A practical anytime algorithm for bipartizing networks

Transcription

1 A practical anytime algorithm for bipartizing networks Timothy D. Goodrich, Blair D. Sullivan Department of Computer Science, North Carolina State University October 1, 2017 Goodrich, Sullivan Practical graph bipartization October 1, / 53

2 Application Applications in Near-Term Quantum Computing Goodrich, Sullivan Practical graph bipartization October 1, / 53

3 Example: Quantum annealing Goodrich, Sullivan Practical graph bipartization October 1, / 53

4 Example: Quantum annealing arg min ~ s X (i,j) E (G ) Jij si sj + X i V (G ) hi si for Jij, hi R and si { 1, +1} Goodrich, Sullivan Practical graph bipartization October 1, / 53

5 Example: Quantum annealing arg min ~ s X (i,j) E (G ) Jij si sj + X i V (G ) hi si for Jij, hi R and si { 1, +1} Goodrich, Sullivan Practical graph bipartization October 1, / 53

6 Example: Quantum annealing TRIAD Goodrich, Sullivan Practical graph bipartization October 1, / 53

7 Example: Quantum circuits Mapping a logical circuit to physical qubit registers. Goodrich, Sullivan Practical graph bipartization October 1, / 53

8 Example: Quantum circuits Mapping a logical circuit to physical qubit registers. Goodrich, Sullivan Practical graph bipartization October 1, / 53

9 Example: Quantum circuits Mapping a logical circuit to physical qubit registers. Goodrich, Sullivan Practical graph bipartization October 1, / 53

10 Example: Quantum circuits Mapping a logical circuit to physical qubit registers. Goodrich, Sullivan Practical graph bipartization October 1, / 53

11 Example: Quantum circuits Mapping a logical circuit to physical qubit registers. Goodrich, Sullivan Practical graph bipartization October 1, / 53

12 Example: Quantum circuits Mapping a logical circuit to physical qubit registers. Goodrich, Sullivan Practical graph bipartization October 1, / 53

13 Example: Quantum circuits Qubits will need to be swapped to different registers a routing problem. Goodrich, Sullivan Practical graph bipartization October 1, / 53

14 Common theme: Bipartite hardware Physical constraints typically require bipartite hardware topologies. Goodrich, Sullivan Practical graph bipartization October 1, / 53

15 Structure Graph Bipartization with Odd Cycle Transversals Goodrich, Sullivan Practical graph bipartization October 1, / 53

16 Odd cycle transversals Minimum odd cycle transversal vertex edit distance to bipartite structure. Goodrich, Sullivan Practical graph bipartization October 1, / 53

17 Odd cycle transversals Minimum odd cycle transversal vertex edit distance to bipartite structure. Goodrich, Sullivan Practical graph bipartization October 1, / 53

18 Odd cycle transversals Minimum odd cycle transversal vertex edit distance to bipartite structure. Goodrich, Sullivan Practical graph bipartization October 1, / 53

19 Odd cycle transversals k-oct Input: Parameter: Question: A graph G = (V, E) and a non-negative integer k k Is there a set S V of size at most k such that G \ S is bipartite? Goodrich, Sullivan Practical graph bipartization October 1, / 53

20 Odd cycle transversals k-oct Input: Parameter: Question: A graph G = (V, E) and a non-negative integer k k Is there a set S V of size at most k such that G \ S is bipartite? * Difficult in general... (NP-hard and MAX SNP-hard) *...but Fixed-Parameter Tractable * e.g. O(3 k kmn) run time algorithm for OCT size k Goodrich, Sullivan Practical graph bipartization October 1, / 53

21 Existing solvers Various exact solvers: * Hüffner s combinatorial MinOCT solver MinOCT Goodrich, Sullivan Practical graph bipartization October 1, / 53

22 Existing solvers Various exact solvers: * Hüffner s combinatorial MinOCT solver * Akiba-Iwata s vertex cover solver MinOCT MinVC Goodrich, Sullivan Practical graph bipartization October 1, / 53

23 Existing solvers Various exact solvers: * Hüffner s combinatorial MinOCT solver * Akiba-Iwata s vertex cover solver * IBM s CPLEX integer linear program solver MinOCT MinVC maximize c T x subject to Ax b x 0 and x Z n ILP Goodrich, Sullivan Practical graph bipartization October 1, / 53

24 Exact serial results Instance Hüffner Akiba-Iwata CPLEX Name V E S time (sec) time (sec) time (sec) bqpgka bqpgka bqpgka NA bqpgka NA NA bqpgka NA bqpgka NA bqpgka NA bqpgka Note: NA values did not terminate within 600 seconds. Akiba-Iwata and CPLEX both dominate Hüffner when minoct grows. Goodrich, Sullivan Practical graph bipartization October 1, / 53

25 Parallel results Instance 1-Core 12-Core 24-Core Name V E S time (sec) time (sec) time (sec) bqpgka bqpgka bqpgka bqpgka bqpgka bqpgka bqpgka bqpgka CPLEX sticks with serial reduction rules...usually. Goodrich, Sullivan Practical graph bipartization October 1, / 53

26 Heuristics DFS-based heuristic BFS-based heuristic IndSet-based heuristic Luby s-based heuristic Goodrich, Sullivan Practical graph bipartization October 1, / 53

27 Heuristic results Instance Heuristics Akiba-Iwata CPLEX Name V E S time (sec) error time (sec) time (sec) bqpgka % bqpgka % bqpgka % bqpgka % NA bqpgka % bqpgka % bqpgka % bqpgka % Note: NA values did not terminate within 600 seconds. Heuristics typically find a solution with less than 5% error. Goodrich, Sullivan Practical graph bipartization October 1, / 53

28 Optimization (Research) A Practical Anytime Algorithm Goodrich, Sullivan Practical graph bipartization October 1, / 53

29 Algorithm Requirements * Practical: Attention paid to heuristics that will not affect theoretical run time, but improve the practical performance. * Anytime: Suffices to find a quick estimation, but we might have extra compute cycles to put towards improvements. * Approximation: Upper and lower bounds are iteratively improved, providing an approximation ratio at each step. * Parallel: The quantum compiler may be running on a large node (or multiple nodes), can we utilize the hardware? Goodrich, Sullivan Practical graph bipartization October 1, / 53

30 Iterative compression Overall strategy: 1. Trivially find solution of size k + 1 on a subgraph of G. 2. Compress solution to size k or return NO. 3. Extend the subgraph and solution by one vertex. 4. Repeat until the subgraph is G. Example: Let k = 3. Goodrich, Sullivan Practical graph bipartization October 1, / 53

31 Iterative compression Overall strategy: 1. Trivially find solution of size k + 1 on a subgraph of G. 2. Compress solution to size k or return NO. 3. Extend the subgraph and solution by one vertex. 4. Repeat until the subgraph is G. Example: Let k = 3. Goodrich, Sullivan Practical graph bipartization October 1, / 53

32 Iterative compression Overall strategy: 1. Trivially find solution of size k + 1 on a subgraph of G. 2. Compress solution to size k or return NO. 3. Extend the subgraph and solution by one vertex. 4. Repeat until the subgraph is G. Example: Let k = 3. Goodrich, Sullivan Practical graph bipartization October 1, / 53

33 Iterative compression Overall strategy: 1. Trivially find solution of size k + 1 on a subgraph of G. 2. Compress solution to size k or return NO. 3. Extend the subgraph and solution by one vertex. 4. Repeat until the subgraph is G. Example: Let k = 3. Goodrich, Sullivan Practical graph bipartization October 1, / 53

34 Iterative compression Overall strategy: 1. Trivially find solution of size k + 1 on a subgraph of G. 2. Compress solution to size k or return NO. 3. Extend the subgraph and solution by one vertex. 4. Repeat until the subgraph is G. Example: Let k = 3. Goodrich, Sullivan Practical graph bipartization October 1, / 53

35 Iterative compression Overall strategy: 1. Trivially find solution of size k + 1 on a subgraph of G. 2. Compress solution to size k or return NO. 3. Extend the subgraph and solution by one vertex. 4. Repeat until the subgraph is G. Example: Let k = 3. Goodrich, Sullivan Practical graph bipartization October 1, / 53

36 Iterative compression Overall strategy: 1. Trivially find solution of size k + 1 on a subgraph of G. 2. Compress solution to size k or return NO. 3. Extend the subgraph and solution by one vertex. 4. Repeat until the subgraph is G. Example: Let k = 3. Goodrich, Sullivan Practical graph bipartization October 1, / 53

37 Iterative compression Overall strategy: 1. Trivially find solution of size k + 1 on a subgraph of G. 2. Compress solution to size k or return NO. 3. Extend the subgraph and solution by one vertex. 4. Repeat until the subgraph is G. Example: Let k = 3. Goodrich, Sullivan Practical graph bipartization October 1, / 53

38 Iterative compression Overall strategy: 1. Trivially find solution of size k + 1 on a subgraph of G. 2. Compress solution to size k or return NO. 3. Extend the subgraph and solution by one vertex. 4. Repeat until the subgraph is G. Example: Let k = 3. Goodrich, Sullivan Practical graph bipartization October 1, / 53

39 Iterative compression Overall strategy: 1. Trivially find solution of size k + 1 on a subgraph of G. 2. Compress solution to size k or return NO. 3. Extend the subgraph and solution by one vertex. 4. Repeat until the subgraph is G. Example: Let k = 3. Goodrich, Sullivan Practical graph bipartization October 1, / 53

40 Compression routine (Reed et al. and Hüffner) Obtain a new, disjoint solution S by subdividing edges incident to S. Goodrich, Sullivan Practical graph bipartization October 1, / 53

41 Compression routine (Reed et al. and Hüffner) Obtain a new, disjoint solution S by subdividing edges incident to S. Goodrich, Sullivan Practical graph bipartization October 1, / 53

42 Compression routine (Lokshtanov et al.) Try all 3 k+1 ways of reassigning the old OCT set. Goodrich, Sullivan Practical graph bipartization October 1, / 53

43 Compression routine (Lokshtanov et al.) v 1 v 2 v 3 v 4 L L L L L L L R L L R L L L R R L R L L.. S S S S Try all 3 k+1 ways of reassigning the old OCT set. Goodrich, Sullivan Practical graph bipartization October 1, / 53

44 Heuristic improvements We have identified several improvements: * Small cut reduction rules * LP-based reductions rules * Exact algorithms for small treewidth subgraphs * BFS-based branching rules * Triangle-based branching rules * Dynamic MinCut solver * Gray code partition alignment Goodrich, Sullivan Practical graph bipartization October 1, / 53

45 Heuristic improvements We have identified several improvements: * Small cut reduction rules * LP-based reductions rules * Exact algorithms for small treewidth subgraphs * BFS-based branching rules * Triangle-based branching rules Dynamic MinCut solver Gray code partition alignment Goodrich, Sullivan Practical graph bipartization October 1, / 53

46 Improvement: Dynamic minimum cut k+1 vertices {}}{ L L L L L L L R L L R L L L R R. S S S S 3 k+1 partitions Reuse most of the max-flow-min-cut network after the first partition. Goodrich, Sullivan Practical graph bipartization October 1, / 53

47 Improvement: Dynamic minimum cut k+1 vertices {}}{ L L L L L L L R L L R L L L R R. S S S S } } O(km) min-cut } O(km) min-cut } O(km) min-cut O(km) min-cut } O(km) min-cut Reuse most of the max-flow-min-cut network after the first partition. Goodrich, Sullivan Practical graph bipartization October 1, / 53

48 Improvement: Dynamic minimum cut k+1 vertices {}}{ L L L L L L L R L L R L L L R R. S S S S } } O(km) min-cut } O(m) min-cut } O(m) min-cut O(m) min-cut } O(m) min-cut Reuse most of the max-flow-min-cut network after the first partition. Goodrich, Sullivan Practical graph bipartization October 1, / 53

49 Improvement: Gray codes for better alignment k+1 vertices {}}{ L L L L L L L R L R L R L R L L. S S S S } } O(km) min-cut } O(m) min-cut } O(m) min-cut O(m) min-cut } O(m) min-cut Utilize gray codes to minimize changes between partition assignments. Goodrich, Sullivan Practical graph bipartization October 1, / 53

50 A note on parallelism k+1 vertices {}}{ L L L L. L R L R L R L L. L L L R. L L L R. S S S S Thread 1 Thread 2 Thread t Gray codes are still compatible with parallelism. Goodrich, Sullivan Practical graph bipartization October 1, / 53

51 Trade-off between parallelism and branching Current approach is fundamentally lower bounded by Ω(2 k ) * Even if some partitions are trivial, we generate them all * Can we smartly generate partitions? Idea: Branch to generate a (shorter) list of partitions * Removes the Ω(2 k ) lower bound * Can still distribute to get CPU parallelism * But we no longer get the nice overlaps between partitions Goodrich, Sullivan Practical graph bipartization October 1, / 53

52 Conclusion Takeaways * Near-term quantum hardware requires bipartizing software networks * CPLEX does well in a pinch * Promising results in heuristics + tuning guarantees Code to be open sourced Winter * Algorithms in Modern C++, experiments in Python scripts * Docker support * Hosted at Goodrich, Sullivan Practical graph bipartization October 1, / 53

53 Thank you for listening! Goodrich, Sullivan Practical graph bipartization October 1, / 53