Exact Combinatorial Algorithms for Graph Bisection

Transcription

1 Exact Combinatorial Algorithms for Graph Bisection Daniel Delling Andrew V. Goldberg Ilya Razenshteyn Renato Werneck Microsoft Research Silicon Valley DIMACS Challenge Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 1 / 26

2 Minimum Bisection Input: undirected, unweighted graph G = (V, E) Output: partition V into sets A and B such that 1 A, B V /2 2 the number of edges between A and B is minimized Variants: ɛ-unbalanced, weights, more parts... Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 2 / 26

3 Example Instance [3524 nodes, 5560, solution 30] Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 3 / 26

4 Example Instance [3524 nodes, 5560, solution 30] Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 3 / 26

5 Motivation Applications: load balancing for parallel computing preprocessing step for some road network algorithms divide-and-conquer (e.g. VLSI design) Solutions: NP-hard, O(log n) best approximation [Räcke 08]. Heuristics: numerous fast and good partitioners often tailored to specific graph classes (e.g. road networks) no approximation/optimality guarantees Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 4 / 26

6 Motivation Applications: load balancing for parallel computing preprocessing step for some road network algorithms divide-and-conquer (e.g. VLSI design) Solutions: NP-hard, O(log n) best approximation [Räcke 08]. Heuristics: numerous fast and good partitioners often tailored to specific graph classes (e.g. road networks) no approximation/optimality guarantees We want exact algorithms! Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 4 / 26

7 Branch-and-Bound Branch-and-bound: implicit enumeration. Subproblem = partial assignment (A, B), with A, B V represents bisections (A +, B + ) with A A + and B B + Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 5 / 26

8 Branch-and-Bound Branch-and-bound: implicit enumeration. Subproblem = partial assignment (A, B), with A, B V represents bisections (A +, B + ) with A A + and B B + Branch: pick v A B, create subproblems (A {v}, B) and (A, B {v}). Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 5 / 26

9 Branch-and-Bound Branch-and-bound: implicit enumeration. Subproblem = partial assignment (A, B), with A, B V represents bisections (A +, B + ) with A A + and B B + Branch: pick v A B, create subproblems (A {v}, B) and (A, B {v}). Bound: L: lower bound on all bisections consistent with (A, B) U: best known bisection (updated on-line) if L U, done with (A, B); otherwise branch Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 5 / 26

10 Branch-and-Bound Branch-and-bound: implicit enumeration. Subproblem = partial assignment (A, B), with A, B V represents bisections (A +, B + ) with A A + and B B + Branch: pick v A B, create subproblems (A {v}, B) and (A, B {v}). Bound: L: lower bound on all bisections consistent with (A, B) U: best known bisection (updated on-line) if L U, done with (A, B); otherwise branch Crucial ingredient: computing lower bounds. Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 5 / 26

11 Lower Bounds Known bounds: linear programming [FMdSWW98, Sen01] hundreds of nodes quadratic programming [HPZ11] up to 3000 nodes semidefinite programming [AFHM08, Arm07] up to 6000 nodes multicommodity flows [SST03] hundreds of nodes degree-based combinatorial [Fel05] tens of nodes (random) Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 6 / 26

12 Lower Bounds Known bounds: linear programming [FMdSWW98, Sen01] hundreds of nodes quadratic programming [HPZ11] up to 3000 nodes semidefinite programming [AFHM08, Arm07] up to 6000 nodes multicommodity flows [SST03] hundreds of nodes degree-based combinatorial [Fel05] tens of nodes (random) We want to do better. Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 6 / 26

13 Summary Our result: exact combinatorial algorithm for graph bisection works well for graphs with a small minimum bisection road networks, VLSI instances, meshes... solves much larger instances than previous approaches Main contributions: new lower bounds branching rules novel decomposition technique Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 7 / 26

14 Flow Lower Bound Goal: lower-bound all bisections consistent with (A, B). Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 8 / 26

15 Flow Lower Bound Goal: lower-bound all bisections consistent with (A, B). Known bound: min-cut (max-flow) between A and B. Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 8 / 26

16 Flow Lower Bound Goal: lower-bound all bisections consistent with (A, B). Known bound: min-cut (max-flow) between A and B. pros: simple, upper bound when balanced; Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 8 / 26

17 Flow Lower Bound Goal: lower-bound all bisections consistent with (A, B). Known bound: min-cut (max-flow) between A and B. pros: simple, upper bound when balanced; cons: weak if A B, typically very unbalanced. Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 8 / 26

18 Packing Lower Bound A, B V : current partial assignment Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 9 / 26

19 Packing Lower Bound A, B V : current partial assignment A + : an extension of A of size V /2 Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 9 / 26

20 Packing Lower Bound A, B V : current partial assignment A + : an extension of A of size V /2 Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 9 / 26

21 Packing Lower Bound A, B V : current partial assignment A + : an extension of A of size V /2 Fact: min A + cut(a +, B) is a valid lower bound for (A, B) cut(a +, B): min-cut between A + and B. Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 9 / 26

22 Packing Lower Bound A, B V : current partial assignment A + : an extension of A of size V /2 Fact: min A + cut(a +, B) is a valid lower bound for (A, B) cut(a +, B): min-cut between A + and B. We must reason about the worst possible extension A +. Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 9 / 26

23 Packing Lower Bound Basic algorithm: Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 10 / 26

24 Packing Lower Bound Basic algorithm: partition free nodes into connected cells adjacent to B Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 10 / 26

25 Packing Lower Bound Basic algorithm: partition free nodes into connected cells adjacent to B if a subset A + of V of size V /2 hits k cells... Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 10 / 26

26 Packing Lower Bound Basic algorithm: partition free nodes into connected cells adjacent to B if a subset A + of V of size V /2 hits k cells......there is a flow of at least k units between A + and B. Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 10 / 26

27 Packing Lower Bound Basic algorithm: partition free nodes into connected cells adjacent to B if a subset A + of V of size V /2 hits k cells......there is a flow of at least k units between A + and B. Lower bound for (A, B) = worst-case A + : hits the fewest possible cells Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 10 / 26

28 Packing Lower Bound Basic algorithm: partition free nodes into connected cells adjacent to B if a subset A + of V of size V /2 hits k cells......there is a flow of at least k units between A + and B. Lower bound for (A, B) = worst-case A + : hits the fewest possible cells adversary picks entire cells, from biggest to smallest Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 10 / 26

29 Packing Lower Bound Basic algorithm: partition free nodes into connected cells adjacent to B if a subset A + of V of size V /2 hits k cells......there is a flow of at least k units between A + and B. Lower bound for (A, B) = worst-case A + : hits the fewest possible cells adversary picks entire cells, from biggest to smallest partition should have balanced cells (greedy + local search) Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 10 / 26

30 Flow + Packing To combine flow and packing, remove flow edges before computing cells. Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 11 / 26

31 Flow + Packing To combine flow and packing, remove flow edges before computing cells. Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 11 / 26

32 Performance Proving 30 is a lower bound: search tree: 1.3M nodes time: 50 minutes 1 [3524 vertices, 5560 edges, answer 30] Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 12 / 26

33 Branching Rule and Forced Assignments Branch on vertices likely to increase L the most: 1 far from A and B (to produce better cells) 2 well connected to other vertices (to increase flow) 3 contained in large packing cells Forced assignments: use logical implications to fix some vertices to A or B works if upper and lower bounds are close discards many potential branching nodes Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 13 / 26

34 Branching Rule and Forced Assignments Proving 30 is a lower bound: search tree: 1.3M nodes time: 50 minutes 1 [3524 vertices, 5560 edges, answer 30] Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 14 / 26

35 Branching Rule and Forced Assignments Proving 30 is a lower bound: search tree: 1.3M nodes 90K nodes time: 50 minutes 3.5 minutes 1 [3524 vertices, 5560 edges, answer 30] Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 14 / 26

36 Increasing Degrees Algorithm is very sensitive to the degrees of assigned nodes. Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 15 / 26

37 Increasing Degrees Algorithm is very sensitive to the degrees of assigned nodes. Idea: branch on entire regions. Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 15 / 26

38 Increasing Degrees Algorithm is very sensitive to the degrees of assigned nodes. Idea: branch on entire regions. Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 15 / 26

39 Increasing Degrees Algorithm is very sensitive to the degrees of assigned nodes. Idea: branch on entire regions. Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 15 / 26

40 Increasing Degrees Algorithm is very sensitive to the degrees of assigned nodes. Idea: branch on entire regions. Problem: a region can cross the minimum bisection. Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 15 / 26

41 Increasing Degrees Algorithm is very sensitive to the degrees of assigned nodes. Idea: branch on entire regions. Problem: a region can cross the minimum bisection. Solution: decomposition! Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 15 / 26

42 Decomposition Given an upper bound U: 1 split edges into U + 1 disjoint sets E 1, E 2,..., E U+1 Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 16 / 26

43 Decomposition Given an upper bound U: 1 split edges into U + 1 disjoint sets E 1, E 2,..., E U+1 Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 16 / 26

44 Decomposition Given an upper bound U: 1 split edges into U + 1 disjoint sets E 1, E 2,..., E U+1 Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 16 / 26

45 Decomposition Given an upper bound U: 1 split edges into U + 1 disjoint sets E 1, E 2,..., E U+1 Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 16 / 26

46 Decomposition Given an upper bound U: 1 split edges into U + 1 disjoint sets E 1, E 2,..., E U+1 Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 16 / 26

47 Decomposition Given an upper bound U: 1 split edges into U + 1 disjoint sets E 1, E 2,..., E U+1 Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 16 / 26

48 Decomposition Given an upper bound U: 1 split edges into U + 1 disjoint sets E 1, E 2,..., E U+1 2 for every i, contract E i and run branch-and-bound Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 16 / 26

49 Decomposition Given an upper bound U: 1 split edges into U + 1 disjoint sets E 1, E 2,..., E U+1 2 for every i, contract E i and run branch-and-bound 3 return best solution found Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 16 / 26

50 Decomposition Given an upper bound U: 1 split edges into U + 1 disjoint sets E 1, E 2,..., E U+1 2 for every i, contract E i and run branch-and-bound 3 return best solution found Some E i will cross the optimum bisection; at least one will not! Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 16 / 26

51 Decomposition Given an upper bound U: 1 split edges into U + 1 disjoint sets E 1, E 2,..., E U+1 2 for every i, contract E i and run branch-and-bound 3 return best solution found Some E i will cross the optimum bisection; at least one will not! E i should be a set of clumps (high degree, well spread). Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 16 / 26

52 Decomposition Proving 30 is a lower bound: search tree: 90K nodes time: 3.5 minutes 1 [3524 vertices, 5560 edges, answer 30] Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 17 / 26

53 Decomposition Proving 30 is a lower bound: search tree: 90K nodes 1369 nodes time: 3.5 minutes 2 seconds 1 [3524 vertices, 5560 edges, answer 30] Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 17 / 26

54 Experiments Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 18 / 26

55 Examples VLSI instance: I I search tree: 20K nodes time: 9 minutes [34046 vertices, edges, answer 80] Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 19 / 26

56 Examples VLSI instance: I I search tree: 20K nodes time: 9 minutes [34046 vertices, edges, answer 80] Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 19 / 26

57 Walshaw Instances Standard benchmark for graph partitioning (ɛ = 0). instance n m opt BB nodes time [s] add uk elt whitaker fe 4elt elt data [ : distributed execution using DryadOpt] Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 20 / 26

58 Walshaw Instances Standard benchmark for graph partitioning (ɛ = 0). instance n m opt BB nodes time [s] add uk elt whitaker fe 4elt elt data [ : distributed execution using DryadOpt] Optimum bisections were known before, but without proofs. Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 20 / 26

59 Other Challenge Instances series instance n m opt BB nodes time [s] clustering karate chesapeake dolphins lesmis polbooks football power delaunay delaunay n delaunay n delaunay n delaunay n streets luxembourg Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 21 / 26

60 Other Challenge Instances series instance n m opt BB nodes time [s] clustering karate chesapeake dolphins lesmis polbooks football power delaunay delaunay n delaunay n delaunay n delaunay n streets luxembourg We can solve very large instances with small bisections. Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 21 / 26

61 Examples NW (9th Challenge): 264 branch-and-bound nodes, 25 minutes [1.2M vertices, 1.4M edges, answer 18] Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 22 / 26

62 Instances from exact literature State-of-the-art approaches: [Arm07]: semidefinite programming [HPZ11]: quadratic programming instance n m opt time [s] [Arm07] [HPZ11] KKT putt01 m mesh gap taq gap taq Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 23 / 26

63 Instances from exact literature State-of-the-art approaches: [Arm07]: semidefinite programming [HPZ11]: quadratic programming instance n m opt time [s] [Arm07] [HPZ11] KKT putt01 m mesh gap taq gap taq Excellent performance for small bisections. Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 23 / 26

64 Examples Gargoyle: 2.6M branch-and-bound nodes, 13 hours [10002 vertices, edges, answer 175] Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 24 / 26

65 Examples Gargoyle: 2.6M branch-and-bound nodes, 13 hours [10002 vertices, edges, answer 175] Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 24 / 26

66 Examples Feline (mesh): 150K branch-and-bound nodes, 75 minutes [20629 vertices, edges, answer 148] Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 25 / 26

67 Examples Feline (mesh): 150K branch-and-bound nodes, 75 minutes [20629 vertices, edges, answer 148] Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 25 / 26

68 Conclusion New algorithm for minimum graph bisection packing bound decomposition Cut size matters Potential applications: evaluate/improve heuristics Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 26 / 26

69 Conclusion New algorithm for minimum graph bisection packing bound decomposition Cut size matters Potential applications: evaluate/improve heuristics Thank you! Renato Werneck (Microsoft Research) Exact Combinatorial Algorithms for Graph Bisection DIMACS Challenge 26 / 26