Formal Complexity Analysis of RoboFlag Drill & Communication and Computation in Distributed Negotiation Algorithms in Distributed Negotiation Algorithms Carla P. Gomes Cornell University

Formal Complexity Analysis of RoboFlag Drill (joint work with Matt Earl and Raff D’Andrea)

Formal Complexity Analysis of Roboflag Drill Question: What is the computational complexity of Roboflag Drill?

Find the simplest particular case of Roboflag Drill for which we can formally prove that the task is NP-complete – Roboflag Drill Base – Find a known NP-complete problem, Q – Reduce Q to Roboflag Drill Base, using a polynomial time reduction Formal Complexity Analysis of Roboflag Drill

Input: Set of attackers initial location velocity (constant) direction (constant) One defender initial location velocity (constant) direction – piecewise linear Goal area Question: Can the defender intersect all the attackers before they reach the goal area? RoboFlag Drill Base

NP-Complete Problem Q: TDET - Scheduling tasks with time depend execution times Input: Set of tasks release time deadline processing time – dependent on start time; One processor Question: Can we schedule all the task on the single processor, so that they are all processed before the deadlines? NP-complete problem, Q, to be reduced to RoboFlag Drill Base

NP-complete in the strong sense Attackers all equidistant from the goal area: Polynomial (becomes NP-Complete with only two different distances) Fixed number of attackers: Fixed Parameter Complexity Class RoboFlag Drill Base (conjectures)

Communication and Computation in Distributed Negotiation Algorithms (joint work with Cesar Fernandez, Bhaskar Krishnamachari, and Bart Selman)

The behaviour of Distributed Negotiation algorithms DN algorithms solve a problem through a distributed computational search process Agents exchange messages for reaching a global solution A4A4 A1A1 A2A2 A3A3 m1m1 m2m2 m3m3 Alteration of the arrival order of messages by:  Active introduction of random delays by the agents  Introduction of random delays because of the network traffic The arrival order of the messages determines the decisions of A 4

Distributed Negotiation Problems (DNP) DNP:  Set of agents: A 1, A 2,..., A n  Set of local problems: P 1, P 2,..., P n P i belongs to A i : only A i can modify the variables of P i  Global Problem among variables of different P i ´s Goal: Solve the local and global problems simultaniously Simplest model: One variable per agent and no local problems

SensorDNP - a benchmark problem Constraints:  Sensors: can track at most one target. Not all the sensors are compatible between them  Targets: need three compatible sensors Goal Goal: track every target with three compatible sensors

DNP algorithms ABT ABT: Static priority order. AWC AWC: Dynamic priority order and min-conflict heuristic. Two types of messages sent by an agent:  ok?: inform neighbors about its own assignment  nogood: ask a higher priority agent to backtrack Solution found: no agent changes its assignment or asks another agent to backtrack Solution not found: top-priority agent asked to backtrack We consider only complete algorithms: they always find a solution if there exists one

DNP algorithms - randomization and restarting Modifications to the DNP algorithms: Active Randomization For every agent: with probability p deliver the next message with increased delay r Restarting: For the top-priority agent: If timeout then 1. Change at random its assignment 2. Inform neighbors about change

Network traffic models and delay distributions  Low data load: fixed (deterministic) delays  Heavy data load and:  Traditional single user session sources: Exponential delay distributions  Aggregate data sources: Log-normal and Fractional Gaussian Noise delay distributions Our results: delays introduced by the network can improve the performance of DNP algorithms

Exponential delay links: results  Instances tested:  3 mobiles and 15 sensors  Inter-agent communication links: exponentially distributed delays  15 instances for each value of P c : Compatibility level between sensors (0 to 1) P v : Visibility level of sensors (0 to 1) P C and P V model the level of resources available

Phase Transition in SensorDNP Sharp transition to solvable instances at critical level of resources

Mean complexity Peak in complexity around phase transition region Worse for low level of compatibility (P c ) P sat = 0.2 P sat = 0.8

Active delaying of messages Inter-agent communication links with fixed delay Reduction on number of messages in almost all cases Reduction on solution time for low values of r Results on a hard soluble instance

Comparing different delay distributions Performance is improved when using restarting using restarting Cost distributions when solving a hard soluble instance

Summary  Formal Complexity Analysis of RoboFlag Drill – Studying reduction from TDET (Tasks with Time Dependent Execution Times)  Distributed Negotiation Algorithms Phase transition phenomena with corresponding peak in complexity for distributed negotiation protocols; Controlled randomization can increase performance of negotiation protocols dramatically.