Weighted Spanning Tree Constraint with Explanations

Transcription

1 Weighted Spanning Tree Constraint with Explanations CPAIOR 016 Diego de Uña, Graeme Gange, Peter Schachte and Peter J. Stuckey Presented by Peter J. Stuckey May 1, 016 University of Melbourne CIS Department NICTA/Data61 Victoria

2 Table of Contents 1. Background Spanning Trees in the Real World Variants of the Minimum Spanning Tree The WST constraint. Explaining Kruskal s algorithm Kruskal s algorithm with mandatory and forbidden edges Example Propagation rule. Experimental Results Peter J. Stuckey WST Constraint with Explanations CPAIOR 016

3 Background

4 Spanning Trees in the Real World Off-shore Wind Farm Planning (Klein et al., 015): Minimize the amount of cable needed to connect a network of turbines such that: The degree of each turbine is less than. The number of turbines on each branch connected to the main transformer is less than 7. Cables between turbines don t cross each other. Electrical and computer networks: there is a maximum number of cables that can be connected to a single switch (Wang et al., 199). Data Compression uses the Spanning trees with bounds on the diameter (Bookstein et al., 1991). Peter J. Stuckey WST Constraint with Explanations CPAIOR 016

5 Variants of the Minimum Spanning Tree Most variants of the Minimum Spanning Tree are NP-hard: Capacitated MST: where a root is chosen and each subtree under the root has a maximum number of nodes (Papadimitriou et al., 198). Degree-constrained MST: minimum or maximum degree of nodes (Narula et al., 1980). Constrained MST: two weights per edge (costs of building and maintenance, for instance) with a bound on one and an objective on the other (Ravi et al., 1996). Diameter-constrained MST: bound on the diameter of the tree (Achuthan et al., 199).... Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 5

6 The WST constraint Given: a graph G = (N, E) a weight function ws : E N a set B containing a Boolean variable c e for each edge. an integer variable w the constraint wst(n, E, ws, B, w) ensures that B describes a sub-tree T of G that spans all nodes in G and is of cost w. c e = true e T Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 6

7 Previous work & Constribution Brief literature review on the WST constraint: The MST constraint (Dooms et al., 006): ensure a graph is an MST. Cannot compromise cost for side constraints. The WST constraint (Dooms et al., 007)(Régin, 008): first appearance of the constraint we study. No explanations Contributions: Explanations for the lower bound of the WST constraint. Explanations for a propagation rule previously introduced. Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 7

8 Explaining Kruskal s algorithm

9 Kruskal s algorithm revisited Kruskal s algorithm main idea: Add cheapest edge until tree is formed. Use a priority queue to retrieve edges. Before adding an edge, ensure it does not form a cycle. Use Union-Find to test acyclicity. Observations: When adding an edge e = (u, v), if u and v are already connected, all edges between them are cheaper (or equal to) than e. If we skip an edge e = (u, v), a more expensive (or equal) edge will be used to connect u and v. Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 9

10 Kruskal s algorithm with mandatory and forbidden edges It is easy to modify Kruskal s algorithm to incorporate information about which edges MUST be in the tree and which edges CANNOT be in the tree. Mandatory edges: Pre-add mandatory edges as long as they don t form cycles. Forbidden edges: Simply skip them. Some more expensive group of edges will connect the endnodes. Peter J. Stuckey WST Constraint with Explanations CPAIOR

11 Explaining Kurskal s algorithm We need to explain failure when the cost of the cheapest possible Spanning Tree given the current decisions is above the limit w. 1. Run Kruskal s algorithm while identifying: a subset of forbidden edges to add in the explanations, a substitute edge for each edge. Identify mandatory edges that could be substituted by cheaper edges such that the total cost was below the limit. Substitute edge e s is a substitute of e T iff replacing e with e s in T yields a spanning tree. Peter J. Stuckey WST Constraint with Explanations CPAIOR

12 Explaining Kurskal s algorithm: Step 1 (Example) Why is the cost of the spanning tree greater than 7 given the current decisions? Explanation = c b h 1 a 5 f d g e i Fig. 1: Initial graph Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 1

13 Explaining Kurskal s algorithm: Step 1 (Example) Why is the cost of the spanning tree greater than 7 given the current decisions? Explanation = c b h 1 a 5 f d g e i Fig. 1: Mandatory edges pre-added. Forbidden edges appear dotted in grey Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 1

14 Explaining Kurskal s algorithm: Step 1 (Example) Why is the cost of the spanning tree greater than 7 given the current decisions? Explanation = c b h 1 a 5 f d g e i Fig. 1: Following Kruskal s algorithm: add edge (h, a) Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 1

15 Explaining Kurskal s algorithm: Step 1 (Example) Why is the cost of the spanning tree greater than 7 given the current decisions? Explanation = c b h 1 a 5 f d g e i Fig. 1: Following Kruskal s algorithm: add edge (e, i) Peter J. Stuckey WST Constraint with Explanations CPAIOR

16 Explaining Kurskal s algorithm: Step 1 (Example) Why is the cost of the spanning tree greater than 7 given the current decisions? Explanation = c b h 1 a 5 f d g e i Fig. 1: Following Kruskal s algorithm: add edge (f, g) Peter J. Stuckey WST Constraint with Explanations CPAIOR

17 Explaining Kurskal s algorithm: Step 1 (Example) Why is the cost of the spanning tree greater than 7 given the current decisions? Explanation = c b h 1 a 5 f d g e i Fig. 1: Following Kruskal s algorithm: cannot add edge (b, c) Peter J. Stuckey WST Constraint with Explanations CPAIOR

18 Explaining Kurskal s algorithm: Step 1 (Example) Why is the cost of the spanning tree greater than 7 given the current decisions? Explanation = h f g Forbidden edge 1 We will have to connect b to c through some more expensive b a edges, thus e we add it to the explanation: c (b,c) 5 10 c d 100 i Fig. 1: Following Kruskal s algorithm: cannot add edge (b, c) Peter J. Stuckey WST Constraint with Explanations CPAIOR

19 Explaining Kurskal s algorithm: Step 1 (Example) Why is the cost of the spanning tree greater than 7 given the current decisions? Explanation = c (b,c) c b h 1 a 5 f d g e i Fig. 1: Following Kruskal s algorithm: add edge (b, a) Peter J. Stuckey WST Constraint with Explanations CPAIOR

20 Explaining Kurskal s algorithm: Step 1 (Example) Why is the cost of the spanning tree greater than 7 given the current decisions? Explanation = c (b,c) c b h 1 a 5 f d g e i Fig. 1: Following Kruskal s algorithm: cannot add edge (g, e) Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 0

21 Explaining Kurskal s algorithm: Step 1 (Example) Why is the cost of the spanning tree greater than 7 given the current decisions? Explanation = c (b,c) Substitute edge h Node g and e are already connected. 1 Edge (g, e) could be used instead of (g, f ), (f, a), (a, d) b or (d, e). We say (g, e) substitutes those edges. c a 5 f d g e i Fig. 1: Following Kruskal s algorithm: cannot add edge (g, e) Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 1

22 Explaining Kurskal s algorithm: Step 1 (Example) Why is the cost of the spanning tree greater than 7 given the current decisions? Explanation = c (b,c) c b h 1 a 5 f d g e i Fig. 1: Following Kruskal s algorithm: add edge (a, f ) Peter J. Stuckey WST Constraint with Explanations CPAIOR 016

23 Explaining Kurskal s algorithm: Step 1 (Example) Why is the cost of the spanning tree greater than 7 given the current decisions? Explanation = c (b,c) c b h 1 a 5 f d g e i Fig. 1: Following Kruskal s algorithm: add edge (c, a) Peter J. Stuckey WST Constraint with Explanations CPAIOR 016

24 Explaining Kurskal s algorithm: Step 1 (Example) Why is the cost of the spanning tree greater than 7 given the current decisions? Explanation = c (b,c) c b h 1 a 5 f d g e i Fig. 1: Following Kruskal s algorithm: add edge (a, d) Peter J. Stuckey WST Constraint with Explanations CPAIOR 016

25 Explaining Kurskal s algorithm: Step 1 (Example) Why is the cost of the spanning tree greater than 7 given the current decisions? Explanation = c (b,c) c b h 1 a 5 f d g e i Fig. 1: Following Kruskal s algorithm: add edge (d, e) Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 5

26 Explaining Kurskal s algorithm: Step 1 (Example) Why is the cost of the spanning tree greater than 7 given the current decisions? Explanation = c (b,c) h Total cost = f 11 > 7 We must fail. b a 5 c d g e i Fig. 1: Following Kruskal s algorithm: add edge (d, e) Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 6

27 Explaining Kruskal s algorithm: Step (Example) Processing the mandatory edges: For each mandatory edge that has a possible substitute, pretend we substitute the edge (relax it). If the new cost is lower than the limit explanation Else continue decreasing the cost In our example: 1 (f, a) + (g, e) = 1 + = 0 > 7 continue 0 (a, d) + (g, e) = = 8 > 7 continue 8 (d, e) + (g, e) = = 1 < 7 explanation The final explanation is: c (b,c) c (d,e) w > 7 Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 7

28 Explaining Kruskal s algorithm (Example) Final explanation: c (b,c) c (d,e) w > 7 c b h 1 a 5 f d g e i Fig. : Following Kruskal s algorithm: add edge (d, e) Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 8

29 Summary of the algorithm 1. Pre-add mandatory edges. Add edges e = (u, v) as per Kruskal s algorithm if u and v not yet connected and e is forbidden: add c e to the explanation if u and v connected:.1 update substitutes in the path connecting them. if some edge in the path is more expensive than e and e is forbidden add c e to the explanation. Relax mandatory edges e with their substitutes e s until we get to a cost lower than the limit: if cost w e + w es limit: cost cost w e + w es else add c e to the explanation Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 9

30 Explaining propagation We have computed the cheapest Spanning Tree T given the current decisions of cost W. Let e = (u, v) be a possible edge not in T. Its support is the most expensive edge in T in the path from u to v. If replacing the support of e with e in T yields a tree that is too expensive, e cannot be part of T (Régin, 008). Our explanations are also applicable to this! Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 0

31 Experimental Results

32 Diameter Constrained Spanning Trees We compare NoProp: using a decomposition with explanation NoExpl: using a global propagator without explanation NaiveExpl: using a global propagator with naive explanations Expl: using a global propagator with good explanations Choco: using Choco and its WST propagator NRS: published results for CP Optimizer with WST propagator Peter J. Stuckey WST Constraint with Explanations CPAIOR 016

33 Diameter-Constrained MST Instance NoProp NoExpl NaiveExpl Expl Choco NRS (IBM OPL) N E D Nodes Time Nodes Time Nodes Time Nodes Time Nodes Time Nodes Time Total Peter J. Stuckey WST Constraint with Explanations CPAIOR 016

34 Non-Crossing Minimal Spanning Trees classes of Euclidean distance complete graphs C101, C01, R101, RC101 (Solomon benchmarks for VRP) Modify lengths of edges by random coefficient w[e] = coef * randomweight + (1-coef) * originalweight Compare results of NoExpl versus Expl reduction in nodes in search Results for time are correlated Peter J. Stuckey WST Constraint with Explanations CPAIOR 016

35 Non-crossing edges MST on Euclidian Graphs reduct reduct coef coef -1.0 C101 C01 R101 RC101 (a) Instances with 5 nodes, 00 edges -1.0 C101 C01 R101 RC101 (b) Instances with 0 nodes, 5 edges reduct reduct coef coef -1.0 C101 C01 R101 RC101 (c) Instances with 5 nodes, 595 edges -1.0 C101 C01 R101 RC101 (d) Instances with 0 nodes, 780 edges Fig. : Reduction in nodes provided by explanations. The closer the reduction is to 1, the more beneficial explanations were. Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 5

36 Conclusion Efficient algorithm for computing explanations for WST propagations Does not produce minimal explanations Beneficial in constrained spanning tree problems Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 6

37 Thank you for your attention. Questions? Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 7

38 Why do we need to update the cost? (just in case) d e a b c 10 1 Fig. : Example of when updating the cost is absolutely necessary Say limit is 5. This is feasible (MST is = ). If (a, b) and (b, c) are mandatory though, the best solution is = 7, = 7 empty explanation = 7, = c (b,c) explanation Peter J. Stuckey WST Constraint with Explanations CPAIOR 016 8