abstract cores in implicit hitting set maxsat solving
play

Abstract Cores in Implicit Hitting Set MaxSat Solving Jeremias Berg - PowerPoint PPT Presentation

Feb 21, 2024 •309 likes •1.46k views

Abstract Cores in Implicit Hitting Set MaxSat Solving Jeremias Berg 1 Fahiem Bacchus 2 Alex Poole 2 1 HIIT, Dept. Computer Science, University of Helsinki, Finland 2 University of Toronto, Department of Computer Science, Canada SAT 2020 Online

  1. Preliminaries: MaxSat Optimisation extension of Boolean Satisfiability (SAT) An instance consists of: ◮ a set of hard clauses and τ = {¬ b 1 , b 2 , ¬ b 3 } ◮ a set of soft clauses. ◮ a weight function wt over the soft clauses. F H = { ( b 1 ∨ b 2 ) , ( b 2 ∨ b 3 ) } F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) } Find τ that: ◮ satisfies all hard clauses and cost ( τ ) = 1 ◮ maximises the number of satisfied soft clauses. F B = { b 1 , b 2 , b 3 } Assume w.l.o.g. that all soft clauses are unit negative literals ◮ Blocking variables: the variables in soft clauses. ◮ F B the set of blocking variables J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 6 / 20

  2. Preliminaries: UNSAT Cores Central in modern MaxSAT solving: F H = { ( b 1 ∨ b 2 ) , ( b 2 ∨ b 3 ) } κ ⊂ F S is an core if F H ∧ κ is F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) } unsatisfiable F B = { b 1 , b 2 , b 3 } κ can be represented as a clause over F B that is entailed by F H or as a linear inequality J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 7 / 20

  3. Preliminaries: UNSAT Cores Central in modern MaxSAT solving: F H = { ( b 1 ∨ b 2 ) , ( b 2 ∨ b 3 ) } κ ⊂ F S is an core if F H ∧ κ is F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) } unsatisfiable F B = { b 1 , b 2 , b 3 } κ can be represented as a clause over F B that is entailed by F H κ = { ( ¬ b 1 ) , ( ¬ b 2 ) } or as a linear inequality J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 7 / 20

  4. Preliminaries: UNSAT Cores Central in modern MaxSAT solving: F H = { ( b 1 ∨ b 2 ) , ( b 2 ∨ b 3 ) } κ ⊂ F S is an core if F H ∧ κ is F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) } unsatisfiable F B = { b 1 , b 2 , b 3 } κ can be represented as a clause over F B that is entailed by F H κ = { ( ¬ b 1 ) , ( ¬ b 2 ) } or as a linear inequality κ = ( b 1 ∨ b 2 ) J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 7 / 20

  5. Preliminaries: UNSAT Cores Central in modern MaxSAT solving: F H = { ( b 1 ∨ b 2 ) , ( b 2 ∨ b 3 ) } κ ⊂ F S is an core if F H ∧ κ is F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) } unsatisfiable F B = { b 1 , b 2 , b 3 } κ can be represented as a clause over F B that is entailed by F H κ = { ( ¬ b 1 ) , ( ¬ b 2 ) } or as a linear inequality κ = ( b 1 ∨ b 2 ) κ = b 1 + b 2 ≥ 1 J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 7 / 20

  6. Preliminaries: UNSAT Cores Central in modern MaxSAT solving: F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) } κ ⊂ F S is an core if F H ∧ κ is F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) } unsatisfiable F B = { b 1 , b 2 , b 3 } κ can be represented as a clause over F B that is entailed by F H κ = { ( ¬ b 1 ) , ( ¬ b 2 ) } or as a linear inequality κ = ( b 1 , b 2 ) κ = b 1 + b 2 ≥ 1 In the rest of the presentation, we represent clauses ( b 1 ∨ b 2 ) as ( b 1 , b 2 ) J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 7 / 20

  7. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } F B = { b 1 , b 2 , b 3 , b 4 } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  8. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } UB = ∞ LB = 0 CORES = ∅ BESTSOL = ∅ J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  9. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB UB = ∞ LB = 0 CORES = ∅ BESTSOL = ∅ J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  10. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB Compute min-cost hitting set hs hs = Min-Hs ( F B , ∅ ) Min-Hs ( F B , CORES ) : UB = ∞ minimize: � b ∈F B b LB = 0 subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES return: { b | b set to 1 in opt. soln } CORES = ∅ BESTSOL = ∅ J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  11. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB Compute min-cost hitting set hs hs = Min-Hs ( F B , ∅ ) Min-Hs ( F B , CORES ) : UB = ∞ minimize: � b ∈F B wt ( b ) b Weighted Case LB = 0 subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES return: { b | b set to 1 in opt. soln } CORES = ∅ BESTSOL = ∅ J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  12. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB Compute min-cost hitting set hs Update LB hs = ∅ Min-Hs ( F B , CORES ) : UB = ∞ minimize: � b ∈F B b LB = |∅| subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES return: { b | b set to 1 in opt. soln } CORES = ∅ BESTSOL = ∅ J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  13. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB Compute min-cost hitting set hs Update LB hs = ∅ Set up assumptions A = F B \ hs Min-Hs ( F B , CORES ) : UB = ∞ minimize: � b ∈F B b LB = 0 subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES return: { b | b set to 1 in opt. soln } CORES = ∅ BESTSOL = ∅ J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  14. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB Compute min-cost hitting set hs Update LB sat-assume ( F H , ¬A ) Set up assumptions Extract cores until SAT A = { b 1 , b 2 , b 3 , b 4 } K = {} Min-Hs ( F B , CORES ) : UB = ∞ minimize: � b ∈F B b LB = 0 subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES return: { b | b set to 1 in opt. soln } CORES = ∅ BESTSOL = ∅ J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  15. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB Compute min-cost hitting set hs Update LB sat-assume ( F H , ¬A ) Set up assumptions � � Extract cores until SAT A = { � b 2 , b 3 , b 4 } b 1 , � K = { ( b 1 , b 2 ) } Min-Hs ( F B , CORES ) : UB = ∞ minimize: � b ∈F B b LB = 0 subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES return: { b | b set to 1 in opt. soln } CORES = ∅ BESTSOL = ∅ J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  16. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB Compute min-cost hitting set hs Update LB sat-assume ( F H , ¬A ) Set up assumptions Extract cores until SAT A = { b 3 , b 4 } K = { ( b 1 , b 2 ) } Min-Hs ( F B , CORES ) : UB = ∞ minimize: � b ∈F B b LB = 0 subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES return: { b | b set to 1 in opt. soln } CORES = ∅ BESTSOL = ∅ J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  17. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB Compute min-cost hitting set hs Update LB sat-assume ( F H , ¬A ) Set up assumptions � � Extract cores until SAT A = { � b 4 } b 3 , � K = { ( b 1 , b 2 ) , ( b 3 , b 4 ) } Min-Hs ( F B , CORES ) : UB = ∞ minimize: � b ∈F B b LB = 0 subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES return: { b | b set to 1 in opt. soln } CORES = ∅ BESTSOL = ∅ J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  18. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB Compute min-cost hitting set hs Update LB sat-assume ( F H , ¬A ) Set up assumptions Extract cores until SAT A = {} K = { ( b 1 , b 2 ) , ( b 3 , b 4 ) } Min-Hs ( F B , CORES ) : UB = ∞ minimize: � b ∈F B b LB = 0 subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES return: { b | b set to 1 in opt. soln } CORES = ∅ BESTSOL = ∅ J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  19. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB Compute min-cost hitting set hs Update LB sat-assume ( F H , ¬A ) Set up assumptions Extract cores until SAT A = {} K = { ( b 1 , b 2 ) , ( b 3 , b 4 ) } τ = {¬ b 1 , b 2 , ¬ b 3 , b 4 } Min-Hs ( F B , CORES ) : UB = ∞ minimize: � b ∈F B b LB = 0 subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES return: { b | b set to 1 in opt. soln } CORES = ∅ BESTSOL = ∅ J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  20. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB Compute min-cost hitting set hs Update LB Set up assumptions Extract cores until SAT Update UB K = { ( b 1 , b 2 ) , ( b 3 , b 4 ) } τ = {¬ b 1 , b 2 , ¬ b 3 , b 4 } Min-Hs ( F B , CORES ) : UB = cost ( τ ) minimize: � b ∈F B b LB = 0 subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES return: { b | b set to 1 in opt. soln } CORES = ∅ BESTSOL = {¬ b 1 , b 2 , ¬ b 3 , b 4 } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  21. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB Compute min-cost hitting set hs Update LB Set up assumptions Extract cores until SAT Update UB K = { ( b 1 , b 2 ) , ( b 3 , b 4 ) } Add cores to CORES τ = {¬ b 1 , b 2 , ¬ b 3 , b 4 } Min-Hs ( F B , CORES ) : UB = 2 minimize: � b ∈F B b LB = 0 subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES return: { b | b set to 1 in opt. soln } CORES = { ( b 1 , b 2 ) , ( b 3 , b 4 ) } BESTSOL = {¬ b 1 , b 2 , ¬ b 3 , b 4 } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  22. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB Compute min-cost hitting set hs Update LB Set up assumptions Extract cores until SAT Update UB Add cores to CORES UB = 2 LB = 0 CORES = { ( b 1 , b 2 ) , ( b 3 , b 4 ) } BESTSOL = {¬ b 1 , b 2 , ¬ b 3 , b 4 } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  23. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB Compute min-cost hitting set hs Update LB hs = Min-Hs ( F B , { ( b 1 , b 2 ) , ( b 3 , b 4 ) } ) Set up assumptions Extract cores until SAT Update UB Add cores to CORES Min-Hs ( F B , CORES ) : UB = 2 minimize: � b ∈F B b LB = 0 subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES return: { b | b set to 1 in opt. soln } CORES = { ( b 1 , b 2 ) , ( b 3 , b 4 ) } BESTSOL = {¬ b 1 , b 2 , ¬ b 3 , b 4 } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  24. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB Compute min-cost hitting set hs Update LB hs = { b 1 , b 4 } Set up assumptions Extract cores until SAT Update UB Add cores to CORES UB = 2 LB = |{ b 1 , b 4 }| CORES = { ( b 1 , b 2 ) , ( b 3 , b 4 ) } BESTSOL = {¬ b 1 , b 2 , ¬ b 3 , b 4 } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  25. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB Compute min-cost hitting set hs Update LB hs = { b 1 , b 4 } Set up assumptions Extract cores until SAT Update UB Add cores to CORES return BESTSOL UB = 2 LB = 2 CORES = { ( b 1 , b 2 ) , ( b 3 , b 4 ) } BESTSOL = {¬ b 1 , b 2 , ¬ b 3 , b 4 } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  26. Solving (unweighted) MaxSat with IHS F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Basic-IHS ( F ) F S = { ( ¬ b 1 ) , ( ¬ b 2 ) , ( ¬ b 3 ) , ( ¬ b 4 ) } Initialize F B = { b 1 , b 2 , b 3 , b 4 } while LB < UB Compute min-cost hitting set hs Update LB hs = { b 1 , b 4 } Set up assumptions Extract cores until SAT Update UB Add cores to CORES return BESTSOL LB need to be increased UB = 2 to optimum before termination LB = 2 CORES = { ( b 1 , b 2 ) , ( b 3 , b 4 ) } BESTSOL = {¬ b 1 , b 2 , ¬ b 3 , b 4 } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 8 / 20

  27. Abstract Cores J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 9 / 20

  28. Drawbacks of IHS Davies [2013] Main motivation for our work: There exists MaxSAT instances on which IHS needs an exponential number of cores. J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 10 / 20

  29. Drawbacks of IHS Davies [2013] Main motivation for our work: There exists MaxSAT instances on which IHS needs an exponential number of cores. n � � � F H = CNF ( b i ≥ r ) i = 1 F S = { ( ¬ b 1 ) , . . . , ( ¬ b n ) } F B = { b 1 , . . . , b n } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 10 / 20

  30. Drawbacks of IHS Davies [2013] Main motivation for our work: There exists MaxSAT instances on which IHS needs an exponential number of cores. n = 8 , r = 4 n � � κ 1 = ( b i 1 , b i 2 , b i 3 , b i 4 , b i 5 ) � F H = CNF ( b i ≥ r ) is a core for any i 1 , i 2 , i 3 , i 4 , i 5 i = 1 F S = { ( ¬ b 1 ) , . . . , ( ¬ b n ) } F B = { b 1 , . . . , b n } Intuition: Any κ ⊂ F S s.t | κ | = ( n − r ) + 1 is a core. J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 10 / 20

  31. Drawbacks of IHS Davies [2013] Main motivation for our work: There exists MaxSAT instances on which IHS needs an exponential number of cores. n = 8 , r = 4 n � � κ 1 = ( b i 1 , b i 2 , b i 3 , b i 4 , b i 5 ) � F H = CNF ( b i ≥ r ) is a core for any i 1 , i 2 , i 3 , i 4 , i 5 i = 1 F S = { ( ¬ b 1 ) , . . . , ( ¬ b n ) } κ 2 = ( b i 1 , b i 2 , b i 3 , b i 4 ) is not a core for any i 1 , i 2 , i 3 , i 4 F B = { b 1 , . . . , b n } Intuition: Any κ ⊂ F S s.t | κ | = ( n − r ) + 1 is a core. J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 10 / 20

  32. Drawbacks of IHS Davies [2013] Main motivation for our work: There exists MaxSAT instances on which IHS needs an exponential number of cores. n = 8 , r = 4 n � � κ 1 = ( b i 1 , b i 2 , b i 3 , b i 4 , b i 5 ) � F H = CNF ( b i ≥ r ) is a core for any i 1 , i 2 , i 3 , i 4 , i 5 i = 1 F S = { ( ¬ b 1 ) , . . . , ( ¬ b n ) } κ 2 = ( b i 1 , b i 2 , b i 3 , b i 4 ) is not a core for any i 1 , i 2 , i 3 , i 4 F B = { b 1 , . . . , b n } Intuition: Any κ ⊂ F S s.t | κ | = ( n − r ) + 1 is a core. n � � IHS needs to extract all ( n − r )+ 1 of them. J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 10 / 20

  33. Drawbacks of IHS Davies [2013] Main motivation for our work: There exists MaxSAT instances on which IHS needs an exponential number of cores. n = 8 , r = 4 n � � κ 1 = ( b i 1 , b i 2 , b i 3 , b i 4 , b i 5 ) � F H = CNF ( b i ≥ r ) is a core for any i 1 , i 2 , i 3 , i 4 , i 5 i = 1 F S = { ( ¬ b 1 ) , . . . , ( ¬ b n ) } κ 2 = ( b i 1 , b i 2 , b i 3 , b i 4 ) is not a core for any i 1 , i 2 , i 3 , i 4 F B = { b 1 , . . . , b n } Intuition: Blocking variables are exchangeable : Any κ ⊂ F S s.t | κ | = ( n − r ) + 1 is a core. cores are defined by the number of them, n � � IHS needs to extract all ( n − r )+ 1 not the identity of them of them. J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 10 / 20

  34. Our Focus Research Question(s) Does there exists a compact representation of large sets of cores that IHS can reason over? J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 11 / 20

  35. Abstract cores Idea: What happens if we introduce literals that count the number of blocking variables set to true? (similar to variables that have been successfully used in core-guided solvers) J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 12 / 20

  36. Abstract cores Idea: What happens if we introduce literals that count the number of blocking variables set to true? (similar to variables that have been successfully used in core-guided solvers) AB = { b 1 , . . . , b 5 } ⊂ F B n � � � F H = b i ≥ r ) CNF ( i = 1 F S = { ( ¬ b 1 ) , . . . , ( ¬ b n ) } F B = { b 1 , . . . , b n } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 12 / 20

  37. Abstract cores Idea: What happens if we introduce literals that count the number of blocking variables set to true? (similar to variables that have been successfully used in core-guided solvers) AB = { b 1 , . . . , b 5 } ⊂ F B n � � Define s AB [ i ] � F H = b i ≥ r ) CNF ( i = 1 F S = { ( ¬ b 1 ) , . . . , ( ¬ b n ) } F B = { b 1 , . . . , b n } s AB [ i ] ↔ �� � b ∈ AB b ≥ i J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 12 / 20

  38. Abstract cores Idea: What happens if we introduce literals that count the number of blocking variables set to true? (similar to variables that have been successfully used in core-guided solvers) AB = { b 1 , . . . , b 5 } ⊂ F B n � � Define s AB [ i ] � F H = b i ≥ r ) CNF ( i = 1 F S = { ( ¬ b 1 ) , . . . , ( ¬ b n ) } F B = { b 1 , . . . , b n } s AB [ i ] ↔ �� � b ∈ AB b ≥ i Note : Can be encoded as CNF J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 12 / 20

  39. Abstract cores Idea: What happens if we introduce literals that count the number of blocking variables set to true? (similar to variables that have been successfully used in core-guided solvers) AB = { b 1 , . . . , b 5 } ⊂ F B n � � Define s AB [ i ] � F H = b i ≥ r ) CNF ( ( b 7 , s AB [ 3 ] , b n ) Consider: i = 1 F S = { ( ¬ b 1 ) , . . . , ( ¬ b n ) } F B = { b 1 , . . . , b n } s AB [ i ] ↔ �� � b ∈ AB b ≥ i J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 12 / 20

  40. Abstract cores Idea: What happens if we introduce literals that count the number of blocking variables set to true? (similar to variables that have been successfully used in core-guided solvers) AB = { b 1 , . . . , b 5 } ⊂ F B n � � Define s AB [ i ] � F H = b i ≥ r ) CNF ( ( b 7 , s AB [ 3 ] , b n ) Consider: i = 1 F S = { ( ¬ b 1 ) , . . . , ( ¬ b n ) } F B = { b 1 , . . . , b n } ( b 7 , b 1 , b 2 , b 3 , b n ) s AB [ i ] ↔ �� � b ∈ AB b ≥ i J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 12 / 20

  41. Abstract cores Idea: What happens if we introduce literals that count the number of blocking variables set to true? (similar to variables that have been successfully used in core-guided solvers) AB = { b 1 , . . . , b 5 } ⊂ F B n � � Define s AB [ i ] � F H = b i ≥ r ) CNF ( ( b 7 , s AB [ 3 ] , b n ) Consider: i = 1 F S = { ( ¬ b 1 ) , . . . , ( ¬ b n ) } F B = { b 1 , . . . , b n } ( b 7 , b 1 , b 2 , b 3 , b n ) s AB [ i ] ↔ �� � b ∈ AB b ≥ i ( b 7 , b 3 , b 4 , b 2 , b n ) J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 12 / 20

  42. Abstract cores Idea: What happens if we introduce literals that count the number of blocking variables set to true? (similar to variables that have been successfully used in core-guided solvers) AB = { b 1 , . . . , b 5 } ⊂ F B n � � Define s AB [ i ] � F H = b i ≥ r ) CNF ( ( b 7 , s AB [ 3 ] , b n ) Consider: i = 1 F S = { ( ¬ b 1 ) , . . . , ( ¬ b n ) } F B = { b 1 , . . . , b n } ( b 7 , b 1 , b 2 , b 3 , b n ) ( b 7 , b 4 , b 2 , b 1 , b n ) s AB [ i ] ↔ �� � b ∈ AB b ≥ i ( b 7 , b 3 , b 4 , b 2 , b n ) J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 12 / 20

  43. Abstract cores Idea: What happens if we introduce literals that count the number of blocking variables set to true? (similar to variables that have been successfully used in core-guided solvers) AB = { b 1 , . . . , b 5 } ⊂ F B n � � Define s AB [ i ] � F H = b i ≥ r ) CNF ( ( b 7 , s AB [ 3 ] , b n ) Consider: i = 1 F S = { ( ¬ b 1 ) , . . . , ( ¬ b n ) } F B = { b 1 , . . . , b n } ( b 7 , b 1 , b 2 , b 3 , b n ) ( b 7 , b 4 , b 2 , b 1 , b n ) s AB [ i ] ↔ �� � b ∈ AB b ≥ i ( b 7 , b 3 , b 4 , b 2 , b n ) J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 12 / 20

  44. Abstract cores Idea: What happens if we introduce literals that count the number of blocking variables set to true? (similar to variables that have been successfully used in core-guided solvers) AB = { b 1 , . . . , b 5 } ⊂ F B Terminology : AB is an abstraction set n � � Define s AB [ i ] � F H = b i ≥ r ) CNF ( ( b 7 , s AB [ 3 ] , b n ) s AB [ i ] is an abstraction variable Consider: i = 1 F S = { ( ¬ b 1 ) , . . . , ( ¬ b n ) } The definition of s AB [ i ] is F B = { b 1 , . . . , b n } s AB [ i ] ↔ �� � b ∈ AB b ≥ i J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 12 / 20

  45. Abstract cores Idea: What happens if we introduce literals that count the number of blocking variables set to true? (similar to variables that have been successfully used in core-guided solvers) AB = { b 1 , . . . , b 5 } ⊂ F B Terminology : AB is an abstraction set n � � Define s AB [ i ] � F H = b i ≥ r ) CNF ( ( b 7 , s AB [ 3 ] , b n ) s AB [ i ] is an abstraction variable Consider: i = 1 F S = { ( ¬ b 1 ) , . . . , ( ¬ b n ) } The definition of s AB [ i ] is F B = { b 1 , . . . , b n } s AB [ i ] ↔ �� � b ∈ AB b ≥ i Abstract Core: a clause over abstraction and blocking variables that is entailed by F H and the definitions of abstraction variables J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 12 / 20

  46. Abstract cores Idea: What happens if we introduce literals that count the number of blocking variables set to true? (similar to variables that have been successfully used in core-guided solvers) AB = { b 1 , . . . , b 5 } ⊂ F B Terminology : AB is an abstraction set n � � Define s AB [ i ] � F H = b i ≥ r ) CNF ( s AB [ i ] is an abstraction variable i = 1 F S = { ( ¬ b 1 ) , . . . , ( ¬ b n ) } Could use other definitions The definition of s AB [ i ] is Summations successfull F B = { b 1 , . . . , b n } s AB [ i ] ↔ �� � b ∈ AB b ≥ i in core-guided solvers. Abstract Core: a clause over abstraction and blocking variables that is entailed by F H and the definitions of abstraction variables J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 12 / 20

  47. Abstract cores are expressive Proposition An abstract core containing the abstraction variables { s AB 1 [ j 1 ] , . . . , s AB k [ j k ] } is equivalent to the conjunction of k | AB i | � � � | AB i | − j i + 1 i = 1 regular cores. J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 13 / 20

  48. Abstract cores are expressive Proposition An abstract core containing the abstraction variables { s AB 1 [ j 1 ] , . . . , s AB k [ j k ] } is equivalent to the conjunction of k | AB i | � � � | AB i | − j i + 1 i = 1 regular cores. Two Questions remain: How to compute abstraction sets? 1 How to extract and reason over abstract cores in IHS? 2 J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 13 / 20

  49. Computing Abstraction Sets Ideally Identify a set S ⊂ F B of exchangeable blocking variables. J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 14 / 20

  50. Computing Abstraction Sets LB: 0 Ideally Identify a set S ⊂ F B of exchangeable blocking variables. b 4 b 5 In practice Form abstraction sets over blocking variables that appear b 1 b 3 frequently in cores together. b 2 J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 14 / 20

  51. Computing Abstraction Sets Core: ( b 1 , b 2 ) LB: 0 Ideally Identify a set S ⊂ F B of exchangeable blocking variables. b 4 b 5 In practice Form abstraction sets over blocking variables that appear b 1 b 3 frequently in cores together. 1 b 2 J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 14 / 20

  52. Computing Abstraction Sets Core: ( b 2 , b 4 ) LB: 0 Ideally Identify a set S ⊂ F B of exchangeable blocking variables. b 4 b 5 In practice Form abstraction sets over blocking variables that appear b 1 b 3 1 frequently in cores together. 1 b 2 J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 14 / 20

  53. Computing Abstraction Sets Core: ( b 3 , b 5 ) LB: 0 Ideally Identify a set S ⊂ F B of exchangeable blocking variables. b 4 b 5 In practice 1 Form abstraction sets over blocking variables that appear b 1 b 3 1 frequently in cores together. 1 b 2 J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 14 / 20

  54. Computing Abstraction Sets Core: ( b 1 , b 2 , b 4 ) LB: 0 Ideally Identify a set S ⊂ F B of exchangeable blocking variables. b 4 b 5 In practice 1 1 Form abstraction sets over blocking variables that appear b 1 b 3 2 frequently in cores together. 2 b 2 J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 14 / 20

  55. Computing Abstraction Sets Clustering LB: 0 Ideally Identify a set S ⊂ F B of exchangeable blocking variables. b 4 b 5 In practice 1 1 Form abstraction sets over blocking variables that appear b 1 b 3 2 frequently in cores together. 2 Recall: IHS needs to increase b 2 LB to optimum J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 14 / 20

  56. Computing Abstraction Sets Ideally Identify a set S ⊂ F B of exchangeable blocking variables. b 4 b 5 In practice 1 1 Form abstraction sets over In weighted case blocking variables that appear edges only between b 1 b 3 2 frequently in cores together. blocking variables with the same weight. 2 Recall: IHS needs to increase b 2 LB to optimum J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 14 / 20

  57. IHS with abstract core reasoning F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Abstract-IHS ( F ) F B = { b 1 , b 2 , b 3 , b 4 } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  58. IHS with abstract core reasoning F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Abstract-IHS ( F ) Initialize F B = { b 1 , b 2 , b 3 , b 4 } UB = ∞ LB = 0 AB = ∅ CORES = ∅ BESTSOL = ∅ J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  59. IHS with abstract core reasoning F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) } Abstract-IHS ( F ) Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } UB = ∞ LB = 0 AB = ∅ CORES = ∅ BESTSOL = ∅ J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  60. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB UB = ∞ LB = 0 AB = { AB = { b 2 , b 3 }} CORES = ∅ BESTSOL = ∅ J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  61. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs hs = Min-Abs ( F B , ∅ , AB ) b ∈ AB b − k · s AB [ k ] ≥ 0 � Min-Abs ( F B , CORES , AB ) : b ∈ AB b − | AB | · s AB [ k ] < k � minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  62. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs hs = ∅ Update LB LB = |∅| UB = ∞ Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = ∅ ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = ∅ return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  63. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs hs = ∅ Update LB Set up assumptions A = ABSTRACT ( F B , hs , AB ) = { b 1 , s AB [ 1 ] , b 4 } AB = { b 2 , b 3 } ABSTRACT ( F B , hs , AB ) A ← { b | b ∈ F B − hs } foreach AB ∈ AB do A ← A − { b | b ∈ AB } A ← A ∪ { s AB [ | AB ∩ hs | + 1 ] } return A J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  64. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs sat-assume ( F H , ¬A ) Update LB A = { b 1 , s AB [ 1 ] , b 4 } Set up assumptions Extract cores until SAT K = {} UB = ∞ LB = 0 Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = ∅ ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = ∅ return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  65. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs sat-assume ( F H , ¬A ) Update LB ✟ A = { b 1 , ✟✟✟ Set up assumptions s AB [ 1 ] , b 4 } Extract cores until SAT K = { ( s AB [ 1 ]) } UB = ∞ LB = 0 Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = ∅ ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = ∅ return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  66. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs sat-assume ( F H , ¬A ) Update LB Set up assumptions A = { b 1 , b 4 } Extract cores until SAT K = { ( s AB [ 1 ]) } UB = ∞ LB = 0 Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = ∅ ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = ∅ return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  67. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs sat-assume ( F H , ¬A ) Update LB Set up assumptions A = { b 1 , b 4 } Extract cores until SAT K = { ( s AB [ 1 ]) } τ = {¬ b 1 , b 2 , b 3 , ¬ b 4 } UB = ∞ LB = 0 Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = ∅ ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = ∅ return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  68. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs Update LB Set up assumptions Extract cores until SAT K = { ( s AB [ 1 ]) } Update UB τ = {¬ b 1 , b 2 , b 3 , ¬ b 4 } UB = cost ( τ ) LB = 0 Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = ∅ ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = {¬ b 1 , b 2 , b 3 , ¬ b 4 } return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  69. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs Update LB Set up assumptions Extract cores until SAT K = { ( s AB [ 1 ]) } Update UB τ = {¬ b 1 , b 2 , b 3 , ¬ b 4 } Add cores to CORES UB = 2 LB = 0 Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = { ( s AB [ 1 ]) } ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = {¬ b 1 , b 2 , b 3 , ¬ b 4 } return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  70. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs Update LB Set up assumptions Extract cores until SAT Update UB Add cores to CORES UB = 2 LB = 0 Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = { ( s AB [ 1 ]) } ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = {¬ b 1 , b 2 , b 3 , ¬ b 4 } return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  71. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs Update LB Set up assumptions Extract cores until SAT Update UB Add cores to CORES UB = 2 LB = 0 Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = { ( s AB [ 1 ]) } ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = {¬ b 1 , b 2 , b 3 , ¬ b 4 } return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  72. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs hs = Min-Abs ( F B , { ( s AB [ 1 ]) } , AB ) Update LB Set up assumptions Extract cores until SAT Update UB Add cores to CORES UB = 2 LB = 0 Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = { ( s AB [ 1 ]) } ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = {¬ b 1 , b 2 , b 3 , ¬ b 4 } return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  73. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs hs = { b 2 } Update LB Set up assumptions Extract cores until SAT Update UB Add cores to CORES LB = |{ b 2 }| UB = 2 Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = { ( s AB [ 1 ]) } ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = {¬ b 1 , b 2 , b 3 , ¬ b 4 } return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  74. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs hs = { b 2 } Update LB A = { b 1 , s AB [ 2 ] , b 4 } Set up assumptions Extract cores until SAT Update UB Add cores to CORES ABSTRACT ( F B , hs , AB ) A ← { b | b ∈ F B − hs } foreach AB ∈ AB do A ← A − { b | b ∈ AB } A ← A ∪ { s AB [ | AB ∩ hs | + 1 ] } return A J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  75. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs sat-assume ( F H , ¬A ) Update LB A = { b 1 , s AB [ 2 ] , b 4 } Set up assumptions Extract cores until SAT K = {} Update UB Add cores to CORES UB = 2 LB = 1 Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = { ( s AB [ 1 ]) } ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = {¬ b 1 , b 2 , b 3 , ¬ b 4 } return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  76. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs sat-assume ( F H , ¬A ) Update LB ✟ b 1 , ✟✟✟ Set up assumptions � s AB [ 2 ] , � � A = { � b 4 } Extract cores until SAT K = { ( b 1 , s AB [ 2 ] , b 4 ) } Update UB Add cores to CORES UB = 2 LB = 1 Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = { ( s AB [ 1 ]) } ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = {¬ b 1 , b 2 , b 3 , ¬ b 4 } return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  77. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs Update LB Set up assumptions Extract cores until SAT K = { ( b 1 , s AB [ 2 ] , b 4 ) } Update UB τ = {¬ b 1 , b 2 , b 3 , ¬ b 4 } Add cores to CORES UB = 2 LB = 1 Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = { ( s AB [ 1 ]) } ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = {¬ b 1 , b 2 , b 3 , ¬ b 4 } return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  78. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs Update LB Set up assumptions Extract cores until SAT K = { ( b 1 , s AB [ 2 ] , b 4 ) } Update UB τ = {¬ b 1 , b 2 , b 3 , ¬ b 4 } Add cores to CORES UB = 2 LB = 1 Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = { ( s AB [ 1 ]) , ( b 1 , s AB [ 2 ] , b 4 ) } ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = {¬ b 1 , b 2 , b 3 , ¬ b 4 } return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  79. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs Update LB Set up assumptions Extract cores until SAT Update UB Add cores to CORES UB = 2 LB = 1 Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = { ( s AB [ 1 ]) , ( b 1 , s AB [ 2 ] , b 4 ) } ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = {¬ b 1 , b 2 , b 3 , ¬ b 4 } return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  80. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs hs = Min-Abs ( F B , CORES , AB ) Update LB Set up assumptions Extract cores until SAT Update UB Add cores to CORES UB = 2 LB = 1 Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = { ( s AB [ 1 ]) , ( b 1 , s AB [ 2 ] , b 4 ) } ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = {¬ b 1 , b 2 , b 3 , ¬ b 4 } return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

  81. IHS with abstract core reasoning Abstract-IHS ( F ) F H = { ( b 1 , b 2 ) , ( b 2 , b 3 ) , ( b 3 , b 4 ) , � 2 � ( b 2 + b 3 ≥ i ) → s AB [ i ] � i = 1 CNF } Initialize while LB < UB F B = { b 1 , b 2 , b 3 , b 4 } Update AB Compute min-cost hitting set hs hs = { b 2 , b 3 } Update LB Set up assumptions Extract cores until SAT Update UB Add cores to CORES LB = |{ b 2 , b 3 }| UB = 2 Min-Abs ( F B , CORES , AB ) : AB = { AB = { b 2 , b 3 }} minimize: � b ∈F B b subject to: � b ∈ κ b ≥ 1 ∀ κ ∈ CORES CORES = { ( s AB [ 1 ]) , ( b 1 , s AB [ 2 ] , b 4 ) } ↔ s AB [ k ] ∀ AB ∈ AB �� b ∈ AB b ≥ k � BESTSOL = {¬ b 1 , b 2 , b 3 , ¬ b 4 } return: { b | b set to 1 in opt. soln } J. Berg (HIIT, U. Helsinki) Abstract Cores in IHS July 2020 15 / 20

Recommend

Community-based Partitioning for MaxSAT Solving Ruben Martins Vasco Manquinho In es Lynce

Community-based Partitioning for MaxSAT Solving Ruben Martins Vasco Manquinho In es Lynce

Community-based Partitioning for MaxSAT Solving Ruben Martins Vasco Manquinho In es Lynce IST/INESC-ID, Technical University of Lisbon, Portugal July 10, 2013 What is Maximum Satisfiability? CNF Formula: x 2 x 2 x 1 x 3 x

730 views • 43 slides
SAT by MaxSAT From NP to Beyond NP and Back Again Joao Marques-Silva (joint work with A.

SAT by MaxSAT From NP to Beyond NP and Back Again Joao Marques-Silva (joint work with A.

SAT by MaxSAT From NP to Beyond NP and Back Again Joao Marques-Silva (joint work with A. Ignatiev and A. Morgado) University of Lisbon, Portugal RTiKC Dagstuhl Seminar Schloss Dagstuhl, Germany September 2017 1 / 33 Why SAT by MaxSAT?

1.44k views • 97 slides
MaxSAT Resolution and Subcube Sums Yuval Filmus Meena Mahajan Gaurav Sood Marc Vinyals SAT

MaxSAT Resolution and Subcube Sums Yuval Filmus Meena Mahajan Gaurav Sood Marc Vinyals SAT

MaxSAT Resolution and Subcube Sums Yuval Filmus Meena Mahajan Gaurav Sood Marc Vinyals SAT 2020 The MaxSAT problem Input: F , a CNF formula ( n variables, m clauses). Output: MaxSAT ( F ) = x { 0 , 1 } n { number of clauses

413 views • 25 slides
Challenges in managing implicit and abstract provenance data: experiences with ProvManager

Challenges in managing implicit and abstract provenance data: experiences with ProvManager

Workshop on the Theory and Practice of Provenance 2011 Challenges in managing implicit and abstract provenance data: experiences with ProvManager Anderson Marinho, Marta Mattoso, Cludia Werner, Vanessa Braganholo, Leonardo Murta

785 views • 12 slides
Inflow-Implicit/Outflow-Explicit Finite Volume Methods for Solving Advection Equations Karol

Inflow-Implicit/Outflow-Explicit Finite Volume Methods for Solving Advection Equations Karol

Inflow-Implicit/Outflow-Explicit Finite Volume Methods for Solving Advection Equations Karol Mikula Department of Mathematics Slovak University of Technology, Bratislava, Slovakia http://www.math.sk/mikula Joint work with Mario Ohlberger,

425 views • 30 slides
Implicit Bias Implicit bias Implicit bias refers to attitudes or stereotypes that affect our

Implicit Bias Implicit bias Implicit bias refers to attitudes or stereotypes that affect our

Implicit Bias Implicit bias Implicit bias refers to attitudes or stereotypes that affect our understanding, actions and decisions in an unconscious manner. It's different from suppressed thoughts we might conceal to keep the peace; it's

634 views • 16 slides
Implicit Surfaces Implicit Surfaces An implicit surface is simply an iso-contour CIS 781 of a

Implicit Surfaces Implicit Surfaces An implicit surface is simply an iso-contour CIS 781 of a

Implicit Surfaces Implicit Surfaces An implicit surface is simply an iso-contour CIS 781 of a scalar function f(x,y,z)=0 . Roger Crawfis The term is usually used when modeling a surface, whereas iso-contour is used when visualizing a

388 views • 12 slides
PII Oliveira Peres S Theorem 2 2012 t act for all ca and 7 positive constants reversible s.t ca

PII Oliveira Peres S Theorem 2 2012 t act for all ca and 7 positive constants reversible s.t ca

and hitting times for Markov chains Mixing between mixing times and Overview 1 up to constants Equivalence of large sets times hitting different sizes of sets comparison for times 2 flitting Refined mixing and hitting equivalence 3 in

176 views • 4 slides
HW/SW Codesign w/ FPGAsMicroprocessors/Embedded Cores ECE 495/595 Microprocessors/Embedded Cores

HW/SW Codesign w/ FPGAsMicroprocessors/Embedded Cores ECE 495/595 Microprocessors/Embedded Cores

HW/SW Codesign w/ FPGAsMicroprocessors/Embedded Cores ECE 495/595 Microprocessors/Embedded Cores (Slides from Patrick Schaumonts course notes) The most successful programmable component of the past decades is, without doubt, the microprocessor

594 views • 44 slides
HW/SW Codesign w/ FPGAsGeneral Purpose Embedded Cores ECE 495/595 G.P. Embedded Cores (A

HW/SW Codesign w/ FPGAsGeneral Purpose Embedded Cores ECE 495/595 G.P. Embedded Cores (A

HW/SW Codesign w/ FPGAsGeneral Purpose Embedded Cores ECE 495/595 G.P. Embedded Cores (A Practical Intro. to HW/SW Codesign, P. Schaumont) The most successful programmable component on silicon is the microprocessor Fueled by a well-balanced mix of

467 views • 21 slides
Implicit Extremes and Implicit MaxStable Laws Stilian Stoev ( sstoev@umich.edu ) University of

Implicit Extremes and Implicit MaxStable Laws Stilian Stoev ( sstoev@umich.edu ) University of

Problem Formulation Limit Theory Implicit Extreme Value Laws Domains of Attraction An Example Implicit Extremes and Implicit MaxStable Laws Stilian Stoev ( sstoev@umich.edu ) University of Michigan, Ann Arbor September 19, 2014 Joint work

808 views • 34 slides
NVIDIA QUADRO RTX NVIDIA TURING GPU Turing SM RT Cores Turing SM RT Cores Up to 10 Giga

NVIDIA QUADRO RTX NVIDIA TURING GPU Turing SM RT Cores Turing SM RT Cores Up to 10 Giga

NVIDIA QUADRO RTX NVIDIA TURING GPU Turing SM RT Cores Turing SM RT Cores Up to 10 Giga Rays/sec Up to 16 TFLOPS + 16 TIPS Ray Triangle Intersection Concurrent FP &amp; INT Execution BVH Traversal Unified L1 Cache Variable Rate Shading

510 views • 29 slides
Illusionist: Transforming Lightweight Cores into Aggressive Cores on Demand Amin Ansari 1 ,

Illusionist: Transforming Lightweight Cores into Aggressive Cores on Demand Amin Ansari 1 ,

Illusionist: Transforming Lightweight Cores into Aggressive Cores on Demand Amin Ansari 1 , Shuguang Feng 2 , Shantanu Gupta 3 , Josep Torrellas 1 , and Scott Mahlke 4 1 University of Illinois, Urbana-Champaign 2 Northrop Grumman Corp. 3 Intel

349 views • 23 slides
SAT Preprocessing for MUS Extraction and MaxSAT Anton Belov Joint work with: Matti J arvisalo,

SAT Preprocessing for MUS Extraction and MaxSAT Anton Belov Joint work with: Matti J arvisalo,

SAT Preprocessing for MUS Extraction and MaxSAT Anton Belov Joint work with: Matti J arvisalo, Ant onio Morgado, Joao Marques-Silva University College Dublin, Ireland University of Helsinki, Finland IST/INESC-ID, Lisbon, Portugal EPCL

1.37k views • 124 slides
MLIC: A MaxSAT-Based framework for learning interpretable classification rules Dmitry Malioutov 1

MLIC: A MaxSAT-Based framework for learning interpretable classification rules Dmitry Malioutov 1

MLIC: A MaxSAT-Based framework for learning interpretable classification rules Dmitry Malioutov 1 Kuldeep S. Meel 2 1 IBM Research, USA 2 School of Computing, National University of Singapore CP 2018 1 / 24 The Rise of Artificial Intelligence

407 views • 39 slides
Using Expert Knowledge in How We Can Use . . . Implicit Expert . . . How We Can Use . . .

Using Expert Knowledge in How We Can Use . . . Implicit Expert . . . How We Can Use . . .

Seismic Inverse Problem Drawbacks of the . . . It Is Necessary to Take . . . How We Can Use . . . Explicit Expert . . . Using Expert Knowledge in How We Can Use . . . Implicit Expert . . . How We Can Use . . . Solving the Seismic Inverse

167 views • 16 slides
Cores so efetivas na codificao de informao? Percepo de Cores Sistema Visual

Cores so efetivas na codificao de informao? Percepo de Cores Sistema Visual

29/8/2008 Cores so efetivas na codificao de informao? Percepo de Cores Sistema Visual Sinais visuais Sinais eltricos Radiaes Sinais luminosos Fvea ou mancha amarela Cones sensveis a comprimento de onda curto,

494 views • 34 slides
Notes Solving Nonlinear Systems Most thoroughly explored in the context of optimization

Notes Solving Nonlinear Systems Most thoroughly explored in the context of optimization

Notes Solving Nonlinear Systems Most thoroughly explored in the context of optimization For systems arising in implicit time integration of stiff problems: Must be more efficient than taking k substeps of an explicit method ruling

73 views • 4 slides
PROGRAMMING TENSOR CORES: NATIVE VOLTA TENSOR CORES WITH CUTLASS Andrew Kerr, Timmy Liu, Mostafa

PROGRAMMING TENSOR CORES: NATIVE VOLTA TENSOR CORES WITH CUTLASS Andrew Kerr, Timmy Liu, Mostafa

PROGRAMMING TENSOR CORES: NATIVE VOLTA TENSOR CORES WITH CUTLASS Andrew Kerr, Timmy Liu, Mostafa Hagog, Julien Demouth, John Tran March 20, 2019 PROGRAMMING TENSOR CORES IN CUDA mma.sync (new instruction in CUDA 10.1) Feeding the Data Path

615 views • 47 slides
IMLI: An Incremental Framework for MaxSAT-Based Learning of Interpretable Classification Rules

IMLI: An Incremental Framework for MaxSAT-Based Learning of Interpretable Classification Rules

IMLI: An Incremental Framework for MaxSAT-Based Learning of Interpretable Classification Rules Bishwamittra Ghosh Joint work with Kuldeep S. Meel 1 Applications of Machine Learning 2 Example Dataset 3 Representation of an interpretable

1.24k views • 55 slides
Exploiting the Power of MIP Solvers in MAXSAT Jessica Davies 1 and Fahiem Bacchus 2 1 MIAT, INRA,

Exploiting the Power of MIP Solvers in MAXSAT Jessica Davies 1 and Fahiem Bacchus 2 1 MIAT, INRA,

Exploiting the Power of MIP Solvers in MAXSAT Jessica Davies 1 and Fahiem Bacchus 2 1 MIAT, INRA, Toulouse, France 2 Department of Computer Science, University of Toronto Outline 1. Background 2. The MaxHS Approach 3. Exploiting CPLEX 4.

630 views • 43 slides
MaxSAT and Related Optimization Problems Joao Marques-Silva 1 , 2 1 University College Dublin,

MaxSAT and Related Optimization Problems Joao Marques-Silva 1 , 2 1 University College Dublin,

MaxSAT and Related Optimization Problems Joao Marques-Silva 1 , 2 1 University College Dublin, Ireland 2 IST/INESC-ID, Lisbon, Portugal SAT/SMT Summer School 2012 FBK, Trento, Italy Example Problem: Minimum Vertex Cover The problem: Graph

1.66k views • 145 slides
Hitting sets and VC-dimension Nicolas Bousquet 1 e 2 Joint work with St ephan Thomass 1

Hitting sets and VC-dimension Nicolas Bousquet 1 e 2 Joint work with St ephan Thomass 1

Hitting sets and VC-dimension Nicolas Bousquet 1 e 2 Joint work with St ephan Thomass 1 Universit e Montpellier II, LIRMM 2 LIP, ENS Lyon Hitting sets and packings Integrality gap VC-dimension k -majority tournaments Erd os-P osa

852 views • 83 slides
Hunting Deadlocks Efficiently in Micro-Architectural Models of Communication Fabrics Freek

Hunting Deadlocks Efficiently in Micro-Architectural Models of Communication Fabrics Freek

Hunting Deadlocks Efficiently in Micro-Architectural Models of Communication Fabrics Freek Verbeek and Julien Schmaltz Growing number of cores (W. Tichy - Keynote ICST 2011) AMD Opteron 12 cores Sun Niagara3 16 cores Intel 8 cores ~1.8 Bill.

833 views • 59 slides

More recommend