PBS: A Pseudo-Boolean Solver and Optimizer

Transcription

1 PBS: A Pseudo-Boolean Soler and Optimizer Fadi A. Aloul, Arathi Ramani, Igor L. Marko, Karem A. Sakallah Uniersity of Michigan 2002 Fadi A. Aloul, Uniersity of Michigan

2 Motiation SAT Solers Apps: Verification, Routing, ATPG, Timing Analysis Problem Type: CSP Problem Format: CNF Example: Chaff, GRASP, SATO Generic ILP Solers Apps: Routing, Planning, Scheduling Problem Type: CSP/Optimization Problem Format: ILP Example: CPLEX, LP_Sole Specialized 0/1 ILP Solers Apps: Verification, Routing, Binate Coering Problem Type: CSP/Optimization Problem Format: CNF/PB (0/1 ILP) Example: Satire, BSOLO, OPBDP, WSAT 2002 Fadi A. Aloul, Uniersity of Michigan

3 Motiation SAT Solers Generic ILP Solers Channel i 012 Specialized 0-1 ILP Solers Many applications require Counting Constraints that impose upper/lower bounds on number of objects Introduce a new specialized 0-1 ILP SAT soler Describe Pseudo-Boolean (PB) search algorithms Adapt SAT applications expressed in pure CNF to CNF/PB format Empirically demonstrate effectieness in EDA applications 2002 Fadi A. Aloul, Uniersity of Michigan

4 Outline Boolean Satisfiability adances Processing Pseudo-Boolean constraints Applications CSP Optimization Experimental ealuation Conclusions 2002 Fadi A. Aloul, Uniersity of Michigan

5 Backtrack Search (DPLL) Init Decision Engine Succeed Deduction Engine Fail SAT No Conflict Exist Diagnosis Engine UNS Yes Fail 2002 Fadi A. Aloul, Uniersity of Michigan Succeed

6 Decision Strategy Init Decision Engine Deduction Engine No Conflict Exist UNS Succeed Yes Diagnosis Engine Fail Fail SAT Succeed Significantly improes the search performance Classified as: Static Dynamic Chaff introduced dynamic VSIDS: 2002 Fadi A. Aloul, Uniersity of Michigan Shown to be effectie on most benchmarks Selects most common literal and emphasizes ariables in recent conflicts

7 Improed BCP Decision Engine Deduction Engine No Init Conflict Exist Succeed Diagnosis Engine UNS Yes Fail Fail SAT Succeed Keeps track of any two unresoled literals in each clause instead of keeping track of all literals Leads to significant improements oer conentional BCP [Moskewicz et al., Zhang et al.] 2002 Fadi A. Aloul, Uniersity of Michigan

8 Conflict Diagnosis and Clause Deletion Init Decision Engine Succeed Deduction Engine No Conflict Exist Diagnosis Engine UNS Yes Fail Fail SAT Succeed Add conflict-induced clauses to aoid regenerating similar conflicts in future parts of the search process Very effectie in expediting the search process Allows non-chronological backtracking 1UIP learning scheme shown to perform best among other learning schemes [Zhang et al.] 2002 Fadi A. Aloul, Uniersity of Michigan

9 Random Restarts and Backtracking Init Decision Engine Succeed Deduction Engine No Conflict Exist Diagnosis Engine UNS Yes Fail Fail SAT Succeed Soler often gets stuck in local non-useful search space Random restarts periodically unassigns all decisions and randomly selects a new decision sequence Restarts ensures that different sub-trees are searched at eery restart Randomization can be combined with backtracking 2002 Fadi A. Aloul, Uniersity of Michigan

10 Outline Boolean Satisfiability adances Processing Pseudo-Boolean constraints Applications CSP Optimization Experimental ealuation Conclusions 2002 Fadi A. Aloul, Uniersity of Michigan

11 Pseudo-Boolean Constraints c1 x1 L cn xn ~ c i, g Z ~ { =,, } x i Literals g Clauses can be generalized as a PB constraint: (x y) (x y 1) None of the presented algorithms rely on the integrality of ci and can be implemented for floating-point ci 2002 Fadi A. Aloul, Uniersity of Michigan

12 2002 Fadi A. Aloul, Uniersity of Michigan Motiating Example Objectie: limit the true assignments to k ars out of the n ars Solution: 1 k n ( 1) k CNF: clauses Each of size PB: single PB constraint at most 2 out of 1, 2, 3, 4, 5, can be true Pure CNF: PB form: ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

13 PB Constraint Data Structure Struct PBConstraint { Goal n; constraint type ~; list of ci and xi s; initlhs; // sum of all ci s LHS; // alue of LHS based on current ariable assignment maxlhs; // maximal possible alue of LHS gien the current ariable assignment } For efficiency: Sort the list of cixi in order of increasing ci Conert all negatie ci to positie: i.e. c1x1 c2x2 n c1x1 c2(1 x2) n c1x1 c2x2 n c Fadi A. Aloul, Uniersity of Michigan

14 Algorithms for PB Search Assigning i to 1: For each literal xi of i If positie xi, LHS = ci If negatie xi, maxlhs -= ci Unassigning i from 1: For each literal xi of i If positie xi, LHS -= ci If negatie xi, maxlhs = ci PB constraint state: type SAT: LHS goal UNS: maxlhs < goal type SAT: maxlhs goal UNS: LHS > goal 2002 Fadi A. Aloul, Uniersity of Michigan 5x16x23x3 12 LHS = 0 maxlhs = 14 5x16x23x3 12 LHS = 5 maxlhs = 14 5x16x23x3 12 LHS = 5 maxlhs = 8 SATISFIABLE

15 Algorithms for PB Search Identifying implications type if ci > goal LHS, xi = 0 Implied by literals in PB assigned to 1 type if ci > maxlhs goal, xi =1 Implied by literals in PB assigned to 0 5x16x23x3 12 LHS = 0 maxlhs = 14 goal - LHS = 12 5x16x23x3 12 LHS = 8 maxlhs = 14 goal - LHS = 4 Imply x2= Fadi A. Aloul, Uniersity of Michigan

16 Algorithms for PB Search Identifying implications type if ci > goal LHS, xi = 0 Implied by literals in PB assigned to 1 type if ci > maxlhs goal, xi =1 Implied by literals in PB assigned to 0 5x16x23x3 10 LHS = 0 maxlhs = 14 maxlhs - goal = 4 Imply x1=x2= Fadi A. Aloul, Uniersity of Michigan

17 Outline Boolean Satisfiability adances Processing Pseudo-Boolean constraints Applications CSP Optimization Experimental ealuation Conclusions 2002 Fadi A. Aloul, Uniersity of Michigan

18 Applications - CSP Global Routing 2-D grid of cells arranged in rows/columns S S S Cell boundaries are edges E E Capacity C is associated with each edge (no more than C routes can pass) E E S Goal: route number of 2-pin connections in the grid with edge capacities Generate satisfiable instances using randomized flooding 2002 Fadi A. Aloul, Uniersity of Michigan

19 Global Routing Formulation Connectiity constraints (for each net) Exactly one edge selected at start/end point If cell is a mid-point, either two or no edges are selected Capacity constraints A net can use a single track across an edge No two nets can use the same track across an edge S E E S W W N N E E Create a ariable for each edge/net 2 x 12 = 24 ariables ( N W )( N E)( W ( W N E) E) 2002 Fadi A. Aloul, Uniersity of Michigan

20 Global Routing Formulation Connectiity constraints (for each net) Exactly one edge selected at start/end point If cell is a mid-point, either two or no edges are selected Capacity constraints A net can use a single track across an edge No two nets can use the same track across an edge S E E S W W N N E E Create a ariable for each edge/net 2 x 12 = 24 ariables ( N E ( N W )( N E W )( N E W ) E W ) 2002 Fadi A. Aloul, Uniersity of Michigan

21 Global Routing Formulation Connectiity constraints (for each net) Exactly one edge selected at start/end point If cell is a mid-point, either two or no edges are selected Capacity constraints 30 Nets A net can use a single 10 Cap track across an edge #Cl = 55M No two nets can use the same track across an s. edge 1 PB S E E S W W N N E E Create a ariable for each edge/net 2 x 12 = 24 ariables # Cl purecnf CNF/PB # Nets = Cap 1 i Cap all _ nets 2002 Fadi A. Aloul, Uniersity of Michigan

22 Global Routing Formulation Connectiity constraints (for each net) Exactly one edge selected at start/end point If cell is a mid-point, either two or no edges are selected Capacity constraints Additional A net can use Variablesa single track across & an Clauses edge No two nets can use the same track across an edge 2002 Fadi A. Aloul, Uniersity of Michigan S E E S N1 N1 N2 N2 W1 W1 W2 W2 E1 E1 E2 E2 Create Cap ariables per edge/net 2 x 2 x 12 = 48 ariables # Cl = purecnf Cap 2 # Nets 2 (# Nets) ( Cap)

23 Applications - optimization Max-ONEs Seeks an assignment that Satisfies all constraints Maximizes the number of ariables assigned to true Useful to represent Max-Clique problems Vertex Coer can be reduced to Min-ONEs Use a single PB constraint of type that includes each ariable with coefficient 1 Iteratiely increase the lower bound until the problem becomes unsatisfiable Extendable to Weighted Max-ONEs 2002 Fadi A. Aloul, Uniersity of Michigan

24 Applications - optimization Max-SAT Finds an assignment that Satisfies maximum possible number of clauses Generalization of SAT Proides more info for unsatisfiable instances Used to represent Max-CUT problems Expressed using a single PB constraint Soled using PBS Addressed indirectly using WalkSAT 2002 Fadi A. Aloul, Uniersity of Michigan

25 Outline Boolean Satisfiability adances Processing Pseudo-Boolean constraints Applications CSP Optimization Experimental ealuation Conclusions 2002 Fadi A. Aloul, Uniersity of Michigan

26 Experimental Setup Platform: Pentium-II 300 MHz with 512MB RAM running Linux Runtime limit: 5000 sec PBS Implemented in C PBS settings: VSIDS decision heuristic Optimized BCP Random Restarts 1 st UIP conflict analysis learning scheme Clause deletion/random backtracking disabled 2002 Fadi A. Aloul, Uniersity of Michigan

27 Global Routing Experiment Instance CNF pseudo-boolean pure CNF PBS Speedup V C #PB PBS SATIRE OPBDP V C Chaff Satire OPBDP Chaff grout grout grout grout grout grout grout grout grout grout grout grout grout grout grout Fadi A. Aloul, Uniersity of Michigan

28 MaxONE Experiment Benchmark Instance ONEs SATIRE OBPDP Satisfiable Max- PBS Speedup V C PBS SATIRE OPBDP DIMACS aim-50-1_6-yes aim-100-1_6-yes aim-200-2_0-yes1_ ii8b jnh jnh par par8-2-c Beijing 3blocks QG qg qg Planning bw_a bw_b bw_c Fadi A. Aloul, Uniersity of Michigan

29 Conclusions Adapting SAT apps to use CNF/PB constraints leads to memory saings and runtime reductions Proposed new specialized 0-1 ILP soler, PBS Confirmed effectieness on real world examples: Global routing consistency instances Max-ONEs optimization problems (extendable to Max-SAT, Min-ONEs) 2002 Fadi A. Aloul, Uniersity of Michigan

30 Future Works Compare state-of-the-art Generic ILP solers, such as CPLEX, to specialized 0-1 ILP solers Apply PBS to Max-SAT and Min-ONEs problems Study applications to Max-Clique, Max Independent Set, and Min Vertex Coer 2002 Fadi A. Aloul, Uniersity of Michigan