Approximation Algorithms for Path-Planning Problems with Nikhil Bansal, Avrim Blum and Adam Meyerson Shuchi Chawla Carnegie Mellon University
Approximation Algorithms for
Path-Planning Problems
with
Nikhil Bansal, Avrim Blum and Adam Meyerson
Shuchi Chawla
Carnegie Mellon University
Shuchi Chawla, Carnegie Mellon University2
The Trick-o-Treaters Problem
Collect as much candy as possible within 6pm and 8pm
More candy more popularity with the kids
Have limited time to look for the lost wallet different places have different likelihoods of containing it
Some complicating constraints Limited amount of time Cannot necessarily visit all locations
Path-planning: Given graph (metric) G, construct a path satisfying
some constraints and optimizing some function
The Lost-Wallet Problem
Shuchi Chawla, Carnegie Mellon University3
Path-planning in the real world
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 University4
A reward-time trade-off
Given graph (metric) G, construct a “short” path that visits “many” nodes
Classic formulation – Traveling SalesmanFind the shortest tour covering all locations
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 University5
A reward-time trade-off
Given graph (metric) G, construct a “short” path that visits “many” nodes
Classic formulation – Traveling SalesmanFind 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 University6
A reward-time trade-off
Given graph (metric) G, construct a “short” path that visits “many” nodes
Classic formulation – Traveling SalesmanFind the shortest tour covering all locations
Budget the path-length and maximize reward Orienteering 4 [Blum C Karger+03]
3 [Bansal Blum C Meyerson 04]
Time Window 3log2n [Bansal Blum C Meyerson 04]
Impose a reward quota and minimize length
k-Path 2 + [Chaudhury Godfrey Rao+ 03]
Shuchi Chawla, Carnegie Mellon University7
The rest of this talk
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 University8
Orienteering and k-Path
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 University9
Why is Orienteering difficult?
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!
sOPT(d)
APPROX
Shuchi Chawla, Carnegie Mellon University10
Why is Orienteering difficult?
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
sOPT(d)
APPROX
Shuchi Chawla, Carnegie Mellon University11
Why is Orienteering difficult?
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
s tOPTAPPROX
Shuchi Chawla, Carnegie Mellon University12
Why is Orienteering difficult?
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+
Min-Excess Path Problem
Shuchi Chawla, Carnegie Mellon University13
The Min-Excess Problem
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 University14
A 3-approximation to Orienteering
Construct a path from s to t, that has length D collects maximum reward
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
s t
Excess of one subpath · (1+2+3)/3Can afford an excess up to D – ℓwhite = 1+2+3
1 2
3
Excess of path P (P) = dP(u,v)– d(u,v)
Using an r-approx for Min-excess ( r Z+ ), we get an r-approximation for s-t Orienteering
v1
v2 OPT
APPROX
Open: Given an r-approx for min-excess (r 2 R +), can we get r-approx to Orienteering?
Shuchi Chawla, Carnegie Mellon University15
So far…
A 3-approximation for Orienteering You
An O(log2n) approx for the Time-Window Problem
Orienteering with deadlines Incorporating release-dates
Extensions and Open Problems
Coming up…
learnt how to look for your lost walletshould’ve
Shuchi Chawla, Carnegie Mellon University16
The Time-Window Problem
Find a path visiting many nodes in their time-window
school bus routing FEDEX dial-a-ride service newspaper delivery
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 University17
The next step: Deadline-TSP
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 University18
A segmentation of OPT
Time
Dead
line
Shuchi Chawla, Carnegie Mellon University19
Deadline-TSP
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 University20
A 2-dimensional view
Time
Dead
line
minimal vertices
“Disjoint Rectangles”
Shuchi Chawla, Carnegie Mellon University21
The Rectangle Argument
Approximate reward contained in a family of disjoint rectangles Every pair of rectangles is non-overlapping in
BOTH dimensions
We construct O(log n) families of disjoint rectangles1. These cover ALL the reward in OPT2. We can approximate the best of them
We get an O(log n)-approximation
Shuchi Chawla, Carnegie Mellon University22
The Rectangle Argument
1. There are O(log n) families of disjoint rectangles that cover all the reward in OPT
Time
Dead
line
Shuchi Chawla, Carnegie Mellon University23
The Rectangle Argument
1. There are O(log n) families of disjoint rectangles that cover all the reward in OPT
Time
Dead
line
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
Shuchi Chawla, Carnegie Mellon University24
The Rectangle Argument
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 University25
The Rectangle Argument
2. We can approximate the best disjoint family
Time
Dead
line
Shuchi Chawla, Carnegie Mellon University26
The O(log n)-approximation
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 rectangles1. These cover ALL the reward in OPT2. We can approximate the best of them
Obtain an O(log n)-approximation
Shuchi Chawla, Carnegie Mellon University27
From Deadlines to Time-Windows
Nodes have deadlines as well as release times
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
s tOPT
ℓ(OPT) = L
v
Require ℓ(s,v) R(v) ℓ(t,v) L-R(v)
D(v) = L-R(v)
st
Shuchi Chawla, Carnegie Mellon University28
A Bicriteria Approximation
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 University29
Deadline TSP 3 log n
An overview of our results
Time-Window Problem 3 log2n
ApproximationProblem
Orienteering 3
Time-Window Problem - bicriteria
reward: log 1/ deadlines: 1+
Shuchi Chawla, Carnegie Mellon University30
Future Directions
Better approximations constant factor for Time-Windows? special metrics such as trees or planar graphs
Hardness of approximation log-hardness for Time-Windows?
Asymmetric Path-planning the graph is directed; still obeys triangle inequality
Shuchi Chawla, Carnegie Mellon University31
Questions?