12/7/2006 Massachusetts Institute of Technology Forward - PDF document

12/7/2006 Massachusetts Institute of Technology Forward Conflict-Directed Search • Backward conflict-directed search uses conflicts to select backtrack points and as a cache used to prune nodes. Generalized Conflict Learning for – dependency-directed backtracking [Stallman-Sussman-77] – conflict-directed backjumping [Prosser-93] Hybrid Discrete/Linear Optimization – dynamic backtracking [Ginsberg-93] – LPSAT [Wolfman-Weld-99]. • Forward conflict-directed search guides the forward step of search Hui Li and Brian Williams away from regions of the state space that are ruled out by known conflicts – Conflict-directed A* [Williams-Nayak-AAAI96, Williams-Ragno-JDAM]. Computer Science and Artificial Intelligence Laboratory – Assumption-based DDBT [deKleer-Williams AAAI86, IJCAI89], Factor Out Failure[Freuder-IJCAI-95 Massachusetts Institute of Technology – Candidate Generation [deKleer-Williams-AIJ87, Reiter-AIJ87] Oct. 5 th , 2005 � Introduce Generalized Forward Conflict-directed Search on Hybrid Discrete/Linear Optimization – Experiments on cooperative vehicle plan execution problems demonstrates that the approach significantly outperforms branch and bound using conflicts on backtracking . 2 90’s Self-Repairing Explorers Solve COPs Using Outline Forward, Conflict-directed Best First Search • Optimal Satisfiability Problem • Deep Space 1 Remote Agent Experiment (May, 1999) • Context [¬] x ij = v ij • Review of Conflict-directed A* • OpSat uses conflicts (nogoods) in • The GCD-BB algorithm the forward direction , to substantially improve Best-first Search (CD A*) • Empirical Evaluation [Williams-Nayak-AAAI96,Williams-Ragno-JDAM]. • Conclusion • Livingstone Model-based Execution Increasing Conflict 1 System [Williams & Nayak, AAAI96] Cost Infeasible Infeasible State estimates State goals Conflict 2 Mode Mode Model: Estimation: Reconfiguration: HMMs + Feasible Tracks likely Tracks least-cost CSPs Conflict 3 States state goals Observations Commands Plant 3 4 00’s Plan-driven Agile Systems Solve Hybrid Discrete/Linear 00’s Plan-driven Agile Systems Solve Hybrid Discrete/Linear Optimization Problems via Forward Conflict-directed Search Optimization Problems via Forward Conflict-directed Search • [Leaute & Williams, AAAI05] • Hybrid Discrete/Linear • [Hoffman & Williams, ICAPS05] • Hybrid Discrete/Linear Gait Poses To put out the Burbank wildfires, . . . Optimization Problems Optimization Problems UAV1 Starts at home; – Disjunctive Linear Programs (DLPs) – Disjunctive Linear Programs (DLPs) {Gets fuel & water; drops on fire-1} [1, 5]; l1 [Balas-ADM-79] [Balas-ADM-79] r1 r2 l1 l1 r2 r2 l2 {Gets fuel & water; drops on fire-2} [2, 6]; – Binary Integer Programming Foot placement – Binary Integer Programming Returns home. – LCNF [Wolfman-IJCAI-99] l1 l2 – LCNF [Wolfman-IJCAI-99] Lat Fwd r1 r2 – Mixed Logical Linear Programs – Mixed Logical Linear Programs (MLLPs) [Hooker-JDAM-99] (MLLPs) [Hooker-JDAM-99] • How do we generalize forward • How do we generalize forward conflict-directed search to conflict-directed search to HDLOPs? HDLOPs? � Generalized Conflict-directed � Generalized Conflict-directed Branch and Bound (GCD-BB) Branch and Bound (GCD-BB) 1

12/7/2006 Generalized Conflict-directed B&B Disjunctive Linear Programs [Balas 79] Extends traditional Branch & Bound: Definition: 1. Constructs conflicts from search tree nodes found – Infeasible or i j – Sub-optimal clause 2. Uses conflicts to guide forward search away from infeasible and sub-optimal states. Example: 3. Performs induced unit clause relaxation – Deduces (some) entailed unit clauses. Demonstrates substantial performance improvement – For both Best-first and depth-first Branch & Bound search, – In terms of speed and memory usage. Compared with – BIPs – BFS and B&B without conflicts – Backup on conflicts. 7 8 Outline Optimal Satisfiability Problems [¬] x ij = v ij • Context f i G U • Review of Conflict-directed A* • Diagnosis Or1 .92 .08 • The GCD-BB algorithm A Or2 1 .91 .09 Or1 X • Empirical Evaluation F 0 B Or3 .93 .07 1 And1 • Conclusion C And1 .94 .06 Or2 1 Y And2 .95 .05 D G 1 1 And2 E Or3 Z f i 1 0 0 = ∏ A-G 1.0 1.0 f x ( ) f x ( ) i i X-Z i 10 9 Conflict-directed A* Conflict-directed A* • Select optimal state outside conflicts at each step. • Select optimal state outside conflicts at each step. • Feasible subregions described by kernel assignments. � Use conflicts to search for kernel assignment containing the best cost candidate. Increasing Increasing Cost Cost Conflict 1 Conflict 1 Infeasible Infeasible Infeasible Conflict 2 Conflict 2 Feasible Feasible Kernel 3 Kernel 1 Conflict 3 Conflict 3 [Williams-Nayak-AAAI96, Williams-Ragno-JDAM0?, Kernel 2 deKleer-Williams-IJCA89] 11 2

12/7/2006 •The kernel assignments are the minimal coverings of Generating candidate after two iterations… the constituent kernels. [deKleer-Williams-AIJ87, Reiter-AIJ87] • The best kernel is found through A* search of the • Diagnosis covering tree. [Williams-Nayak-AAAI96, Williams-Ragno-JDAM] A 1 Or1 X (Kernels resolve ALL conflicts - Constituent Kernels F 0 ALL states outside ALL conflicts) B 1 And1 1st iteration C 1 Or2 Y D G 1 1 {A1=U, M1=U , M2=U} And2 {A1=U} {M1=U} {M2=U} 2nd iteration E Or3 Z 0 {A1=U, A2=U, Conflicts: {O1=G, O2=G, A1=G} is inconsistent M3=U M1=U, M3=U} {O1=G, A1=G A2=G} is inconsistent 3rd iteration A2=U Constituent Kernels: {O1=U, O2=U, A1=U} at least one holds {A1=U} {M2=U, M3=U} {M1=U} {M1=U, A2=U} (Resolve one conflict - {O1=U, A1=U A2=U} at least one holds All states outside conflict) 13 1. Branch and Bound for DLPs Outline 1. Branch on subproblems. 2. Maintain running best soln in incumbent. Binary variables 3. Bound cost using relaxed problems . Clauses 4. Prune infeasible and • Context suboptimal branches. For DLPs For DLPs • Review of Conflict-directed A* min -x-3y s.t. x ≤ 200 root y ≤ 200 For BIPs minimize -x-3y • The GCD-BB algorithm s.t. x ≤ 200 x ≤ 100 y ≤ 30 y ≤ 200 min -x-3y y ≤ 30 V x ≤ 100 min -x-3y A 2 A 1 y ≤ 20 V x ≤ 10 s.t. x ≤ 200 1. Branch and Bound for DLPs s.t. x ≤ 200 y ≤ 200 x ≥ 80 V x ≥ 30 V y ≤ 0 y ≤ 20 x ≤ 10 y ≤ 200 x ≤ 10 x ≤ 100 y ≤ 30 min -x-3y min -x-3y 2. Induced Unit Clause Relaxation s.t. x ≤ 200 min -x-3y B 2 B 1 s.t. x ≤ 200 y ≤ 200 y ≤ 200 s.t. x ≤ 200 x ≤ 100 Incumbent x ≤ 100 y ≤ 200 3. Generalized Conflict Learning x ≤ 10 y ≤ 30 y ≤ 20 -160 x ≤ 10 y ≤ 0 x ≥ 80 an upper bound x ≥ 30 4. Forward Conflict-directed Search C 3 C 2 C 1 • Empirical Evaluation min -x-3y min -x-3y min -x-3y s.t. x ≤ 200 s.t. x ≤ 200 s.t. x ≤ 200 y ≤ 200 y ≤ 200 y ≤ 200 x ≤ 100 x ≤ 100 • Conclusion x ≤ 100 x ≤ 10 x ≤ 10 x ≤ 10 y ≤ 0 x ≥ 30 x ≥ 80 15 16 2. Induced Unit Clause Relaxation 2. Induced Unit Clause Relaxation Example: • BIP relaxation : DP at a sub-problem before relaxation: Binary constraint x ∈ {0,1} → Continuous constraint 0 ≤ x ≤ 1 Relax to max x +3y propositional s.t. x ≤ 200 max x +3y Problem: adding binary variables and constraints increases clauses Unit propagate y ≤ 200 s.t. a a 1 the dimensionality of the search problem. x ≤ 100 b y ≤ 5 v x ≥ 100 c d v ¬ c x > 80 v x ≥ 30 v y ≤ 0 • Simple DLP relaxation : e v f v g Remove all non-unit clauses from the DLP Relaxed Problem: Problem: ignoring all non-unit clauses forms a weak max x +3y max x +3y relaxation. max x +3y s.t. a s.t. a a 1 a 1 s.t. x ≤ 200 b b � Induced unit clause relaxation : y ≤ 200 c c Relax non- Reintroduce x ≤ 100 d d unit clauses the linear – Strengthen by adding entailed unit clauses y ≤ 5 e v f v g inequalities – Approach: Relax DLP to a propositional theory + apply unit propagation …. (alternatively, failed literals) 17 18 3