The Weighted Average Constraint Alessio Bonfietti - PowerPoint PPT Presentation

The Weighted Average Constraint Alessio Bonfietti <alessio.bonfietti@unibo.it> Michele Lombardi <michele.lombardi2@unibo.it> DEIS, University of Bologna

Main Topic Paper topic: a global constraint for weighted average expressions P n − 1 i =0 W i · V i average([ W i ] , [ V i ] , Y ) ↔ Y = P n − 1 i =0 W i Equivalent for integer variables: P n − 1 ! i =0 W i · V i average([ W i ] , [ V i ] , Y ) ↔ Y = round P n − 1 i =0 W i

Motivation Context: workload dispatching for high performance computing Server room Multi-core Platforms

Motivation Context: workload dispatching for high performance computing § Jobs arrive in batches § Jobs are assigned to different machines/cores § Local scheduling (by the OS) § Obj: maximize worst core efficiency Mapped Jobs ... Avg power ... temperature efficiency

Motivation Context: assignment problems with balancing issues customer (size = demand) facility (size = capacity) Single Source Capacitated Facility Problem § Assign customers to facility § Meet capacity constraints ...with Fair Travel Times § Balance the average travel time per facility

Which Model? Modeling Choices: § Assignment variables: n = #customers/jobs m = #facilities/cores X i ∈ { 0 ..m − 1 } ∀ i = 0 . . . n − 1 § For each facility/core k : P n − 1 i =0 ( X i = k ) · power i POWER = By abstracting a little bit: P n − 1 i =0 ( X i = k ) P n − 1 i =0 W i · v i Y = P n − 1 i =0 ( X i = k ) · ttime i P n − 1 i =0 W i TTIME = P n − 1 i =0 ( X i = k )

Which Model? P n − 1 i =0 W i · v i How do we model this expression? Y = P n − 1 i =0 W i § Fixed denominator P n − 1 i =0 W i · v i spread and deviation Y = sum constraint! to improve filtering w § Just post it! n − 1 n − 1 X X Likely weak propagation... W i · v i = Y · W i i =0 i =0

Which Model? P n − 1 i =0 X i · v i How do we model this expression? Y = P n − 1 i =0 X i § Otherwise, we need a new global constraint: average([ W i ] , [ V i ] , Y ) Here it is!

Filtering Spring equivalent: average as a bar pulled by metal spring § Weights W i = spring thickness, Values v i = anchor points

Filtering Spring equivalent: average as a bar pulled by metal spring Y V = [ − 3 , 1 , 1 , 3] [0,3] W 0 = [0 , 3] [0,3] [1,2] W 1 = [1 , 3] W 2 = [1 , 2] [1,3] W 3 = [0 , 3] dark light -3 -2 -1 0 1 2 3 § Assumption 1: fixed values (adapted to variable V i ) § Assumption 2: continuous domains (adapted to integer domains)

Pruning the Average Variable Y upper bound = right-most position for the bar § Minimize all weights § Scan W i from right to left [0,3] [0,3] [1,2] [1,3] -3 -2 -1 0 1 2 3

Pruning the Average Variable Y upper bound = right-most position for the bar § Minimize all weights UB = 1.8 § Scan W i from right to left [0,3] § Maximize W i if: [0,3] v i > current avg [1,2] § Repeat the process [1,3] § WC complexity: O(n) + O(n log(n)) for the ordering -3 -2 -1 0 1 2 3

Pruning the Weight Variables W i upper bound = largest thickness so that the Y boundaries are not crossed Y in [-1.80,-0.25] [0,3] [0,3] [1,2] [1,3] -3 -2 -1 0 1 2 3

Pruning the Weight Variables W i upper bound = largest thickness so that the Y boundaries are not crossed § Maximize W i if: Y in [-1.80,-0.25] v i ≤ max( Y ) [0,3] § Minimize W i if: [0,3] v i > max( Y ) [1,2] § UB if v i > max( Y ) max slack [1,3] § LB if v i < max( Y ) configuration -3 -2 -1 0 1 2 3 Y MS

Pruning the Weight Variables W i upper bound = largest thickness so that the Y boundaries are not crossed § Maximize W i if: Y in [-1.80,-0.25] 2.846 v i ≤ max( Y ) [0,3] § Minimize W i if: [0,3] v i > max( Y ) [1,2] § UB if v i > max( Y ) max slack [1,3] § LB if v i < max( Y ) configuration By solving: num( Y MS ) + (max( W i ) − min( W i )) · v i ≤ max( Y ) den( Y MS ) + (max( W i ) − min( W i )) -3 -2 -1 0 1 2 3 Y MS

Pruning the Weight Variables W i upper bound = largest thickness so that the Y boundaries are not crossed § Maximize W i if: Y in [-1.80,-0.25] 2.846 v i ≤ max( Y ) [0,3] § Minimize W i if: [0,3] v i > max( Y ) [1,2] § UB if v i > max( Y ) max slack [1,3] § LB if v i < max( Y ) configuration § WC complexity: O(n) -3 -2 -1 0 1 2 3 Y MS

Incremental Filtering Problems of this class can grow pretty large: § Thermal Aware Workload Dispatching: 120 to 480 jobs § Fair Capacitated Facility Location: 50 customer, 16-50 locations Incremental filtering can save a lot of computation time § Rules for the fixed values case § Particularly effective for {0,1} weights

Incremental Filtering for the Y Variable Store: § num(Y UB )/den(Y UB ) § Index of the last maximized W i UB = 1.8 [0,3] [2,3] [1,2] [1,3] -3 -2 -1 0 1 2 3

Incremental Filtering for the Y Variable Store: § num(Y UB )/den(Y UB ) § Index of the last maximized W i When a weight changes: [0,3] [2,3] [1,2] [1,3] -3 -2 -1 0 1 2 3

Incremental Filtering for the Y Variable Store: § num(Y UB )/den(Y UB ) § Index of the last maximized W i cur avg = 0.429 When a weight changes: [0,3] § Update current avg [2,3] [1,2] [1,3] -3 -2 -1 0 1 2 3

Incremental Filtering for the Y Variable Store: § num(Y UB )/den(Y UB ) § Index of the last maximized W i cur avg = 0.429 When a weight changes: [0,3] § Update current avg [2,3] [1,2] [1,3] -3 -2 -1 0 1 2 3

Incremental Filtering for the Y Variable Store: § num(Y UB )/den(Y UB ) § Index of the last maximized W i UB = 0.5 When a weight changes: [0,3] [0,3] § Update current avg [2,3] [2,3] § Maximize new W i § No more than n shifts [1,2] [1,2] § No more than n × [1,3] [1,3] dom size updates § WC complexity O(n × dom size) -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3

Experimental Results Experiments on: § Capacitated Facility Location (max worst case average travel time) § Thermal aware workload dispatching: max worst case efficiency Benchmarks: § Problem #1: Single Source instances by Beasley in the OR-Library § Problem #1: custom (publicly available) instances Solution method (goal: testing constraint propagation): § Random restarts with fixed threshold § Random variable and value selection Compare with competitor approaches

Results for Capacitated Facility Location Decomposed)Average) 0.0%& CAP6X& !20.0%& CAP7X& !40.0%& CAP9X& CAP10X& !60.0%& CAP12X& CAP13X& !80.0%& !100.0%& 1& 3& 5& 10& 30& 60&

Results for Thermal Aware Dispatching Decomposed)Average) 0.00%& M1&120& !20.00%& M1&240& !40.00%& M1&480& M2&120& !60.00%& M2&240& M2&480& !80.00%& !100.00%& 1& 3& 5& 10& 30& 60&

Results for Thermal Aware Dispatching Decomposed)Average) 8.00%& 6.00%& 4.00%& M1&120& 2.00%& M1&240& 0.00%& M1&480& !2.00%& M2&120& !4.00%& M2&240& !6.00%& M2&480& !8.00%& !10.00%& 1& 3& 5& 10& 30& 60&

Conclusion Main results § A global constraint for weighted average expressions § Useful for allocation problems with balancing components § (Incremental) Filtering algorithms Future work directions § Apply incremental filtering ideas from the sum constraint § More application scenarios § Devise constraints for other classical inputs to machine learning models

Thanks! Questions? Funded by a DEIS, University of Bologna grant