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Approximation Algorithms for Path-Planning ProblemsPowerPoint Presentation

Approximation Algorithms for Path-Planning Problems

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Approximation Algorithms for Path-Planning Problems. with Nikhil Bansal, Avrim Blum and Adam Meyerson. Shuchi Chawla. The Trick-o-Treaters Problem. Collect as much candy as possible within 6pm and 8pm More candy more popularity with the kids Some complicating constraints

Approximation Algorithms for Path-Planning Problems

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Approximation Algorithms for Path-Planning Problems

with

Nikhil Bansal, Avrim Blum and Adam Meyerson

Shuchi Chawla

- Collect as much candy as possible within 6pm and 8pm
- More candy more popularity with the kids
- Some complicating constraints
- Limited amount of time
- Mr. X always gives twice as much candy as Mrs. Y, but his house is a long detour.

- Orienteering:
Given a metric and a starting point, cover as many “high-reward” locations as possible within a limited amount of time

Shuchi Chawla, Carnegie Mellon University

- A robot-navigation problem
- Deliver packages to certain locations
- Faster delivery => greater happiness
- Limited battery power
- Packages have different deadlines for delivery

- Assembly analysis
- Manufacturing
- Production planning

Shuchi Chawla, Carnegie Mellon University

- Given graph (metric) G, construct a path satisfying some constraints and optimizing some function.
- Classic formulation – Traveling Salesman
Find the shortest tour covering all locations

- Budget the path-length and maximize reward
- Orienteering Hard bound on path length
- Time Window Visit node v within [Rv, Dv]

- Impose a reward quota and minimize length
- k-Path Collect at least k reward

Shuchi Chawla, Carnegie Mellon University

- Given graph (metric) G, construct a path satisfying some constraints and optimizing some function.
- Classic formulation – Traveling Salesman
Find the shortest tour covering all locations

- Budget the path-length and maximize reward
- Orienteering4 [Blum C Karger+03]
3 [Bansal Blum C Meyerson 04]

- Time Window3log2n [Bansal Blum C Meyerson 04]

- Orienteering4 [Blum C Karger+03]
- Impose a reward quota and minimize length
- k-Path 2 + [Chaudhury Godfrey Rao+ 03]

Shuchi Chawla, Carnegie Mellon University

- A 3-approximation for Orienteering
- An O(log2n) approx for the Time-Window Problem
- Orienteering with deadlines
- Incorporating release-dates

- Extensions and Open Problems

Shuchi Chawla, Carnegie Mellon University

- Orienteering : length · D; maximize reward
- k-Path : reward ¸ k ; minimize length
- Complementary problems
- Series of results on k-TSP (related to k-Path)
[BRV99] [Garg99] [AK00] [CGRT03] …

best approx: (2+)

- None for Orienteering until recently!

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!

OPT(d)

s

APPROX

Shuchi Chawla, Carnegie Mellon University

- First attempt – Use distance-based approximations to approximate reward
- Idea – Modify the algorithm itself
- Doesn’t help – moat-growing always goes for shallow fruit
- Orienteering is inherently harder; Perturbation of the input changes the output widely

OPT(d)

s

APPROX

Shuchi Chawla, Carnegie Mellon University

t

s

- Second attempt – approximate subparts of the optimal path and shortcut other parts
- If we stray away from the optimal path by a lot, we may not be able to cover reward that’s far away
- Approximate the “extra” length taken by a path over the shortest path length

OPT

APPROX

Shuchi Chawla, Carnegie Mellon University

Min-Excess Path Problem

- Second attempt – approximate subparts of the optimal path and shortcut other parts
- If we stray away from the optimal path by a lot, we may not be able to cover reward that’s far away
- 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

- Given graph G, start and end nodes s, t, reward on nodes v,target reward k, find a path that collects reward at least k and minimizes(P) =ℓ(P) – d(s,t)
- At optimality, this is exactly the same as the k-path objective of minimizing ℓ(P)
- However, approximation is different: Min-excess is strictly harder than K-path
- There is a (2+)-approximation for Min-Excess
[Blum, C, Karger, Meyerson, Minkoff, Lane, FOCS’03]

- Our algorithm returns a path with length
d(s,t) + (2+) (P)

excess

Shuchi Chawla, Carnegie Mellon University

t

3

s

1

2

- There exists a path from s to t, that
- collects reward at least
- has length D

- Given a 3-approximation to min-excess:
1. Divide into 3 “equal-reward” parts (hypothetically)

2. Approximate the part with the smallest excess

3-approximation to orienteering

- Using an r-approx for Min-excess ( r Z+ ), we get an r-approximation for s-t Orienteering

Excess of path P

(P)

= dP(u,v)– d(u,v)

Open: Given an r-approx for min-excess (r 2R +),

can we get r-approx to Orienteering?

v2

OPT

v1

APPROX

Excess of one path · (1+2+3)/3

Can afford an excess up to (1+2+3)

Shuchi Chawla, Carnegie Mellon University

- A 3-approximation for Orienteering
- An O(log2n) approx for the Time-Window Problem
- Orienteering with deadlines
- Incorporating release-dates

- Extensions and Open Problems

Coming up…

Shuchi Chawla, Carnegie Mellon University

- Find a path visiting many nodes in their time-window
school bus routing bank and postal deliveries

industrial refuse collection newspaper delivery

fuel oil delivery dial-a-ride service

- Widely studied in scheduling and OR literature
- Constant-approx known for points on a line,
few different time-windows;

No approximation known for the general case

- A special case – The Deadline-TSP Problem
- Vertices only have deadlines
- All “release-times” are 0.

Shuchi Chawla, Carnegie Mellon University

- Every vertex has a deadline D(v); Find a path that maximizes nodes v visited before D(v)
- If the last node on the path has the min deadline, use Orienteering to approximate the reward
- Everything visited before the minimum deadline
- Don’t need to bother about deadlines of other nodes

- Does OPT always have a large subpath with the above property?
- There are many subpaths of OPT with the above property that together contain all the reward

NO!

Shuchi Chawla, Carnegie Mellon University

Deadline

Time

Shuchi Chawla, Carnegie Mellon University

- Segment graph into many parts, approximate each using Orienteering and patch them together
- How do we find such a segmentation without knowing the optimal path?
- In order to avoid double-counting of reward, segments should be node-disjoint
- Our result –
There exists a segmentation based only on deadlines, such that the resulting solution is a (3 log n)-approximation

Shuchi Chawla, Carnegie Mellon University

minimal vertices

Deadline

“Disjoint Rectangles”

Time

Shuchi Chawla, Carnegie Mellon University

- Approximate reward contained in a “disjoint” family of rectangles
- Every pair of rectangles is non-overlapping in BOTH dimensions

- We construct O(log n) families of disjoint rectangles
1. These cover ALL the reward in OPT

2. We can approximate the best of them

- We get an O(log n)-approximation

Shuchi Chawla, Carnegie Mellon University

Deadline

Time

- There are O(log n) families of disjoint rectangles that cover all the reward in OPT

Shuchi Chawla, Carnegie Mellon University

If there are between 2b and 2b+1 points in between,

then either the bth or a larger family contains exactly 1 point in the interval

- There are O(log n) families of disjoint rectangles that cover all the reward in OPT

Deadline

Time

Shuchi Chawla, Carnegie Mellon University

2. We can approximate the best disjoint family

- Suppose we know the minimal vertices
- Just try out all the log n families

- Problem - Minimal vertices depend on the optimal tour!
- Solution –
Try all possibilities.

They are ordered by deadlines, so use a simple dynamic program

Shuchi Chawla, Carnegie Mellon University

Deadline

Time

2. We can approximate the best disjoint family

Shuchi Chawla, Carnegie Mellon University

- Approximate reward contained in a “disjoint” family of rectangles
- Every pair of rectangles is non-overlapping in BOTH dimensions

- We construct O(log n) families of disjoint rectangles
1. These cover ALL the reward in OPT

2. We can approximate the best of them

- Obtain an O(log n)-approximation

Shuchi Chawla, Carnegie Mellon University

t

s

s

t

- Nodes have deadlines as well as release times
- Note that release times are dual to deadlines – if we look at the path from the end to the start, release times become deadlines!
- Log-approximation for deadlines log-approximation for release dates
- Algorithm for Time-Windows:
- Run the approximation for Deadline-TSP
- Replace Orienteering by Orienteering with release-dates

- O(log2n)-approximation for the Time-Window problem

ℓ(OPT) = L

D(v) = L-R(v)

OPT

v

Require ℓ(s,v) R(v)

ℓ(t,v) L-R(v)

Shuchi Chawla, Carnegie Mellon University

- Given any > 0,
Get O(log 1/) fraction of reward

Exceed deadlines by a (1+) factor

- O( log Dmax )-approximation
- Constant factor approximation if we can exceed deadlines by a small constant factor
- Nice trade-off:
Halving the extra time taken, increases the approximation factor by only an additive 1

Shuchi Chawla, Carnegie Mellon University

Problem

Approximation

Orienteering

3

Deadline TSP

3 logn

Time-Window Problem

3 log2n

reward: log 1/

deadlines: 1+

Time-Window Problem - bicriteria

Shuchi Chawla, Carnegie Mellon University

- Better approximations
- can we get a constant factor for Time-Windows?
- special metrics such as trees or planar graphs
- hardness of approximation?

- Asymmetric Path-planning
- the graph is directed; still obeys triangle inequality
- polylog-approximations and lower bounds for distance
- need entirely different ideas for asymmetric-Orienteering
- is it log-hard?

Shuchi Chawla, Carnegie Mellon University

Shuchi Chawla, Carnegie Mellon University