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### 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

The focus of our paper

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

time

Shuchi Chawla, Carnegie Mellon University

The focus of our paper

- 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

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

New problem

Best: (2+)-approx

[Garg] [AroraKarpinski] …

Shuchi Chawla, Carnegie Mellon University

Our contributions

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

A Robot Navigation Problem

- 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

Robot Navigation: A probabilistic view

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

Rest of this talk

- 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

Using K-path directly

- 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

Approximating Orienteering

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

Approximating Orienteering

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

From Min-Excess to Orienteering

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

From Min-Excess to Orienteering

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

Solving Min-Excess

- 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

Solving Min-Excess

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

Solving Discounted-Reward TSP

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

A summary of our results

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

Some extensions

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

Shuchi Chawla, Carnegie Mellon University

Future work…

- 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

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