Loading in 5 sec....

Algorithms for Orienteering and Discounted-Reward TSPPowerPoint Presentation

Algorithms for Orienteering and Discounted-Reward TSP

- 404 Views
- Uploaded on
- Presentation posted in: Sports / Games

Algorithms for Orienteering and Discounted-Reward TSP

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Algorithms for Orienteering and Discounted-Reward TSP

Shuchi Chawla

Carnegie Mellon University

Joint work with

Avrim Blum, Adam Meyerson, David Karger, Maria Minkoff and Terran Lane

Orienteering

Dis. Rew. TSP

reward

- Given weighted graph G, root s, reward on nodes v
- Construct a path P rooted at s
- High level objective: Collect large reward in little time
- Orienteering
Maximize reward collected with path of length D

- Discounted-Reward TSP
Reward from node v, if reached at time t is vt

- Orienteering

time

Shuchi Chawla, Carnegie Mellon University

- Given weighted graph G, root s, reward on nodes v
- Construct a path P rooted at s
- High level objective: Collect large reward in little time
- Orienteering
Maximize reward collected with path of length D

- Discounted-Reward TSP
Reward from node v, if reached at time t is vt

A related problem…

- K-Traveling Salesperson
Minimize length while collecting at least K in reward

- Orienteering

No approximation algorithm known previously for the rooted non-geometric version

New problem

Best: (2+)-approx

[Garg] [AroraKarpinski] …

Shuchi Chawla, Carnegie Mellon University

Orienteering

Discounted-Reward TSP

Problem

Source/Reduction

Approximation

K-path (CP)

[Chaudhuri et al’03]

2+

Min-Excess Path (EP)

1.5 CP – 0.5

2.5+

2+

1+[EP]

4

(1+EP)(1+1/EP)EP

8.12+

6.75+

Shuchi Chawla, Carnegie Mellon University

- Task: deliver packages to locations in a building
- Faster delivery => greater happiness
- Classic formulation – Traveling Salesperson Problem
Find the shortest tour covering all locations

- Uncertainty in robot’s lifetime/behavior
- battery failure; sensor error…
- Robot may fail before delivering all packages

- Deliver as many packages as possible
- Some packages have higher priority than others

Shuchi Chawla, Carnegie Mellon University

Discounted-Reward TSP

Orienteering

- At every time step, the robot has a fixed probability (1-) of failing
- If a package with value is delivered at time t, the expected reward is t
- Goal: Construct a path such that the total discounted reward collected is maximized

“Discounted Reward”

Alternately, robot has a fixed battery life D

Goal: Construct path of length at most D that collects maximum reward

Shuchi Chawla, Carnegie Mellon University

- The Min-Excess problem
- Using Min-Excess to solve Orienteering
- Solving Min-Excess
- Using Min-Excess to solve Discounted-Reward TSP
- Extensions and open problems

Shuchi Chawla, Carnegie Mellon University

- First attempt – Use distance-based approximations to approximate reward
- Let OPT(d) = max achievable reward with length d
- A 2-approx for distance implies that ALG(d) ¸ OPT(d/2)
- However, we may have OPT(d/2) << OPT(d)
- Bad trade-off between distance and reward!
- Same problem with Discounted-Reward TSP

Shuchi Chawla, Carnegie Mellon University

t

s

- Using a distance-based approx
- Divide the optimal path into many segments
- Approximate the max reward segment using distance saved byshort-cutting other segments

- If min-distance between s and v is d, we spend at least d in going to v, regardless of the path

Shuchi Chawla, Carnegie Mellon University

Min-Excess Path Problem

- Using a distance-based approx
- Divide the optimal path into many segments
- Approximate the max reward segment using distance saved byshort-cutting other segments

- If min-distance between s and v is d, we spend at least d in going to v, regardless of the path
- Approximate the “extra” length taken by a path over the shortest path length
- If OPT obtains k reward with length d+, ALG should obtain the same reward with length d+

Shuchi Chawla, Carnegie Mellon University

2t/3

t

s

excess = t/3

- There exists a path from s to t, that
- collects reward at least
- has length · D
- t is the farthest from s among all nodes in the path

- Excess at node v = “v” = extra time taken to reach v = dPv – dv

t· D-dt

new excess = t/3

Can afford an excess up to t

Shuchi Chawla, Carnegie Mellon University

2t/3

excess = t/3

- There exists a path from s to t, that
- collects reward at least
- has length · D
- t is the farthest from s among all nodes in the path

- For any integer r, 9 a path from s to v that
- collects reward /r
- has excess · (D-dt)/r · (D-dv)/r

- Using an r-approx for Min-excess, we get an r-
approximation

- Note: If t is not the farthest node, a similar
analysis gives an r+1 approximation

t

s

t· D-dt

new excess = t/3

Can afford an excess up to t

Shuchi Chawla, Carnegie Mellon University

- OPT = d+; k-path gives us ALG = (d+)
We want ALG = d +

- Note: When ¼ d, (d+) ¼d + O()
- Idea: When is large, approximate using k-path
- What if << d ?
- Small path is almost like a shortest path
or “its distance from s mostly increases monotonically”

Shuchi Chawla, Carnegie Mellon University

Approximate

Dynamic Program

- OPT = d+; k-path gives us ALG = (d+)
We want ALG = d +

- Note: When ¼ d, (d+) ¼d + O()
- Idea: When is large, approximate using k-path
- What if << d ?
- Small path is almost like a shortest path
or “its distance from s mostly increases monotonically”

- Idea: Completely monotone path use dynamic programming!

Patch segments using dynamic programming

t

s

wiggly

wiggly

monotone

monotone

monotone

Shuchi Chawla, Carnegie Mellon University

half life

t

s

v

excess = 1

- WLOG, = ½. Reward of v at time t = vt
- An interesting observation:
OPT collects half of its reward before the first node that has excess 1

- Therefore, approximate the min-excess from s to v
- New path has excess 3. Reward by factor of 23.
- 16-approximation

’ = 2OPT(v,t) > OPT

reward ¸OPT/2

length of entire remaining path decreases by 1

Shuchi Chawla, Carnegie Mellon University

Orienteering

Discounted-Reward TSP

Problem

Source/Reduction

Approximation

K-path (CP)

[Chaudhuri et al’03]

2+

Min-Excess Path (EP)

1.5 CP – 0.5

2+

1+[EP]

4

(1+EP)(1+1/EP)EP

6.75+

Shuchi Chawla, Carnegie Mellon University

- Unrooted versions
- Multiple robots
- Max-reward Steiner tree of bounded size

Shuchi Chawla, Carnegie Mellon University

- Improve the approximations
- 2-approx for Orienteering?

- Robot Navigation
- A highly complex process with various kinds of uncertainty
- Can we model the MDP as a simple graph problem?

- Different deadlines for different packages

Shuchi Chawla, Carnegie Mellon University