Formal Complexity Analysis of RoboFlag Formal Complexity Analysis of RoboFlag DrillDrill

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Communication and ComputationCommunication and Computation

in Distributed Negotiation Algorithmsin Distributed Negotiation Algorithms

Carla P. GomesCarla P. Gomes

Cornell UniversityCornell University

Formal Complexity Analysis of RoboFlag Formal Complexity Analysis of RoboFlag DrillDrill

(joint work with Matt Earl and Raff D’Andrea)

Formal Complexity Analysis of Roboflag DrillFormal 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 DrillFormal Complexity Analysis of Roboflag Drill

Input: Set of attackers initial location

velocity (constant)direction (constant)

One defenderinitial locationvelocity (constant)direction – piecewise linear

Goal areaQuestion: Can the defender intersect all the attackers

before they reach the goal area?

RoboFlag Drill BaseRoboFlag Drill Base

NP-Complete Problem Q:TDET - Scheduling tasks with time depend execution times

Input:Set of tasks

release timedeadlineprocessing time – dependent on start time;

One processorQuestion: 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 NP-complete problem, Q, to be reduced to RoboFlag Drill BaseRoboFlag 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 BaseRoboFlag Drill Base(conjectures)(conjectures)

Communication and ComputationCommunication and Computation

in Distributed Negotiation Algorithmsin Distributed Negotiation Algorithms

(joint work with Cesar Fernandez, Bhaskar Krishnamachari, and Bart Selman)

The behavThe behaviiour of Distributed Negotiation algorithmsour of Distributed Negotiation algorithms

DN algorithms solve a problem through a distributed computational search process

Agents exchange messages for reaching a global solution

A4

A1 A2 A3

m1 m2 m3

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 A4

Distributed Negotiation Problems (DNP)Distributed Negotiation Problems (DNP)

DNP:

Set of agents: A1, A2, ..., An

Set of local problems: P1, P2, ..., Pn

Pi belongs to Ai: only Ai can modify the variables of Pi

Global Problem among variables of different Pi

´sGoal:

Solve the local and global problems simultaniously

Simplest model:

One variable per agent and no local problems

SensorDNP - a benchmark problemSensorDNP - a benchmark problem

Constraints:

Sensors: can track at most one target. Not all the sensors are compatible between them

Targets: need three compatible sensors

GoalGoal: track every target with three compatible

sensors

DNP algorithmsDNP algorithms

ABTABT: Static priority order.

AWCAWC: 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 restartingDNP 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 then1. Change at random its assignment2. Inform neighbors about change

Network traffic models and delay distributionsNetwork 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 Our results: delays introduced by the network can improve the performance of DNP algorithmsimprove the performance of DNP algorithms

Exponential delay links: resultsExponential delay links: results

Instances tested:

3 mobiles and 15 sensors

Inter-agent communication links: exponentially distributed delays

15 instances for each value of Pc : Compatibility level between sensors (0 to 1)

Pv : Visibility level of sensors (0 to 1)

PC and PV model the level of resources available

Phase Transition in SensorDNPPhase Transition in SensorDNP

Sharp transition to solvable instances at critical level of resources

Mean complexityMean complexity

Peak in complexity around phase transition region

Worse for low level of compatibility (Pc)

Psat

= 0.2

Psat

= 0.8

Active delaying of messagesActive 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 distributionsComparing different delay distributions

Performance is improved when using restarting

using restarting

Cost distributions when solving a hard soluble instance

SummarySummary

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.