The Directed Network Design Problem with Relays Odysseus 2018 Markus Leitner 1 c 2 Martin Riedler 3 Mario Ruthmair 1 Ivana Ljubi´ 1 ISOR, University of Vienna, Vienna, Austria 2 ESSEC Business School of Paris, France 3 TU Wien, Vienna, Austria June 8, 2018 Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 1 / 36
Network Design with Relays Models network design problems in transportation and telecommunication. Freight transportation networks: for long haul distance trips, relay points are set along the paths for the exchange of drivers, trucks and trailers. Telecommunication networks: optical signal deteriorates after traversing a certain distance, and has to be re-amplified, i.e., regenerator devices need to be installed. E-mobility networks: batteries of EVs need to be recharged after a certain distance, hence charging stations need to be placed in the network. Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 2 / 36
Tesla Supercharger Network ( ≈ 1200 stations) Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 3 / 36
Network Design with Relays 1 Network Design: Build the network or augment the existing one. 2 Location: Where to place relays, and how many? 3 Routing: How to route each commodity from its source to destination? Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 4 / 36
PROBLEM DEFINITION Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 5 / 36
Directed Network Design with Relays Given: directed graph G = ( V, A ) relay placement costs c : V → Z > 0 arc costs w : A → Z ≥ 0 and arc lengths d : A → Z ≥ 0 set K of O-D pairs (commodities) distance limit λ max ∈ Z > 0 Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 6 / 36
Directed Network Design with Relays Given: directed graph G = ( V, A ) relay placement costs c : V → Z > 0 arc costs w : A → Z ≥ 0 and arc lengths d : A → Z ≥ 0 set K of O-D pairs (commodities) distance limit λ max ∈ Z > 0 Goal: install a subset of relays and arcs of minimum cost s.t. there exists a feasible simple path for each O-D pair from K . an O-D path P is feasible if each subpath of P which is longer than λ max contains a relay Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 6 / 36
Example — Symmetric Instance λ max = 5 , K = { ( A, B ) } 1 1 1 1 5 5 1 1 3 3 A B 5 Instance Acyclic Solution (cost=5) Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 7 / 36
Example — Symmetric Instance λ max = 5 , K = { ( A, B ) } 1 1 1 1 5 5 1 1 3 3 A B 5 Instance Cyclic Solution (cost=1) Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 7 / 36
Previous Work Undirected NDPR: Cabral et al. (2007): Set-covering formulation (each column is an O-D path, including relays) Heuristics: VNS, Xiao and Konak (2017), tabu search, Lin et al. (2014), GAs, Kulturel-Konak and Konak (2008); Konak (2012) Exact algorithms based on B&P&C (columns are segments between the relays): ◮ Yıldız et al. (2018) ◮ Leitner et al. (2018) Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 8 / 36
Previous Work Undirected NDPR: Cabral et al. (2007): Set-covering formulation (each column is an O-D path, including relays) Heuristics: VNS, Xiao and Konak (2017), tabu search, Lin et al. (2014), GAs, Kulturel-Konak and Konak (2008); Konak (2012) Exact algorithms based on B&P&C (columns are segments between the relays): ◮ Yıldız et al. (2018) ◮ Leitner et al. (2018) Directed NDPR: Introduced in Li et al. (2012), exact, 2 models: ◮ compact Node-Arc model ◮ Set-Covering model (similar to Cabral et al. (2007)) ⇒ B&P Heuristic: Li et al. (2017) Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 8 / 36
Our contribution: Directed NDPR: New models based on layered graphs (distance-expanded graphs): ◮ multi-commodity flows ◮ cut-sets Branch-and-Cut (B&C) algorithms for both models Both B&C significantly outperform the previous state-of-the-art from Li et al. (2012) Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 9 / 36
A BASIC FORMULATION Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 10 / 36
Node Arc Formulation from Li et al. (2012) 1 if k = ( i, v ) b k i = ( u, v ) ∈ K − 1 if k = ( u, i ) 0 otherwise v k i = distance of node i from the preceeding relay for commodity k . � 1 if relay is installed at node i y i = i ∈ V 0 otherwise � 1 if arc a is installed x a = a ∈ A 0 otherwise Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 11 / 36
Node Arc Formulation from Li et al. (2012) � � (NA) min c i y i + w a x a i ∈ V a ∈ A � f k � f k a = b k a − ∀ k ∈ K , ∀ i ∈ V (1) i a ∈ δ +( i ) a ∈ δ − ( i ) v k i + d ( i,j ) f k ( i,j ) − λ max (1 − f k ( i,j ) + y j ) ≤ v k ∀ k ∈ K , ∀ ( i, j ) ∈ A (2) j v k i + d ( i,j ) f k ( i,j ) ≤ λ max ∀ k ∈ K , ∀ ( i, j ) ∈ A (3) f k a ≤ x a ∀ k ∈ K , ∀ a ∈ A (4) 0 ≤ v k i ≤ λ max (1 − y i ) ∀ k ∈ K , ∀ i ∈ V (5) v u,v = 0 ∀ ( u, v ) ∈ K (6) u f k a ∈ { 0 , 1 } ∀ k ∈ K , ∀ a ∈ A (7) y i ∈ { 0 , 1 } ∀ i ∈ V (8) 0 ≤ x a ≤ 1 ∀ a ∈ A (9) Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 12 / 36
MODELS ON LAYERED GRAPHS Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 13 / 36
Solution Structure Set S of commodity sources, set T u of targets of source u Single-source case: If S = { u } , there exists an optimal solution which is a Steiner arborescence rooted at u , with leaves from T u . Each O-D path in this tree must be made feasible by installing some relays (when needed). Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 14 / 36
Solution Structure Set S of commodity sources, set T u of targets of source u Single-source case: If S = { u } , there exists an optimal solution which is a Steiner arborescence rooted at u , with leaves from T u . Each O-D path in this tree must be made feasible by installing some relays (when needed). Multiple sources: An optimal solution is a union of Steiner arborescences rooted at u , with required placement of relays when needed. Steiner arborescence: rooted subtree connecting a given set of terminals. Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 14 / 36
Example How to integrate the fact that on some nodes of the Steiner tree relays have to be installed? Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 15 / 36
Basic Idea Create node copies according to feasible distances at which a node can be reached Embed Steiner trees into this network, for each source u 0 4 1 4 2 4 3 4 0 3 1 3 2 3 3 3 0 2 1 2 2 2 2 1 1 2 1 1 0 3 2 3 0 0 1 0 2 0 3 0 1 Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 16 / 36
0 4 1 4 2 4 3 4 0 3 1 3 2 3 3 3 0 2 1 2 2 2 1 1 2 1 0 0 1 0 2 0 3 0 Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 17 / 36
Solution (Single Source) Solution: Steiner tree rooted at 0, each target reached at some layer. Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 18 / 36
Layered Graph Models Model Name Connectivity Aggregation Type L-CUT cutsets per source B&C L-MCF multi-commodity flow none pseudo-compact B&C L-CUT: z u ∀ u ∈ S, ∀ a ∈ A u a ∈ { 0 , 1 } L L-MCF: f uv ∀ ( u, v ) ∈ K , ∀ a ∈ A u ∈ { 0 , 1 } a L Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 19 / 36
Layered Cut Model � � (L-CUT) min c i y i + w a x a i ∈ V a ∈ A Ensure connectivity between the source u and a copy of v ∈ T u : ∀ u ∈ S, ∀ v ∈ T u , � z u { v l | v l ∈ V u L } ⊆ W ⊂ V u a ≥ 1 L , a ∈ δ − ( W ) u / ∈ W Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 20 / 36
Layered Cut Model � � (L-CUT) min c i y i + w a x a i ∈ V a ∈ A Ensure connectivity between the source u and a copy of v ∈ T u : ∀ u ∈ S, ∀ v ∈ T u , � z u { v l | v l ∈ V u L } ⊆ W ⊂ V u a ≥ 1 L , a ∈ δ − ( W ) u / ∈ W Indegree of a node v over all layers is at most one for i �∈ T u , and exactly one for i ∈ T u . � � z u ∈ T u , i � = u a ≤ 1 ∀ u ∈ S, ∀ i / i l ∈ V u ∈ A r a ∈ δ − ( i l ) ,a/ L L � � z u ∀ u ∈ S, ∀ i ∈ T u a = 1 i l ∈ V u ∈ A r a ∈ δ − ( i l ) ,a/ L L Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 20 / 36
Layered Cut Model (cont.) Vertical arcs linked to relays: � z u ( i l ,i 0 ) ≤ y i ∀ u ∈ S, ∀ i ∈ V ( i l ,i 0 ) ∈ A u L Each ( i, j ) ∈ A can be used in at most one layer � z u ( i l ,j m ) ≤ x ( i,j ) ∀ u ∈ S, ∀ ( i, j ) ∈ A ( i l ,j m ) ∈ A u L Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 21 / 36
Layered MCF Model: No linking with z u a needed Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 22 / 36
Comparing the strength of the two models Theorem Formulations L-MCF and L-CUT are equally strong, i.e., the LP-relaxation values of the two models coincide. Ivana Ljubi´ c Directed Network Design with Relays June 8, 2018 23 / 36
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