Steiner Tree Problems with Side Constraints Using Constraint Programming

Size: px

Start display at page:

Download "Steiner Tree Problems with Side Constraints Using Constraint Programming"

Peregrine Dennis
5 years ago
Views:

1 Steiner Tree Problems with Side Constraints Using Constraint Programming AAAI 2016 Diego de Uña, Graeme Gange, Peter Schachte and Peter J. Stuckey February 14, 2016 University of Melbourne CIS Department

2 Table of Contents 1. Background Example of a Steiner Tree Problem The Steiner Tree Problem Constraint Programming 2. Steiner Tree Propagator Steiner Tree Propagator Some of the main ideas: Treeness Some of the main ideas: Lower bounds Diego de Uña STP with Side Constraints Using CP AAAI

3 Background

4 Example of a Steiner Tree Problem Given an electrical network, connect a given subset of the nodes with minimum cost. a b c d e f g h i j k l m Fig. 1: Base solution. All weights are 1 (solid) or (dashed). Cost 10 Diego de Uña STP with Side Constraints Using CP AAAI

5 Example of a Steiner Tree Problem Given an electrical network, connect a given subset of the nodes with minimum cost. Side constraint: for each pair of nodes there need two be 2 edge-disjoint alternative paths. b f j a c d g h k l e i m Fig. 1: Problem with side constraint between b and g Diego de Uña STP with Side Constraints Using CP AAAI

6 Example of a Steiner Tree Problem Given an electrical network, connect a given subset of the nodes with minimum cost. Side constraint: for each pair of nodes there need two be 2 edge-disjoint alternative paths. b f j a c d g h k l e i m Fig. 1: Solution (solid lines): all pairs of black nodes have at least two edge-disjoint alternative routes. Cost 17 Diego de Uña STP with Side Constraints Using CP AAAI

7 The Steiner Tree Problem Definition: Given a weighted graph G and a set M of mandatory nodes (aka terminals). Find a tree T of minimum weight s.t. all the nodes in M are in T. Proved NP-complete by Karp. Generalization of MST. Useful for: Electrical networks (Agrawal et al., 15). VLSI design (Hwang et al., 12). Phylogenetics (Winter, 187). Broadcasting (quality) (Karpinski et al., 2003).... c.f. The Steiner Tree Problem (book, (Hwang et al., 12)). Diego de Uña STP with Side Constraints Using CP AAAI

8 The STP is never pure State of the art in the pure STP achieved by (Polzin et al., 2001). Reductions remove edges that are useless. In the real world: STP is never alone. Examples: Find STP such that... Maximum/minimum degree on the nodes. Mandatory nodes must be leafs. Capacity constraint. Connectivity constraints. Diameter constrained. Geometric constraints.... anything you can imagine. and all combined! Diego de Uña STP with Side Constraints Using CP AAAI

Constraint Programming High-level programming paradigm: Programmer becomes a modeller. Uses a solver to solve his model. Versatile and reusable code. Fast, low memory usage.

9 Constraint Programming High-level programming paradigm: Programmer becomes a modeller. Uses a solver to solve his model. Versatile and reusable code. Fast, low memory usage. Our choice: Modelling language: MiniZinc (Nethercote et al., 2007). Solver: Chuffed (Chu, 2011). Fig. 2: Example of MiniZinc code Diego de Uña STP with Side Constraints Using CP AAAI 2016

10 Constraint Programming Solver (Schulte et al., 2008) A CP solver works by interleaving search and propagation. Search: try all possible values for all variables. = Too slow! Propagation: filter the possible values of variables to not try too many options. = Reduce the search space Example of propagation a {2, 3, 4}, b {1, 2, 3} such that a + 2b 4 a b Search: try a = 2 Propagation: a b b {1, 2, 3} Propagation: a + 2b 4 b {1, 3} Propagators DO NOT solve the problem, just filter domains. LCG: Propagators can generate explanations/clauses. Diego de Uña STP with Side Constraints Using CP AAAI

11 Steiner Tree Propagator

12 Steiner Tree Propagator Graphs in CP: Boolean variables for edges and nodes. If true part of the solution The global constraint steiner tree({n n V }, {e e E}, adj, ends, ws, w) }{{}}{{} the graph variable the graph Ensures that the solution is a tree (checks failure and propagates). Ensures the solution has cost w. Diego de Uña STP with Side Constraints Using CP AAAI

13 Some of the main ideas: Treeness mandatory undecided forbidden reachable: run DFS to detect unreachable nodes. Explain: run DFS from unreachable node find the border. e 5 e 6 e 7 n o fail e 1 n e 2 b c e 6 e 5 g e 7 e 8 d f e 3 o e 4 Fig. 3: Explaining reachability Diego de Uña STP with Side Constraints Using CP AAAI

14 Some of the main ideas: Treeness articulations: bridges and articulations must be in the solution. Explain: During Tarjan s algorithm record mandatory nodes. Run DFS s to find forbidden edges. c h d c h e 4 e 5 e c e 6 b c e 4 e 3 mandatory undecided forbidden a e 2 e 6 e 3 e 5 n b c e 1 e e 4 e 10 d g e h e 7 e 8 f Fig. 3: Explaining articulations and bridges Diego de Uña STP with Side Constraints Using CP AAAI

15 Some of the main ideas: Treeness cycle: prevent and detect cycles by maintaining a UF. Explain: fancy UF remembers connections (no path compression) can retrieve entire paths. e 4 e 5 e 2 e 4 e 5 e 3 e 1 mandatory undecided forbidden e 4 a c e 5 e 3 e 2 b n e 1 Fig. 3: Cycle prevention Diego de Uña STP with Side Constraints Using CP AAAI

16 Some of the main ideas: Lower bounds If the lower bound is higher than w, we can stop the search here. Shortest path based LB (Splb): 1) Contract 2) Connect mandatory pairs by shortest paths Explain: all mandatory edges + all forbidden edges that used to be in a shortest path Fig. 4: Contraction LB(G ) = 1 spc G (u) 2 u M Diego de Uña STP with Side Constraints Using CP AAAI

17 Some of the main ideas: Lower bounds If the lower bound is higher than w, we can stop the search here. LP based lower bound (Lplb): Solve pure STP by LP, row generation (Aneja, 180). Explain: any edge with non-zero reduced cost. W Fig. 4: Cut formulation of the STP minimize ws[e] bool(e) such that: e E W, bool(e) 1 e δ(w ) Diego de Uña STP with Side Constraints Using CP AAAI

18 Some results Conflicts Nodes Propagations Time B Splb (2) Lplb (2) Lplb (n.l) (2) Sp+Lplb (2) Nolb (2) NoProp (3) Choco (10) Table 1: Results for the GoSST. Conflicts Nodes Propagations Time B Splb Lplb Lplb (n.l) (1*) Sp+Lplb Nolb NoProp (8) Choco (3) Table 2: Results for the TSTP. Diego de Uña STP with Side Constraints Using CP AAAI

19 Some results Conflicts Nodes Propagations Time B Splb (2) Lplb (2) Lplb (n.l) (2) Sp+Lplb (2) Nolb (2) NoProp (3) Choco (10) Table 1: Results for the GoSST. Conflicts Nodes Propagations Time B05 Lplb B05 Lplb (n.l) hours Table 2: LCG can save a lot of search! Diego de Uña STP with Side Constraints Using CP AAAI

20 Some (more) results Conflicts Nodes Propagations Time ES10FST ES20FST B Splb Lplb Lplb (n.l.) Sp+Lplb Nolb NoProp (4) Choco Splb (2) Lplb Lplb (n.l) Sp+Lplb Nolb (2) NoProp (11) Choco (3) Splb (2) Lplb Lplb (n.l) (1) Sp+Lplb Nolb (2) NoProp (11) Choco (5) Table 3: Results for the pure Steiner Tree problem. Diego de Uña STP with Side Constraints Using CP AAAI

21 Conclusion Contributions: First CP approach to the problem. Tree propagator with explanations. New fast lower bounding technique based on shortest paths, with explanations. Use of LP formulation as lower bound, added explanations. Diego de Uña STP with Side Constraints Using CP AAAI

22 Thank you for your attention. Questions? Diego de Uña STP with Side Constraints Using CP AAAI

23 Proof of Splb (just in case) In a Steiner Tree, there is a path such that contains 2 terminals and only one node of degree 3 or more. Removing such path from the tree yields a tree. Do: Find 2 mandatory nodes such that they are connected by one such path. Remove that path. Repeat until no pairs are left. The cost of those paths can only be to the cost of the tree. Shortest paths are even cheaper than those paths. Because we count some paths twice, we divide by 2. Diego de Uña STP with Side Constraints Using CP AAAI

Overview on kidney exchange programs

Overview on kidney exchange programs Ana Viana et al ana.viana@inesctec.pt 11 March 2016 Outline Kidney failure: some figures. The past: deceased and living donor transplants. The present: Kidney exchange