Facility Location problem with an additional constraint: • At most H nodes on each path between the root r and an open facility. Application: When survivability of a network with respect to node failures is an issue.
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