ANTs PI meeting, Dec. 17, 2001Washington University / DCMP1 Flexible Methods for Multi-agent Distributed Resource Allocation by Exploiting Phase Transitions Modeling and Analyzing Resource Allocation Problems Using Soft Constraint Satisfaction and Optimization Weixiong Zhang (PI) Kenneth Swanson, Xiaotao Zhang, Peng Wang, Michael P. Moran, Guandong Wang, Zhao Xing, Zhongshen Guo Computational Intelligence Center and Computer Science Department Washington University in St. Louis

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP2 Objectives and Approaches Understanding and characterizing resource allocation problems in ANTs applications. –Modeling methods: soft constraint satisfaction/optimization –Phase transitions and backbones (sources of complexity) –Scalability (e.g. impact of problem structures) Developing general and efficient algorithms for resource allocations –Systematic search methods –Approximation methods –Distributed algorithms –Phase-aware problem solving for good enough/sooner enough solutions

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP3 Work in this period EW challenge problem –Design and develop a moving target tracking system in RadSim –Preliminary working system –Testbed for studying many technical difficulties –(more to come at next PI meeting) Marbles pilot scheduling problems –(main focus of this presentation)

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP4 Current Work on EW Challenge Problem Technical issues under consideration –Scalability how problem structures and agent organization affect complexity –Uncertainty in resource conflict resolution: uncertainty in measurement, communication error, etc. –Scan scheduling for detecting new targets quickly with small amount of energy –Irregular sensor layout We have shown that triangle topology provides the best area coverage What if sensor layout is out of your control – how to quickly form teams

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP5 Marbles Scheduling Problem Main focus of this period Some results

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP6 The Marbles Problem Resource allocation in a task scheduling problem –Schedule as many tasks as possible (to reduce the overall penalty of unscheduled tasks) –Block resource requirement Each task requires a set of resources It cannot be schedule unless all resource requirements are fulfilled –Exclusive resource contention A shared resource may be applicable to multiple requirements But it can be used to fulfill only one requirement R1R1 R2R2 R3R3 T1T1 Q 1,1 011 Q 1,2 100 T2T2 Q 2,1 110 Q 2,2 110 Resource requirements Resources Tasks

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP7 The Problem is Difficult The problem is NP-hard –The decision version is NP-complete –Reduced from set packing (NP-complete) Set packing: –Given a collection S of finite sets of elements, a positive integer K –Decide: if S contains at least K mutually disjoint subsets Reduction: –Map an elements to a resource –Map a subset to a task

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP8 Technical Content Hard and soft constraints Modeling consideration and choices Constraint models –Models in optimization –Models in satisfaction Experimental analysis (phase transitions) Current and future work

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP9 Hard and Soft Constraints Task constraints (soft constraints) –C task: turning on tasks (typically, not all of them can be satisfied at once) Constraint (T i = 1) to represent turning on task T i Weight equal to 1 or its penalty Block resource requirements (hard constraints) –C req: Fulfilling resource requirement of a task if it is on –Weight is more than the total weight of soft constraints Exclusive resource contention (hard constraints) –C res: A resource can only be used by one requirement –Weight is more than the total weight of soft constraints

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP10 Main Modeling Considerations Optimization vs. decision –Optimization: try to turn on all tasks, and then find the maximal number of tasks that can be indeed turned on –Decision: Guess the possible number of tasks that can be turned on, and then verify it. Do a binary search on the number of tasks. (caution: it may not work if tasks are weighted and it is to minimize the overall weight of unscheduled tasks.) General variables versus Boolean variables –CSP/COP (Constraint Optimization Problem) versus SAT/MAX-SAT –K-encoding issue Which choice to take and under what conditions? OptimizationDecision General variablesA COP modelA set of CSP models Boolean variablesA MAX-SAT modelA set of SAT models

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP11 Main Modeling Choices Variable versus values –Resources as variables and requirements as values –Or vice versa –Which one to use? R1R1 R2R2 R3R3 T1T1 Q 1,1 011 Q 1,2 100 T2T2 Q 2,1 110 Q 2,2 110 Resource requirements Resources Tasks

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP12 Main Modeling Choices Expressiveness of a model E.g. Two resources may be assigned to one requirement (but one is used) Should hidden constraints be made explicit? Interaction between modeling considerations and choices and search algorithms

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP13 COP/CSP Models OptimizationDecision COP1 (requirements as variables) CSP1 (requirements as variables) COP2 (resources as variables) CSP2 (resources as variables) COP3 (resources as variables, more explicit than COP2) CSP3 (resources as variables, more explicit than CSP2) MAX-SAT4SAT4 MAX-SAT5 (more explicit than MAX SAT4) SAT5 (more explicit than SAT4) OptimizationDecision General variablesA COP modelA set of CSP models Boolean variablesA MAX-SAT modelA set of SAT models Original ISI Marbles model

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP14 COP2 Model: Resources as Variables t: # of tasks q i : # of resource requirements of task i r: # of resources T i : Boolean variable for task i R k = { Q ij | task i, requirement j } C task = ^ k=1..t (T i = 1) C block = ^ k=1..t C block (T i ) C block (T i )= ^ j=1..q i ((T i =0) V k=1..r (R k =Q ij )) R1R1 R2R2 R3R3 T1T1 Q 1,1 011 Q 1,2 100 T2T2 Q 2,1 110 Q 2,2 110 C task = (T 1 = 1) ^ (T 2 = 1) (T 1 =0) V (R 2 =Q 11 ) V (R 3 =Q 11 ) C block = (T 1 =0) V (R 1 =Q 12 ) (T 2 =0) V (R 1 =Q 21 ) V (R 2 =Q 11 ) (T 2 =0) V (R 1 =Q 22 ) V (R 2 =Q 22 )

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP15 COP3 Model: More Explicit than COP2 t: # of tasks q i : # of resource requirements of task i r: # of resources T i : Boolean variable for task i R k = { Q ij | task i, requirement j} C task = ^ k=1..t (T i = 1) C block = ^ k=1..t C block (T i ) ^ k=1..t C' block (T i ) C block (T i )= ^ j=1..q i ((T i =0) V k=1..r (R k =Q ij )) C' block (T i ) = ^ ((R u ≠ Q ij ) V (R v ≠ Q ij )) R1R1 R2R2 R3R3 T1T1 Q 1,1 011 Q 1,2 100 T2T2 Q 2,1 110 Q 2,2 110 C task = (T 1 = 1) ^ (T 2 = 1) (T 1 =0) V (R 2 =Q 11 ) V (R 3 =Q 11 ) C block = (T 1 =0) V (R 1 =Q 12 ) (T 2 =0) V (R 1 =Q 21 ) V (R 2 =Q 11 ) (T 2 =0) V (R 1 =Q 22 ) V (R 2 =Q 22 ) (R 2 ≠ Q 11 ) V (R 3 ≠ Q 11 ) C' block = (R 1 ≠ Q 21 ) V (R 2 ≠ Q 21 ) (R 1 ≠ Q 22 ) V (R 3 ≠ Q 22 )

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP16 Phase Transitions Marbles problems (8 tasks with 2 requirements each)

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP17 Phase Transitions (2) Marbles problems (8 tasks with 2 requirements each)

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP18 Experimental Results: Systematic search and local search OptimizationDecision COP1 (requirements as variables) CSP1 (requirements as variables) COP2 (resources as variables) CSP2 (resources as variables) COP3 (resources as variables, more explicit than COP2) CSP3 (resources as variables, more explicit than CSP2) MAX-SAT4SAT4 MAX-SAT5 (more explicit than MAX SAT4) SAT5 (more explicit than SAT4) OptimizationDecision General variablesA COP modelA set of CSP models Boolean variablesA MAX-SAT modelA set of SAT models

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP19 Experimental Results : Complete Algorithm MAX SAT Models

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP20 Experimental Results : Complete Algorithm SAT Models

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP21 Experimental Analysis : Complete Algorithm Summary OptimizationDecision Model 43 MAX SAT4 2 SAT4 Model 54 MAX SAT5 1 SAT5

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP22 Experimental Analysis : Local Search Experiment setup –A WalkSAT-like algorithm for all models –Given all models the same total amount of CPU time (adjusted by the numbers of moves and restarts –Measure the final solution quality –CPU time the best solution found the first time Results –Optimization models are better than decision models –COP1 is the best –COP2 is better than COP3 OptimizationDecision COP1 (requirements as variables) CSP1 (requirements as variables) COP2 (resources as variables) CSP2 (resources as variables) COP3 (resources as variables, more explicit than COP2) CSP3 (resources as variables, more explicit than CSP2) MAX-SAT4SAT4 MAX-SAT5 (more explicit than MAX SAT4) SAT5 (more explicit than SAT4)

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP23 Summary Marbles problems are indeed difficult There are phase transition phenomena in Marbles problems Modeling and search algorithms affect each other Good modeling methods can greatly reduce problem-solving time

ANTs PI meeting, Dec. 17, 2001Washington University / DCMP24 Next Steps Marbles scheduling problem –More accurate results on phase transitions –More efficient search algorithms –Large problems –Timed Marbles problems for a long period, e.g., days,weeks and months EW challenge problem –Scalability –Uncertainty –Scan scheduling (larger coverage, less energy) –Irregular layout