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Formal Complexity Analysis of RoboFlag Drill & Communication and Computation

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### Formal Complexity Analysis of RoboFlag Drill

Formal Complexity Analysis of RoboFlag Drill

&

Communication and Computation

in Distributed Negotiation Algorithms

Carla P. Gomes

Cornell University

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

Question:

What is the computational complexity

of Roboflag Drill?

Formal Complexity Analysis of Roboflag DrillFind 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 DrillInput: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 BaseNP-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 BaseNP-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)

A4

m1

m2

m3

A1

A2

A3

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

The arrival order of the messages determines the decisions of A4

- 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

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´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: track every target with three compatible sensors

DNP algorithms

We consider only complete algorithms: they always find a solution if there exists one

- 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

ABT: Static priority order.

AWC: Dynamic priority order and min-conflict heuristic.

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

DNP algorithms - randomization and restarting

Modifications to the DNP algorithms:

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

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 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 (Pc)

Psat = 0.2

Psat = 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

Cost distributions when solving a hard soluble instance

using restarting

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.

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